diff --git "a/LdAyT4oBgHgl3EQfsvlL/content/tmp_files/load_file.txt" "b/LdAyT4oBgHgl3EQfsvlL/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/LdAyT4oBgHgl3EQfsvlL/content/tmp_files/load_file.txt" @@ -0,0 +1,812 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf,len=811 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='00581v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='IT] 2 Jan 2023 1 Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets† Jiaxin Wang, Fang-Wei Fu, Yadi Wei Abstract Bent partitions of V (p) n are quite powerful in constructing bent functions, vectorial bent functions and generalized bent functions, where V (p) n is an n-dimensional vector space over Fp, n is an even positive integer and p is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' It is known that partial spreads is a class of bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [4], [18], two classes of bent partitions whose forms are similar to partial spreads were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [3], more bent partitions Γ1, Γ2, Γ• 1, Γ• 2, Θ1, Θ2 were presented from (pre)semifields, including the bent partitions given in [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In this paper, we investigate the relations between bent partitions and vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any prime p, we show that one can generate certain bent partitions (called bent partitions satisfying Condition C) from certain vectorial dual-bent functions (called vectorial dual-bent functions satisfying Condition A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In particular, when p is an odd prime, we show that bent partitions satisfying Condition C one-to-one correspond to vectorial dual-bent functions satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We give an alternative proof that Γ1, Γ2, Γ• 1, Γ• 2, Θ1, Θ2 are bent partitions in terms of vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We present a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We show that any ternary weakly regular bent function f : V (3) n → F3 (n even) of 2-form can generate a bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When such f is weakly regular but not regular, the generated bent partition by f is not coming from a normal bent partition, which answers an open problem proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We give a sufficient condition on constructing partial difference sets from bent partitions, and when p is an odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Index Terms Bent partitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' bent functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' vectorial bent functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' vectorial dual-bent functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' semifields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' partial difference sets Jiaxin Wang, Fang-Wei Fu and Yadi Wei are with Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China, Emails: wjiaxin@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='cn, fwfu@nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='cn, wydecho@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' †This research is supported by the National Key Research and Development Program of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 2018YFA0704703 and 2022YFA1005001), the National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 12141108, 61971243, 12226336), the Natural Science Foundation of Tianjin (20JCZDJC00610), the Fundamental Research Funds for the Central Universities of China (Nankai University), and the Nankai Zhide Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' INTRODUCTION Boolean bent functions were introduced by Rothaus [21] and were generalized to p-ary bent functions by Kumar, Scholtz and Welch [15], where p is an arbitrary prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Due to applications of p-ary bent functions in cryptography, coding theory, sequence and combinatorics, they have been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We refer to surveys [5], [17] and a book [19] on p-ary bent functions and their generalizations such as vectorial bent functions and generalized bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [10], C¸ es¸melio˘glu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' introduced vectorial dual-bent functions, which is a special class of vectorial bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [7], [8], [22], vectorial dual-bent functions were used to construct partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In particular, Wang and Fu [22] showed that for certain vectorial dual-bent functions F : V (p) n → V (p) s (where V (p) n is an n-dimensional vector space over the prime field Fp), the preimage set of any subset of V (p) s for F forms a partial difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Very recently, bent partitions of V (p) n were introduced [4], [18], which are quite powerful in constructing bent functions, vectorial bent functions and generalized bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The well- known partial spreads is a class of bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [18], Meidl and Pirsic for the first time presented two classes of bent partitions for p = 2 different from partial spreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [4], Anbar and Meidl generalized the contributions in [18] to the case of p being odd and gave the corresponding two classes of bent partitions for odd p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [3], Anbar, Kalaycı and Meidl presented more bent partitions Γ1, Γ2, Γ• 1, Γ• 2, Θ1, Θ2 from (pre)semifields, including the bent partitions given in [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [2], Anbar, Kalaycı and Meidl showed that any union of elements in the bent partition given in [4], [18] forms a partial difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In terms of constructing partial difference sets, certain vectorial dual-bent functions and certain bent partitions seem to play the same role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Therefore, it is interesting to investigate the relations between vectorial dual-bent functions and bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In this paper, we show that by using certain vectorial dual-bent functions (called vectorial dual-bent functions satisfying Condition A), we can construct bent partitions of V (p) n with certain properties (called bent partitions satisfying Condition C) for any prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Particularly, when p is an odd prime, we prove that bent partitions of V (p) n with Condition C one-to-one correspond to vectorial dual-bent functions satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In terms of vectorial dual-bent functions, we provide an alternative proof that Γ1, Γ2, Γ• 1, Γ• 2, Θ1, Θ2 given in [3] are bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We provide a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We prove that any ternary weakly regular bent function f : V (3) n → F3 (n even) of 2-form can generate a bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In the special case January 3, 2023 DRAFT 3 that f is weakly regular but not regular, the generated bent partition by f is not coming from a normal bent partition, which answers an open problem proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By using vectorial dual-bent functions as the link between bent partitions and partial difference sets, we give a sufficient condition on constructing partial difference sets from bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When p is an odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In Section II, we state some needed results on vectorial dual-bent functions and bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In Section III, we present relations between certain bent partitions and certain vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In Section IV, we give a sec- ondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In Section V, we present relations between certain bent partitions and certain partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In Section VI, we make a conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' PRELIMINARIES In this section, we state some basic results on vectorial dual-bent functions and bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' First, we fix some notations used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' p is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' ζp = e 2π√−1 p is a complex primitive p-th root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that ζ2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Fpn is the finite field with pn elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Fn p is the vector space of the n-tuples over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' V (p) n is an n-dimensional vector space over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' ⟨·⟩n denotes a (non-degenerate) inner product of V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In this paper, when V (p) n = Fpn, let ⟨a, b⟩n = Trn 1(ab), where a, b ∈ Fpn, Trn k(·) denotes the trace function from Fpn to Fpk, k | n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' when V (p) n = Fn p, let ⟨a, b⟩n = a · b = �n i=1 aibi, where a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , an), b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , bn) ∈ Fn p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' when V (p) n = V (p) n1 ×· · ·×V (p) nm (n = �m i=1 ni), let ⟨a, b⟩n = �m i=1⟨ai, bi⟩ni, where a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , am), b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , bm) ∈ V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any set A ⊆ V (p) n and u ∈ V (p) n , let χu(A) = � x∈A χu(x), where χu denotes the character χu(x) = ζ⟨u,x⟩n p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Vectorial dual-bent functions A function F : V (p) n → V (p) s is called a vectorial p-ary function, or simply p-ary function when s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The Walsh transform of a p-ary function f : V (p) n → Fp is the complex valued January 3, 2023 DRAFT 4 function defined by Wf(a) = � x∈V (p) n ζf(x)−⟨a,x⟩n p , a ∈ V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (1) A p-ary function f : V (p) n → Fp is called bent if |Wf(a)| = p n 2 for all a ∈ V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that when f is a Boolean bent function, that is, p = 2, then n must be even since in this case, Wf is an integer valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' A vectorial p-ary function F : V (p) n → V (p) s is called vectorial bent if all component functions Fc : V (p) n → Fp, c ∈ V (p) s \\{0} defined as Fc(x) = ⟨c, F(x)⟩s are bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' It is known that if F : V (p) n → V (p) s is vectorial bent, then s ≤ n 2 if p = 2, and s ≤ n if p is an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' If f : V (p) n → Fp is bent, then so are cf, c ∈ F∗ p, that is, any p-ary bent function is vectorial bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For F : V (p) n → V (p) s , the vectorial bentness of F is independent of the inner products of V (p) n and V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The Walsh transform of a p-ary bent function f : V (p) n → Fp satisfies that for any a ∈ V (p) n , when p = 2, we have Wf(a) = 2 n 2 (−1)f∗(a), (2) and when p is an odd prime, we have Wf(a) = \uf8f1 \uf8f2 \uf8f3 ±p n 2 ζf∗(a) p if p ≡ 1 (mod 4) or n is even, ± √ −1p n 2 ζf∗(a) p if p ≡ 3 (mod 4) and n is odd, (3) where f ∗ is a function from V (p) n to Fp, called the dual of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' A p-ary bent function f : V (p) n → Fp is called weakly regular if Wf(a) = εfp n 2 ζf∗(a) p , where εf is a constant independent of a, otherwise f is called non-weakly regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In particular, if εf = 1, f is called regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The (non- )weakly regularity of f is independent of the inner product of V (p) n and if f is weakly regular, εf is independent of the inner product of V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By (2), all Boolean bent functions are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' If f is a p-ary weakly regular bent function, then the dual f ∗ of f is also weakly regular bent with (f ∗)∗(x) = f(−x) (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In 2018, C¸ es¸melio˘glu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' [10] introduced vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' A vectorial p-ary bent function F : V (p) n → V (p) s is called vectorial dual-bent if there exists a vectorial bent function G : V (p) n → V (p) s such that (Fc)∗ = Gσ(c) for any c ∈ V (p) s \\{0}, where (Fc)∗ is the dual of the component function ⟨c, F(x)⟩s and σ is some permutation over V (p) s \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The vectorial bent function G is called a vectorial dual of F and denoted by F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 5 It is known in [10] that the property of being vectorial dual-bent is independent of the inner products of V (p) n and V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that for a vectorial dual-bent function, its vectorial dual is not unique since being vectorial bent and vectorial dual-bent for a function is a property of the vector space consisting of all component functions (see Remark 1 of [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For example, if a p-ary function f (seen as a vectorial function into V (p) 1 , p odd) is vectorial dual-bent under any fixed inner product, then its dual f ∗ is unique, but its vectorial dual is not unique since for any c ∈ F∗ p, cf ∗ is a vectorial dual of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' A p-ary function f : V (p) n → Fp is called an l-form if f(ax) = alf(x) for any a ∈ F∗ p and x ∈ V (p) n , where 1 ≤ l ≤ p − 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By the results in [7], [22], we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proposition 1 ( [7], [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' A p-ary function f with f(0) = 0 is a weakly regular vectorial dual- bent function if and only if f is a weakly regular bent function of l-form with gcd(l−1, p−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In particular, a p-ary function f is a weakly regular vectorial dual-bent function with (cf)∗ = cf ∗ for any c ∈ F∗ p if and only if f is a weakly regular bent function of (p − 1)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In the rest of this subsection, we recall an important class of p-ary bent functions, called Maiorana-McFarland bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let f : Fpn × Fpn → Fp be defined as f(x, y) = Trn 1(αxπ(y)) + g(y), where α ∈ F∗ pn, π is a permutation over Fpn and g : Fpn → Fp is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then f is bent and is called a Maiorana-McFarland bent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The dual f ∗ of f is f ∗(x, y) = Trn 1(−π−1(α−1x)y) + g(π−1(α−1x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (4) All Maiorana-McFarland bent functions are regular (see [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Bent partitions Very recently, the concept of bent partitions of V (p) n were introduced [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n be an even positive integer, K be a positive integer divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , AK} be a partition of V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Assume that every function f for which every i ∈ Fp has exactly K p of sets Aj in Γ in its preimage, is a p-ary bent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then Γ is called a bent partition of V (p) n of depth K and every such bent function f is called a bent function constructed from bent partition Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 6 Let Γ = {U, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , AK} be a partition of V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Assume that every function f with the following properties is bent: (1) Every c ∈ Fp has exactly K p of the sets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , AK in its preimage set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (2) f(x) = c0 for all x ∈ U and some fixed c0 ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then we call Γ a normal bent partition of V (p) n of depth K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Bent partitions are very powerful in constructing bent functions, vectorial bent function and generalized bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In this paper, we focus on the relations between bent partitions and vectorial bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proposition 2 ( [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , Aps} be a bent partition of V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then every function F : V (p) n → V (p) s such that every element i ∈ V (p) s has the elements of exactly one of the sets Aj, 1 ≤ j ≤ ps, in its preimage, is a vectorial bent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' It is known that partial spreads is a class of bent partitions (for instance see Section 2 of [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [4], [18], two classes of explicit bent partitions different from partial spreads were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [3], bent partitions Γ1, Γ2, Γ• 1, Γ• 2, Θ1, Θ2 were presented from certain (pre)semifields, including the bent partitions given in [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We will recall bent partitions Γ1, Γ2, Γ• 1, Γ• 2, Θ1, Θ2 given in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' First, we need to introduce some basic knowledge on (pre)semifields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let ◦ be a binary operation defined on (V (p) n , +) such that (i) x ◦ y = 0 implies x = 0 or y = 0, (ii) (x + y) ◦ z = (x ◦ z) + (y ◦ z), (z ◦ (x + y) = (z ◦ x) + (z ◦ y), respectively), for all x, y, z ∈ V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then (V (p) n , +, ◦) is called a right (left, respectively) prequasifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' If (V (p) n , +, ◦) is a right and a left prequasifield, then it is called a presemifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' If (V (p) n , +, ◦) is a presemifield for which there is an element e ̸= 0 such that e ◦ x = x ◦ e = x for all x ∈ V (p) n , then it is called a semifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let P = (Fpn, +, ◦) be a presemifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then one can obtain presemifields P • = (Fpn, +, •) and P ⋆ = (Fpn, +, ⋆) from P, where • and ⋆ are given by x • y = y ◦ x for all x, y ∈ Fpn, and Trn 1(z(x ◦ y)) = Trn 1(x(z ⋆ y)) for all x, y, z ∈ Fpn, January 3, 2023 DRAFT 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The presemifield P ⋆ is called the dual of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let s be a positive divisor of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' If x ◦ (cy) = c(x ◦ y) holds for any x, y ∈ Fpn, c ∈ Fps, then P is called right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Each presemifield P = (Fpn, +, ◦) can induce a semifield P ′ = (Fpn, +, ∗) via the following transformation: choose any α ∈ F∗ pn and give ∗ by (x ◦ α) ∗ (α ◦ y) = x ◦ y for all x, y ∈ Fpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Lemma 2 of [3], if P is right Fps-linear, then P ′ is also right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The finite field Fpn is a right Fps-linear semifield (that is, ◦ is the field multiplication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For more right Fps-linear (pre)semifields, see Section 3 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Now we recall bent partitions Γ1, Γ2, Γ• 1, Γ• 2, Θ1, Θ2 given in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n, s be positive integers satisfying s | n and gcd(pn−1, ps+p−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Set u = ps+p−1, and let d be an integer with du ≡ 1 mod (pn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let P = (Fpn, +, ◦) be a (pre)semifield such that its dual P ⋆ = (Fpn, +, ⋆) is right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For given x ∈ Fpn, if x = 0, then let ηx = 0, and if x ̸= 0, then let ηx be given by x ⋆ η−1 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Ut = {(x, t ◦ xu) : x ∈ F∗ pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Γ1 = {Ai, i ∈ Fps}, (5) where Ai = � t∈Fpn:Trn s (t)=i Ut if i ̸= i0, Ai0 = � t∈Fpn:Trn s (t)=i0 Ut � U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define U• t = {(x, xu ◦ t) : x ∈ F∗ pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Γ• 1 = {A• i , i ∈ Fps}, (6) where A• i = � t∈Fpn:Trn s (t)=i U• t if i ̸= i0, A• i0 = � t∈Fpn:Trn s (t)=i0 U• t � U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Vt = {(t ◦ xd, x) : x ∈ F∗ pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Γ2 = {Bi, i ∈ Fps}, (7) January 3, 2023 DRAFT 8 where Bi = � t∈Fpn:Trn s (t)=i Vi if i ̸= i0, Bi0 = � t∈Fpn:Trn s (t)=i0 Vi � V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define V • t = {(xd ◦ t, x) : x ∈ F∗ pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Γ• 2 = {B• i , i ∈ Fps}, (8) where B• i = � t∈Fpn:Trn s (t)=i V • i if i ̸= i0, Bi0 = � t∈Fpn:Trn s (t)=i0 V • i � V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Xt = {(tηd x, x) : x ∈ F∗ pn} if t ∈ Fpn, and X = {(x, 0) : x ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Θ1 = {Si, i ∈ Fps}, (9) where Si = � t∈Fpn:Trn s (t)=i Xt if i ̸= i0, Si0 = � t∈Fpn:Trn s (t)=i0 Xt � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Yt = {(x, tηu x) : x ∈ F∗ pn} if t ∈ Fpn, and Y = {(0, y) : y ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Θ2 = {Ti, i ∈ Fps}, (10) where Ti = � t∈Fpn:Trn s (t)=i Yt if i ̸= i0, Ti0 = � t∈Fpn:Trn s (t)=i0 Yt � Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In the finite field case, that is, ◦ and ⋆ are the field multiplication, then Γ1 = Γ• 1 = Θ2, Γ2 = Γ• 2 = Θ1, which reduces to the two classes bent partitions given in [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In fact, for the parameter u in the bent partitions Γ1, Γ• 1, Γ2, Γ• 2, Θ1, Θ2, one can consider the more general form u ≡ pj mod (ps − 1) by the proofs in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' RELATIONS BETWEEN CERTAIN BENT PARTITIONS AND CERTAIN VECTORIAL DUAL-BENT FUNCTIONS Throughout this section, we consider bent partitions and vectorial dual-bent functions satisfy- ing the following conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Condition C: Let n be an even positive integer, s be a positive integer with s ≤ n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {Ai, i ∈ V (p) s } be a bent partition of V (p) n which satisfies that F∗ pAi = Ai for all i ∈ V (p) s January 3, 2023 DRAFT 9 and all bent functions f constructed from Γ are regular (that is, εf = 1) or weakly regular but not regular (that is, εf = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We denote by ε = εf for all bent functions f constructed from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Condition A: Let n be an even positive integer, s be a positive integer with s ≤ n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let F : V (p) n → V (p) s be a vectorial dual-bent function with (Fc)∗ = (F ∗)c, c ∈ V (p) s \\{0} for a vectorial dual F ∗ of F and all component functions being regular or weakly regular but not regular, that is, εFc, c ∈ V (p) s \\{0} are all the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We denote by ε = εFc for all c ∈ V (p) s \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' It is easy to see that the known bent partitions, including partial spreads and Γi, Γ• i , Θi, i = 1, 2 defined by (5)-(10), all satisfy F∗ pAi = Ai, i ∈ V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By the results in [3], [11], [14], all bent functions constructed from partial spreads and Γi, Γ• i , Θi, i = 1, 2 are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, the known bent partitions all satisfy Condition C with ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Moreover, when p = 2, it is easy to see that Condition C is trivial for any bent partition of V (2) n of depth powers of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In this section, we present relations between bent partitions satisfying Condition C and vectorial dual-bent functions satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' First, we need a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n be an even positive integer, s be a positive integer with s ≤ n 2, and F : V (p) n → V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (1) F is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (2) There exist pairwise disjoint sets Wi ⊆ V (p) n , i ∈ V (p) s with � i∈V (p) s Wi = V (p) n and a constant ε ∈ {±1} (ε = 1 if p = 2) such that for any nonempty set I ⊆ V (p) s , χu(DF,I) = pn−sδ{0}(u)|I| + εp n 2 −s(psδWI(u) − |I|), u ∈ V (p) n , (11) where DF,I = {x ∈ V (p) n : F(x) ∈ I}, WI = � i∈I Wi, and for any set S, δS denotes the indicator function of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Proposition 3 of [22] (Note that although Proposition 3 of [22] only considers the case of p being odd, p = 2 also holds), for any u ∈ V (p) n , i ∈ V (p) s we have χu(DF,i) = pn−sδ{0}(u) + p−s � c∈V (p) s \\{0} WFc(−u)ζ−⟨c,i⟩s p , (12) where DF,i = {x ∈ V (p) n : F(x) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 10 (1) ⇒ (2): If F is a vectorial dual-bent function satisfying Condition A (Note that if p = 2, then ε = 1 since all Boolean bent functions are regular), then χu(DF,i) = pn−sδ{0}(u) + εp n 2 −s � c∈V (p) s \\{0} ζ(Fc)∗(−u)−⟨c,i⟩s p = pn−sδ{0}(u) + εp n 2 −s � c∈V (p) s \\{0} ζ(F ∗)c(−u)−⟨c,i⟩s p = pn−sδ{0}(u) + εp n 2 −s � c∈V (p) s \\{0} ζ⟨c,F ∗(−u)−i⟩s p = pn−sδ{0}(u) + εp n 2 −s(psδ{0}(F ∗(−u) − i) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (13) Define Wi = {x ∈ V (p) n : F ∗(−x) = i}, i ∈ V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then Wi � Wj = ∅ for any i ̸= j and � i∈V (p) s Wi = V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By (13), for any nonempty set I ⊆ V (p) s and u ∈ V (p) n we have χu(DF,I) = � i∈I χu(DF,i) = � i∈I pn−sδ{0}(u) + εp n 2 −s(psδWi(u) − 1) = pn−sδ{0}(u)|I| + εp n 2 −s(psδWI(u) − |I|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (2) ⇒ (1): By the assumption on Wi, i ∈ V (p) s , we have that for any x ∈ V (p) n , there exists a unique i ∈ V (p) s such that x ∈ Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define G : V (p) n → V (p) s by G(x) = i if − x ∈ Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By the definition of G, for any u ∈ V (p) n , i ∈ V (p) s we have χu(DF,i) = pn−sδ{0}(u) + εp n 2 −s(psδ{0}(G(−u) − i) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (14) For any c ∈ V (p) s \\{0}, WFc(−u) = � x∈V (p) n ζ⟨c,F (x)⟩s+⟨u,x⟩n p = � i∈V (p) s � x∈V (p) n :F (x)=i ζ⟨c,i⟩s+⟨u,x⟩n p January 3, 2023 DRAFT 11 = � i∈V (p) s ζ⟨c,i⟩s p χu(DF,i) = � i∈V (p) s \\{G(−u)} ζ⟨c,i⟩s p (pn−sδ{0}(u) − εp n 2 −s) + (pn−sδ{0}(u) + εp n 2 −s(ps − 1))ζ⟨c,G(−u)⟩s p = (pn−sδ{0}(u) − εp n 2 −s) � i∈V (p) s ζ⟨c,i⟩s p + εp n 2 ζGc(−u) p = εp n 2 ζGc(−u) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (15) By (15) and the assumption that ε = 1 if p = 2, F is a vectorial bent function with εFc = ε and (Fc)∗ = Gc for any c ∈ V (p) s \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since Fc is a weakly regular bent function, we have that Gc = (Fc)∗ is also weakly regular bent and G is vectorial bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, F is vectorial dual-bent with εFc = ε and (Fc)∗ = (F ∗)c for any c ∈ V (p) s \\{0}, where F ∗ = G, that is, F satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Based on Lemma 1, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let F : V (p) n → V (p) s be a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define Ai = DF,i, i ∈ V (p) s , where DF,i = {x ∈ V (p) n : F(x) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then Γ = {Ai, i ∈ V (p) s } is a bent partition satisfying Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Lemma 1 and its proof, for any i ∈ V (p) s and u ∈ V (p) n , χu(Ai) = χu(DF,i) = pn−sδ{0}(u) + εp n 2 −s(psδ{0}(F ∗(−u) − i) − 1), where ε = 1 if p = 2 since all Boolean bent functions are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any union S of ps−1 sets of {Ai : i ∈ V (p) s }, we have χu(S) = \uf8f1 \uf8f2 \uf8f3 pn−1δ{0}(u) + εp n 2 −1(p − 1), if AF ∗(−u) ⊆ S, pn−1δ{0}(u) − εp n 2 −1, if AF ∗(−u) ⊈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (16) Let f be an arbitrary function such that for every j ∈ Fp, there are exactly ps−1 sets Ai in Γ in its preimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define g(u) = f(AF ∗(−u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that g is a p-ary function from V (p) n to Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then by (16), we have χu(Df,j) = \uf8f1 \uf8f2 \uf8f3 pn−1δ{0}(u) + εp n 2 −1(p − 1), if j = g(u), pn−1δ{0}(u) − εp n 2 −1, if j ̸= g(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (17) January 3, 2023 DRAFT 12 By (17), for any u ∈ V (p) n we have Wf(−u) = � x∈V (p) n ζf(x)+⟨u,x⟩n p = � j∈Fp ζj p � x∈V (p) n :f(x)=j ζ⟨u,x⟩n p = � j∈Fp ζj pχu(Df,j) = � j∈Fp\\{g(u)} ζj p(pn−1δ{0}(u) − εp n 2 −1) + ζg(u) p (pn−1δ{0}(u) + εp n 2 −1(p − 1)) = (pn−1δ{0}(u) − εp n 2 −1) � j∈Fp ζj p + εp n 2 ζg(u) p = εp n 2 ζg(u) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (18) By (18) and ε = 1 if p = 2, f is a weakly regular bent function with εf = ε and f ∗(x) = g(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Wj = {u ∈ V (p) n : g(u) = j}, j ∈ Fp, then Wj, j ∈ Fp are pairwise disjoint and � j∈Fp Wj = V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By (17), for any u ∈ V (p) n and nonempty set J ⊆ Fp we have χu(Df,J) = pn−1δ{0}(u)|J| + εp n 2 −1(pδWJ(u) − |J|), (19) where Df,J = {x ∈ V (p) n : f(x) ∈ J}, WJ = � j∈J Wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By (19) and Lemma 1, f is vectorial dual- bent with (cf)∗ = c(βf ∗), c ∈ F∗ p for some β ∈ F∗ p (since all vectorial duals of f are cf ∗, c ∈ F∗ p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let c = 1, we obtain β = 1, that is, f is vectorial dual-bent with (cf)∗ = cf ∗, c ∈ F∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Proposition 1, f is a (p − 1)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In particular, Fc is a (p − 1)-form for any c ∈ F∗ ps, which yields that F(αx) = F(x) for any α ∈ F∗ p and F∗ pAi = Ai, i ∈ V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Hence, Γ is a bent partition satisfying Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Theorem 1, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n be an even positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let f : V (p) n → Fp be a weakly regular bent function of (p − 1)-form, then {Df,j, j ∈ Fp} is a bent partition of V (p) n , where Df,j = {x ∈ V (p) n : f(x) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Proposition 1, f is a weakly regular vectorial dual-bent function with (cf)∗ = cf ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since n is even, εcf = εf for all c ∈ F∗ p (see Theorem 1 of [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then by Theorem 1, the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 13 A bent partition Γ = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , AK} of depth K is called coming from a normal bent partition if there is U ⊆ Ai for some i such that {U, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , Ai−1, Ai\\U, Ai+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , AK} is a normal bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In [4], there is an open problem: Do bent partitions exist which are not coming from a normal bent partition of depth K > 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In the following, we provide a positive answer for this open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By the definition of l-form, a ternary function f is a 2-form if and only if f(x) = f(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n be an even positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' If f : V (3) n → F3 with f(x) = f(−x) is a ternary weakly regular but not regular bent function (that is, εf = −1), then by Corollary 1, {Df,0, Df,1, Df,2} is a bent partition of depth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' There exist such ternary bent functions f, for instance see [7], [17]: f(x) = Trn 1(αx2), x ∈ F3n, (20) where n is even, α ∈ F∗ 3n is a square element if 4 | n, and α ∈ F∗ 3n is a non-square element if 4 ∤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' f(x) = Trn 1(ax 3n−1 4 +3m+1), x ∈ F3n, (21) where n = 2m, m odd, a = α 3m+1 4 for a primitive element α of F3n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' f(x) = Trn 1(α(x33k+32k−3k+1 + x2)), x ∈ F3n, (22) where n = 4k for an arbitrary positive integer k, α ∈ F∗ 32k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' f(x, y, z) = (g(x) − h(x))z2 + yz + g(x), (x, y, z) ∈ F3n × F3 × F3, (23) where n is even, g and h are distinct bent functions constructed by (20) or (22) if 4 | n, g and h are distinct bent functions constructed by (20) or (21) if 4 ∤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any ternary weakly regular but not regular bent function f : V (3) n → F3 (n even) with f(x) = f(−x), the corresponding bent partition {Df,0, Df,1, Df,2} is not coming from a normal bent partition by Theorem 4 (i) of [4], which provides a positive answer for the above open problem proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We first recall Theorem 4 (i) of [4] and then give an example to illustrate this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Lemma 2 ( [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {U, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' , AK} be a normal bent partition of V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then |U| = p n 2 and |Aj| = pn−p n 2 K , 1 ≤ j ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 14 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let f : F34 → F3 be defined by f(x) = Tr4 1(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then f is ternary weakly regular bent with f(x) = f(−x) and εf = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Corollary 1, {Df,0, Df,1, Df,2} is a bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By the result of Nyberg [20], for any weakly regular p-ary bent function g : V (p) n → Fp with n even, we have {|Dg,i|, i ∈ Fp} = {pn−1 + εfp n 2 −1(p − 1), pn−1 − εfp n 2 −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For our example, |Df,0| = 21, |Df,1| = |Df,2| = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Lemma 2, it is easy to see that {Df,0, Df,1, Df,2} can not be from a normal bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In the following, based on Theorem 1, we give an alternative proof that Γi, Γ• i , Θi, i = 1, 2 defined by (5)-(10) given in [3] are bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let s, n be positive integers with s | n, u be an integer with u ≡ pj0 mod (ps − 1) for some 0 ≤ j0 ≤ s − 1 and gcd(u, pn − 1) = 1, and let d be an integer with du ≡ 1 mod (pn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let P = (Fpn, +, ◦) be a (pre)semifield such that its dual P ⋆ = (Fpn, +, ⋆) is right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For given x ∈ Fpn, if x = 0, then let ηx = 0, and if x ̸= 0, then let ηx be given by x ⋆ η−1 x = 1 (For convention we set η−1 0 = ηpn−2 0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any α ∈ F∗ pn and i0 ∈ Fps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' define F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = Trn s (αa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y)) + i0(1 − xpn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (24) where for given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then a(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and if x ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then a(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) is given by a(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) ◦ xu = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and F •(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = Trn s (αa•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y)) + i0(1 − xpn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (25) where for given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then a•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and if x ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then a•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) is given by xu ◦ a•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = Trn s (αb(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y)) + i0(1 − ypn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (26) where for given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if y = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then b(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and if y ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then b(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) is given by b(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) ◦ yd = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and G•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = Trn s (αb•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y)) + i0(1 − ypn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (27) where for given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if y = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then b•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and if y ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then b•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) is given by yd ◦ b•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and M(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = Trn s (αη−u x y) + i0(1 − xpn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (28) and N(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = Trn s (αxη−d y ) + i0(1 − ypn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) ∈ Fpn × Fpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (29) January 3, 2023 DRAFT 15 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let F, F •, G, G•, M, N be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then they are all vectorial dual-bent functions satisfying Condition A with ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We only prove the result for F and M since the proofs for F •, G, G• are similar to the proof for F, and the proof for N is similar to the proof for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For F: For any c ∈ F∗ ps, we have Fc(x, y) = Trn 1(cαa(x, y)) + Trs 1(ci0)(1 − xpn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any c ∈ F∗ ps and (w, v) ∈ Fpn × Fpn, we have WFcu(w, v) = � x∈F∗ pn � y∈Fpn ζTrn 1 (cuαa(x,y))−Trn 1 (wx+vy) p + ζTrs 1(cui0) p � y∈Fpn ζ−Trn 1 (vy) p = � x∈Fpn � y∈Fpn ζTrn 1 (cuαa(x,y))−Trn 1 (wx+vy) p + pn(ζTrs 1(cui0) p − 1)δ{0}(v) = Wh(w, v) + pn(ζTrs 1(cui0) p − 1)δ{0}(v), where h(x, y) = Trn 1(cuαa(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For given x ∈ Fpn, if x = 0, then let λx = 0, and if x ̸= 0, then let λx be given by x ⋆ λ−1 x = α (For convention we set λ−1 0 = λpn−2 0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define ρ(x) = λ−d x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then ρ is a permutation over Fpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any x ∈ F∗ pn, set z = ρ−1(c−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then λ−d z = ρ(z) = c−1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By du ≡ 1 mod (pn − 1), we have λ−1 z = c−uxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since z ̸= 0 and P ⋆ is right Fps-linear, we have α = z ⋆ λ−1 z = z ⋆ (c−uxu) = c−u(z ⋆ xu), that is, ρ−1(c−1x) ⋆ xu = αcu for any x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, when x ̸= 0, Trn 1(cuαa(x, y)) = Trn 1(a(x, y)(ρ−1(c−1x) ⋆ xu)) = Trn 1(ρ−1(c−1x)(a(x, y) ◦ xu)) = Trn 1(ρ−1(c−1x)y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When x = 0, by a(0, y) = ρ−1(0) = 0, we have Trn 1(cuαa(x, y)) = Trn 1(ρ−1(c−1x)y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Hence, h(x, y) = Trn 1(ρ−1(c−1x)y), which is a Maiorana-McFarland bent function and by (4), Wh(w, v) = pnζ−Trn 1 (cwρ(v)) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Therefore, for any c ∈ F∗ ps, WFcu(w, v) = pn(ζ−Trn 1 (cwρ(v)) p + (ζTrs 1(cui0) p − 1)δ{0}(v)) = pnζ−Trn 1 (cwρ(v))+Trs 1(cui0)(1−vpn−1) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (30) By (30) and ud ≡ 1 mod (pn − 1), we have that for any c ∈ F∗ ps, Fc is a regular bent function with (Fc)∗(x, y) = −Trn 1(cdxρ(y)) + Trs 1(ci0)(1 − ypn−1) = −Trn 1(cdpj0(xρ(y))pj0) + Trs 1(ci0)(1 − ypn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 16 Since u ≡ pj0 mod (ps − 1) and du ≡ 1 mod (pn − 1), we have d ≡ ps−j0 mod (ps − 1) and thus (cd)pj0 = c for any c ∈ F∗ ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Therefore, F is a vectorial bent function with εFc = 1 and (Fc)∗ = Hc for all c ∈ F∗ ps, where H(x, y) = −Trn s ((xρ(y))pj0) + i0(1 − ypn−1) = −(Trn s (xρ(y)))pj0 + i0(1 − ypn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since Fc is regular bent, we have that (Fc)∗ = Hc is also regular bent and H is vectorial bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, F is vectorial dual-bent with εFc = 1 and (Fc)∗ = (F ∗)c for all c ∈ F∗ ps, where F ∗ = H, that is, F satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For M: For any c ∈ F∗ ps, Mc(x, y) = Trn 1(cαη−u x y) + Trs 1(ci0)(1 − xpn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Similar to the discussion for F, for any c ∈ F∗ ps and (w, v) ∈ Fpn × Fpn we have WMc(w, v) = Wg(w, v) + pn(ζTrs 1(ci0) p − 1)δ{0}(v), where g(x, y) = Trn 1(cαη−u x y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let π(x) = η−u x , then π is a permutation over Fpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since g is a Maiorana-McFarland bent function, then by (4), Wg(w, v) = pnζ−Trn 1 (wπ−1(c−1α−1v)) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any given y ∈ F∗ pn, set π−1(c−1α−1y) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then c−1α−1y = π(z) = η−u z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By ud ≡ 1 mod (pn − 1), we have η−1 z = c−dα−dyd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since z ̸= 0 and P ⋆ is right Fps-linear, we have 1 = z ⋆ η−1 z = z ⋆ (c−dα−dyd) = c−d(z ⋆ α−dyd), that is, π−1(c−1α−1y) ⋆ α−dyd = cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For given (x, y) ∈ Fpn × Fpn, if y = 0, then let r(x, y) = 0, and if y ̸= 0, then let r(x, y) be given by r(x, y)◦α−dyd = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When v ̸= 0, we have Trn 1(wπ−1(c−1α−1v)) = Trn 1(π−1(c−1α−1v)(r(w, v)◦ α−dvd)) = Trn 1(r(w, v)(π−1(c−1α−1v) ⋆ α−dvd)) = Trn 1(cdr(w, v)) = Trn 1(c(r(w, v))pj0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When v = 0, since π−1(0) = 0 and r(w, 0) = 0, we have Trn 1(wπ−1(c−1α−1v)) = Trn 1(c(r(w, v))pj0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, −Trn 1(wπ−1(c−1α−1v)) = −Trn 1(c(r(w, v))pj0) and WMc(w, v) = pn(ζ−Trn 1 (c(r(w,v))pj0 ) p + (ζTrs 1(ci0) p − 1)δ{0}(v)) = pnζ−Trn 1 (c(r(w,v))pj0 )+Trs 1(ci0)(1−vpn−1) p , which implies that M is a vectorial dual-bent function with εMc = 1 and (Mc)∗ = (M∗)c for all c ∈ F∗ ps, where M∗(x, y) = −Trn s ((r(x, y))pj0) + i0(1 − ypn−1), January 3, 2023 DRAFT 17 that is, M satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Theorem 1 and Proposition 3, we have that {DF,i, i ∈ Fps}, {DF •,i, i ∈ Fps}, {DG,i, i ∈ Fps}, {DG•,i, i ∈ Fps}, {DM,i, i ∈ Fps} and {DN,i, i ∈ Fps} are bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' It is easy to verify that DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � t∈Fpn:T rn s (αt)=i Ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' � t∈Fpn:T rn s (αt)=i0 Ut � U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' DF •,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � t∈Fpn:T rn s (αt)=i U • t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' � t∈Fpn:T rn s (αt)=i0 U • t � U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' DG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � t∈Fpn:T rn s (αt)=i Vt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' � t∈Fpn:T rn s (αt)=i0 Vt � V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' DG•,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � t∈Fpn:T rn s (αt)=i V • t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' � t∈Fpn:T rn s (αt)=i0 V • t � V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' DM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � t∈Fpn:T rn s (αt)=i Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' � t∈Fpn:T rn s (αt)=i0 Xt � X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' DN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � t∈Fpn:T rn s (αt)=i Yt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' � t∈Fpn:T rn s (αt)=i0 Yt � Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' where Ut = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' t ◦ xu) : x ∈ F∗ pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and U = {(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) : y ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' U• t = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' xu ◦ t) : x ∈ F∗ pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and U = {(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) : y ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Vt = {(t ◦ xd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x) : x ∈ F∗ pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and V = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 0) : x ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' V • t = {(xd ◦ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x) : x ∈ F∗ pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and V = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 0) : x ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Xt = {(tηd x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x) : x ∈ F∗ pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and X = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 0) : x ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Yt = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' tηu x) : x ∈ F∗ pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and Y = {(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) : y ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For the above bent partitions from vectorial dual-bent functions F, F •, G, G•, M, N, by set- ting α = 1, u = ps + p − 1 with gcd(u, pn − 1) = 1, then we can obtain bent partitions Γ1, Γ• 1, Γ2, Γ• 2, Θ1, Θ2 defined by (5)-(10) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus by the above analysis, we provide an alternative derivation that Γ1, Γ• 1, Γ2, Γ• 2, Θ1, Θ2 are bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When p is an odd prime, we show that the converse of Theorem 1 also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let p be an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {Ai, i ∈ V (p) s } be a bent partition of V (p) n satisfying Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define F : V (p) n → V (p) s by F(x) = i if x ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then F is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since F∗ pAi = Ai for any i ∈ V (p) s , all bent functions constructed from Γ are (p−1)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When s = 1, the conclusion follows from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In the following, we consider the case of s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let f be an arbitrary bent function constructed from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='4 of [13], for any u ∈ V (p) n and j ∈ Fp we have χu(Df,j) = \uf8f1 \uf8f2 \uf8f3 pn−1δ{0}(u) + εp n 2 −1(p − 1), if f ∗(u) = j, pn−1δ{0}(u) − εp n 2 −1, if f ∗(u) ̸= j, (31) where Df,j = {x ∈ V (p) n : f(x) = j}, j ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any fixed u ∈ V (p) n , since {χu(Df,j), j ∈ Fp} = {pn−1δ{0}(u) + εp n 2 −1(p − 1), pn−1δ{0}(u) − εp n 2 −1} for any bent function f constructed from Γ, we have that for any fixed u ∈ V (p) n , there exists a unique G(u) ∈ V (p) s such that χu(Ai), i ̸= G(u) are all the same and χu(Ai) ̸= χu(AG(u)), i ̸= G(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that G is a function from V (p) n to V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Moreover by (31), for any fixed u ∈ V (p) n we have χu(Ai) = \uf8f1 \uf8f2 \uf8f3 pn−sδ{0}(u) + εp n 2 −s(ps − 1), if i = G(u), pn−sδ{0}(u) − εp n 2 −s, if i ̸= G(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (32) Define Wi = {u ∈ V (p) n : G(u) = i}, i ∈ V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then obviously Wi, i ∈ V (p) s are pairwise disjoint and � i∈V (p) s Wi = V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By (32), for any u ∈ V (p) n and nonempty set I ⊆ V (p) s we have χu(DF,I) = � i∈I χu(Ai) = pn−sδ{0}(u)|I| + εp n 2 −s(psδWI(u) − |I|), (33) where DF,I = {x ∈ V (p) n : F(x) ∈ I}, WI = � i∈I Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By (33) and Lemma 1, the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When p is an odd prime, from Theorems 1 and 2 we obtain a characterization of bent partitions satisfying Condition C in terms of vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let p be an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {Ai, i ∈ V (p) s } be a partition of V (p) n , where n is even, s ≤ n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define F : V (p) n → V (p) s as F(x) = i if x ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then Γ is a bent partition satisfying Condition C if and only if F is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 19 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' CONSTRUCTING BENT PARTITIONS FROM VECTORIAL DUAL-BENT FUNCTIONS In this section, we construct bent partitions from vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The following theorem provides a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n, m, s be positive integers for which n is even and s ≤ n 2, s | m, s ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any i ∈ Fps, let F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x) : V (p) n → Fps be a vectorial dual-bent function with ((F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x))c)∗ = ((F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x))∗)c and ε(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))c = ε for any c ∈ F∗ ps, where (F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x))∗ is a vectorial dual of F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x) and ε ∈ {±1} is a constant independent of i, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let α, β ∈ Fpm be linearly independent over Fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let R be a permutation over Fpm with R(0) = 0 and T : Fps → Fps be an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define H : V (p) n × Fpm × Fpm → Fps as H(x, y1, y2) = F(T rm s (αR(y1ypm−2 2 ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x) + T rm s (βR(y1ypm−2 2 )) + T (T rm s (αR(y1ypm−2 2 ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then H is a vectorial dual-bent function satisfying Condition A and Γ = {Ai, i ∈ Fps} is a bent partition satisfying Condition C, where Ai = {(x, y1, y2) ∈ V (p) n ×Fpm ×Fpm : H(x, y1, y2) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Denote d(y) = Trm s (βR(y1ypm−2 2 )), e(y) = Trm s ((β − α)R(y1ypm−2 2 )), y = (y1, y2) ∈ Fpm × Fpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any c ∈ F∗ ps and (a, b) = (a, b1, b2) ∈ V (p) n × Fpm × Fpm, we have WHc(a, b) = � x∈V (p) n � y=(y1,y2)∈Fpm×Fpm ζT rs 1(cF (d(y)−e(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))+T rs 1(cd(y))+T rs 1(cT (d(y)−e(y))) p ζ−⟨a,x⟩n−T rm 1 (b1y1+b2y2) p = � i∈Fps � y=(y1,y2)∈Fpm×Fpm:d(y)−e(y)=i � x∈V (p) n ζT rs 1(cF (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))+T rs 1(cd(y))+T rs 1(cT (i)) p ζ−⟨a,x⟩n−T rm 1 (b1y1+b2y2) p = p−s � i∈Fps W(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))c(a)ζT rs 1(cT (i)) p � y=(y1,y2)∈Fpm×Fpm ζT rs 1(cd(y))−T rm 1 (b1y1+b2y2) p � j∈Fps ζT rs 1(cj(i−(d(y)−e(y)))) p = p−s � i∈Fps W(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))c(a)ζT rs 1(cT (i)) p � j∈Fps ζT rs 1(ijc) p � y=(y1,y2)∈Fpm×Fpm ζT rs 1(c((1−j)d(y)+je(y)))−T rm 1 (b1y1+b2y2) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Theorem 3 of [10], for any j ∈ Fps, J(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y) = (1−j)d(y)+je(y) is a partial spread vectorial dual-bent function with ε(J(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='y))c = 1 and ((J(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' y))c)∗ = ((1−j)d∗(y)+je∗(y))c for any c ∈ F∗ ps, January 3, 2023 DRAFT 20 where d∗(y) = Trm s (βR(−ypm−2 1 y2)), e∗(y) = Trm s ((β − α)R(−ypm−2 1 y2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Therefore, WHc(a, b) = pm−s � i∈Fps W(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))c(a)ζT rs 1(cT (i)) p � j∈Fps ζT rs 1(ijc) p ζT rs 1(c((1−j)d∗(b)+je∗(b))) p = pm−sζT rs 1(cd∗(b)) p � i∈Fps W(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))c(a)ζT rs 1(cT (i)) p � j∈Fps ζT rs 1(cj(i−(d∗(b)−e∗(b)))) p = pmζT rs 1(cd∗(b)) p W(F (d∗(b)−e∗(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))c(a)ζT rs 1(cT (d∗(b)−e∗(b))) p = εp n 2 +mζ ((F (T rm s (αR(−bpm−2 1 b2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))c)∗(a)+T rs 1(cT rm s (βR(−bpm−2 1 b2)))+T rs 1(cT (T rm s (αR(−bpm−2 1 b2)))) p = εp n 2 +mζ((F (T rm s (αR(−bpm−2 1 b2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='x))∗)c(a)+T rs 1(cT rm s (βR(−bpm−2 1 b2)))+T rs 1(cT (T rm s (αR(−bpm−2 1 b2)))) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (34) Note that ε = 1 if p = 2 since all Boolean bent functions are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By (34), H is a vectorial bent function with (Hc)∗ = Gc and εHc = ε for any c ∈ F∗ ps, where G(a, b1, b2) = (F(T rm s (αR(−bpm−2 1 b2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x))∗(a) + T rm s (βR(−bpm−2 1 b2)) + T (T rm s (αR(−bpm−2 1 b2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since Hc is weakly regular bent, we have that Gc = (Hc)∗ is also weakly regular bent and G is vectorial bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, H is vectorial dual-bent with (Hc)∗ = (H∗)c and εHc = ε for any c ∈ F∗ ps, where H∗ = G, that is, H satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Theorem 1, the partition Γ generated from H is a bent partition satisfying Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The following explicit construction of bent partitions is an immediate result of Proposition 3 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n, m, s be positive integers with s | n, s | m, s ̸= m, and ui, i ∈ Fps be integers for which for any i ∈ Fps, ui ≡ pji mod (ps − 1) for some 0 ≤ ji ≤ s − 1 and gcd(ui, pn − 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any i ∈ Fps, let di be an integer with uidi ≡ 1 mod (pn − 1), and Pi = (Fpn, +, ◦i) be a (pre)semifield for which its dual P ⋆ i is right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any i ∈ Fps, let F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x1, x2) : Fpn × Fpn → Fps be an arbitrary vectorial dual-bent function constructed by Proposition 3 with u = ui, d = di, P = Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let α, β ∈ Fpm be linearly independent over Fps, R be a permutation over Fpm with R(0) = 0 and T : Fps → Fps be an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Define H : Fpn × Fpn × Fpm × Fpm → Fps as H(x1, x2, y1, y2) = F(T rm s (αR(y1ypm−2 2 ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x1, x2) + T rm s (βR(y1ypm−2 2 )) + T (T rm s (αR(y1ypm−2 2 ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then Γ = {Ai, i ∈ Fps} is a bent partition satisfying Condition C, where Ai = {(x1, x2, y1, y2) ∈ Fpn × Fpn × Fpm × Fpm : H(x1, x2, y1, y2) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 21 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' With the same notation as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that in Theorem 4, by setting vectorial dual-bent functions H constructed by Theorem 5 as building blocks (that is, as F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x)), we can obtain more explicit vectorial dual-bent functions which can generate more bent partitions by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We give an example by using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let p = 3, s = 4, n = m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let α be a primitive element of F38 and β = 1, R be the identity map and T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any i ∈ F34, let F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' x1, x2) = \uf8f1 \uf8f2 \uf8f3 Tr8 4(x−89 1 x2), if i ∈ F∗ 34, Tr8 4(x1x−83 2 ), if i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then H(x1, x2, y2, y2) = (Tr8 4(αy1y6559 2 ))80(Tr8 4(x−89 1 x2 − x1x−83 2 )) + Tr8 4(x1x−83 2 + y1y6559 2 ), and Γ = {DH,i, i ∈ F34} is a bent partition satisfying Condition C, where DH,i = {(x1, x2, y1, y2) ∈ (F38)4 : H(x1, x2, y1, y2) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' RELATIONS BETWEEN BENT PARTITIONS AND PARTIAL DIFFERENCE SETS In this section, by taking vectorial dual-bent functions as the link between bent partitions and partial difference sets, we give a sufficient condition on constructing partial difference sets from bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When p is an odd prime, we characterize bent partitions satisfying Condition C in terms of partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let (G, +) be a finite abelian group of order v and D be a subset of G with k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then D is called a (v, k, λ, µ) partial difference set of G, if the expressions d1 − d2, for d1 and d2 in D with d1 ̸= d2, represent each nonzero element in D exactly λ times, and represent each nonzero element in G \\ D exactly µ times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When λ = µ, then D is called a (v, k, λ) difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that if D is a partial difference set of G with −D = D, then so are D∪{0}, D \\ {0}, G \\ D (see [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' There is an important tool to characterize partial difference sets in terms of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 22 Lemma 3 ( [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let G be an abelian group of order v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Suppose that D is a subset of G with k elements which satisfies −D = D and 0 /∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then D is a (v, k, λ, µ) partial difference set if and only if for each non-principal character χ of G, χ(D) = β ± √ ∆ 2 , where χ(D) = � x∈D χ(x), β = λ − µ, γ = k − µ, ∆ = β2 + 4γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When p is an odd prime or s ≥ 2, we give the value distribution of vectorial dual-bent functions satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let F : V (p) n → V (p) s be a vectorial dual-bent function satisfying Condition A, where p is odd or s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then |DF,F (0)| = pn−s + εp n 2 −s(ps − 1), |DF,i| = pn−s − εp n 2 −s if i ̸= F(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that if f is a weakly regular p-ary bent function, then for any a ∈ Fp, f − a is a weakly regular bent function with (f − a)∗ = f ∗ − a and εf−a = εf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since F is a vectorial dual-bent function with (Fc)∗ = (F ∗)c, c ∈ V (p) s \\{0}, we have that F(x) − F(0) is a vectorial bent function and for any c ∈ V (p) s \\{0}, ((F − F(0))c)∗ = (Fc)∗ − ⟨c, F(0)⟩s = (F ∗)c − ⟨c, F(0)⟩s = (F ∗ − F(0))c, which implies that F(x) − F(0) is a vectorial dual-bent function with ((F − F(0))c)∗ = (F ∗ − F(0))c and ε(F −F (0))c = ε for any c ∈ V (p) s \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By the proof of Theorem 1, F(ax) = F(x) for any a ∈ F∗ p and thus F(x) = F(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Corollary 1 of [22] (Note that although Corollary 1 of [22] only considers the case of p being odd, the conclusion of Corollary 1 of [22] also holds for p = 2, s ≥ 2), we have |DF −F (0),0| = pn−s + εp n 2 −s(ps − 1), |DF −F (0),i| = pn−s − εp n 2 −s if i ̸= 0, that is, |DF,F (0)| = pn−s + εp n 2 −s(ps − 1), |DF,i| = pn−s − εp n 2 −s if i ̸= F(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In the following, we give a characterization of vectorial dual-bent functions satisfying Con- dition A in terms of partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 23 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n be an even positive integer, s be a positive integer with s ≤ n 2, and F : V (p) n → V (p) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (1) F is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (2) When p = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' s = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then the support supp(F) of F defined as supp(F) = {x ∈ V (2) n : F(x) = 1} is a (2n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 2n−1 ± 2 n 2 −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 2n−2 ± 2 n 2 −1) difference set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' and when p is odd or s ≥ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then for any nonempty set I ⊆ V (p) s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='I\\{0} is a (pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' µ) partial difference set for which −DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='I = DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='I and if F(0) ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then k = pn−s|I| + εp n 2 −s(ps − |I|) − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' λ = pn−2s|I|2 + εp n 2 −s(ps − |I|) − 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' µ = pn−2s|I|2 + εp n 2 −s|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (35) and if F(0) /∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' then k = pn−s|I| − εp n 2 −s|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' λ = pn−2s|I|2 + εp n 2 −s(ps − 3|I|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' µ = pn−2s|I|2 − εp n 2 −s|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (36) where DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content='I = {x ∈ V (p) n : F(x) ∈ I} and ε ∈ {±1} is a constant (ε = 1 if p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' It is easy to see that a Boolean function F is a vectorial dual-bent function satisfying Condition A if and only if F is bent, that is, Condition A is trivial for any Boolean bent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By the well-known result that a Boolean function F : V (2) n → F2 is bent if and only if its support supp(F) = {x ∈ V (2) n : F(x) = 1} is a (2n, 2n−1 ± 2 n 2 −1, 2n−2 ± 2 n 2 −1) difference set (see [11]), the conclusion obviously holds for p = 2, s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' In the following, we prove the conclusion for p being odd or s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (1) ⇒ (2): By the proof of Theorem 1, F(−x) = F(x), that is, −DF,I = DF,I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' For any u ∈ V (p) n \\{0}, with the same argument as in the proof of Theorem 2 of [22], χu(DF,I) = \uf8f1 \uf8f2 \uf8f3 εp n 2 − εp n 2 −s|I|, if F ∗(−u) ∈ I, −εp n 2 −s|I|, if F ∗(−u) /∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' where ε = 1 if p = 2 since all Boolean bent functions are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' If F(0) ∈ I, then |DF,I\\{0}| = |DF,I|−1 and χu(DF,I\\{0}) = χu(DF,I)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Proposition 4, |DF,I\\{0}| = (|I|−1)(pn−s−εp n 2 −s)+(pn−s+εp n 2 −s(ps−1)−1) = pn−s|I|+εp n 2 −s(ps−|I|)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Lemma 3, DF,I\\{0} is a (pn, k, λ, µ) partial difference set, where k, λ, µ are given in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 24 If F(0) /∈ I, then |DF,I\\{0}| = |DF,I| and χu(DF,I\\{0}) = χu(DF,I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Proposition 4, |DF,I\\{0}| = |I|(pn−s − εp n 2 −s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Lemma 3, DF,I\\{0} is a (pn, k, λ, µ) partial difference set, where k, λ, µ are given in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (2) ⇒ (1): By Lemma 3, for any u ∈ V (p) n and nonempty set I ⊆ V (p) s we have χu(DF,I) = pn−sδ{0}(u)|I| + εp n 2 − εp n 2 −s|I| or χu(DF,I) = pn−sδ{0}(u)|I| − εp n 2 −s|I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (37) For any i ∈ V (p) s , define Wi = {u ∈ V (p) n : χu(DF,i) = pn−sδ{0}(u) + εp n 2 − εp n 2 −s}, where DF,i = {x ∈ V (p) n : F(x) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We claim that Wi � Wi′ = ∅ for any i ̸= i′ and � i∈V (p) s Wi = V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Indeed, if there exist i ̸= i′ such that Wi � Wi′ ̸= ∅, that is, there exists u ∈ V (p) n such that χu(DF,i) = χu(DF,i′) = pn−sδ{0}(u) + εp n 2 − εp n 2 −s, then χu(DF,i � DF,i′) = 2pn−sδ{0}(u) + 2εp n 2 − 2εp n 2 −s, which contradicts with (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, Wi � Wi′ = ∅ for any i ̸= i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' If there exists u ∈ V (p) n such that u /∈ Wi for any i ∈ V (p) s , that is, χu(DF,i) = pn−sδ{0}(u) − εp n 2 −s for any i ∈ V (p) s , then χu(V (p) n ) = � i∈V (p) s χu(DF,i) = pnδ{0}(u) − εp n 2 , which contradicts with χu(V (p) n ) = � x∈V (p) n ζ⟨u,x⟩n p = pnδ{0}(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, � i∈V (p) s Wi = V (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By the above analysis, we have χu(DF,I) = pn−sδ{0}(u)|I| + εp n 2 −s(psδWI(u) − |I|), (38) where WI = � i∈I Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By (38) and Lemma 1, F is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' The following theorem provides a sufficient condition on constructing partial difference sets from bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let n be an even positive integer and s be a positive integer with s ≤ n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Assume that Γ = {Ai, i ∈ V (p) s } is a bent partition of V (p) n for which the function F : V (p) n → V (p) s defined by F(x) = i if x ∈ Ai is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then when p = 2, s = 1, A0 and A1 are (2n, 2n−1 ± 2 n 2 −1, 2n−2 ± 2 n 2 −1) difference set and (2n, 2n−1 ∓ 2 n 2 −1, 2n−2 ∓ 2 n 2 −1) difference set, respectively, and when p is odd or s ≥ 2, for any nonempty set I ⊆ V (p) s , AI\\{0} = � i∈I Ai\\{0} is a (pn, k, λ, µ) partial difference set, where (k, λ, µ) are given in (35) if 0 ∈ AI and (k, λ, µ) are given in (36) if 0 /∈ AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' January 3, 2023 DRAFT 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Note that if D is a (v, k, λ) difference set of a finite abelian group G, then G\\D is a (v, v − k, v − 2k + λ) difference set of G (for instance see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then the result follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Proposition 3, the bent partition Γ1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Γ2, Γ• 1, Γ• 2, Θ1, Θ2) satisfies the condition in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Theorem 7, any union of sets from Γ1 (resp, Γ2, Γ• 1, Γ• 2, Θ1, Θ2) forms a partial difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Thus, the results given in Corollary 15 of [1] on constructing partial difference sets from Γ1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Γ2, Γ• 1, Γ• 2, Θ1, Θ2) (which includes the results given in Theorem 2 of [2] on constructing partial difference sets from Γ1, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Γ2, in the finite field) can also be illustrated by our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Since the bent partitions constructed in Theorem 5 satisfy the condition in Theorem 7, we have the following corollary from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {Ai, i ∈ Fps} be a bent partition constructed by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then when p = 2, s = 1, A0 and A1 are (2n, 2n−1 ± 2 n 2 −1, 2n−2 ± 2 n 2 −1) difference set and (2n, 2n−1 ∓ 2 n 2 −1, 2n−2 ∓ 2 n 2 −1) difference set, respectively, and when p is odd or s ≥ 2, for any nonempty set I ⊆ Fps, AI\\{0} = � i∈I Ai\\{0} is a (pn, k, λ, µ) partial difference set, where (k, λ, µ) are given in (35) with ε = 1 if 0 ∈ AI and (k, λ, µ) are given in (36) with ε = 1 if 0 /∈ AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We give an example by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {DH,i, i ∈ F34} be the bent partition constructed in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By Corol- lary 2, DH,i is a (1853020188851841, 22876791923520, 282470988879, 282429005040) partial difference set for any i ∈ F∗ 34, DH,0\\{0} is a (1853020188851841, 22876834970240, 282472051759, 282430067922) partial difference set, (DH,0 � DH,1)\\{0} is a (1853020188851841, 45753626893760, 1129760129761, 1129719208806) partial difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When p is an odd prime, we immediately obtain the following characterization of bent partitions of V (p) n satisfying Condition C from Theorems 3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let p be an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Let Γ = {Ai, i ∈ V (p) s } be a partition of V (p) n , where n is even and s ≤ n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Then the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (1) Γ is a bent partition satisfying Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' (2) For any nonempty set I ⊆ V (p) s , AI\\{0} = � i∈I Ai\\{0} is a (pn, k, λ, µ) partial difference January 3, 2023 DRAFT 26 set with −AI = AI, where (k, λ, µ) are given in (35) if 0 ∈ AI and (k, λ, µ) are given in (36) if 0 /∈ AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' CONCLUSION In this paper, we investigated relations between bent partitions and vectorial dual-bent functions (Theorems 1, 2, 3) and gave some new constructions of bent partitions satisfying Condition C (Corollary 1, Theorems 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' We illustrated that for any ternary weakly regular bent function f : V (3) n → F3 (n even) with f(x) = f(−x) and εf = −1, the generated bent partition by f is not coming from a normal bent partition (see Example 1), which answers an open problem proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' By taking vectorial dual-bent functions as the link between bent partitions and partial difference sets, we give a sufficient condition on constructing partial difference sets from bent partitions (Theorem 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' When p is an odd prime, we characterized bent partitions satisfying Condition C in terms of partial difference sets (Theorem 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' REFERENCES [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Anbar, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Meidl, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Pirsic, Vectorial bent functions and partial difference sets, Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' Codes Cryptogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 89, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'} +page_content=' 10, pp.' metadata={'source': 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