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Alexander Pott - One of the best experts on this subject based on the ideXlab platform.
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Non-extendable $$\mathbb{F}_{q}$$-Quadratic Perfect Nonlinear Maps
Open Problems in Mathematics and Computational Science, 2014Co-Authors: Ferruh Özbudak, Alexander PottAbstract:Let q be a power of an odd prime. We give examples of non-extendable \(\mathbb{F}_{q}\)-quadratic Perfect Nonlinear maps. We also show that many classes of \(\mathbb{F}_{q}\)-quadratic Perfect Nonlinear maps are extendable. We give a short survey of some related results and provide some open problems.
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Uniqueness of Fq-quadratic Perfect Nonlinear maps from Fq 3 to Fq2
Finite Fields and Their Applications, 2014Co-Authors: Ferruh Özbudak, Alexander PottAbstract:Let q be a power of an odd prime. We prove that all F"q-quadratic Perfect Nonlinear maps from F"q"^"3 to F"q^2 are equivalent. We also give a geometric method to find the corresponding equivalence explicitly.
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on designs and multiplier groups constructed from almost Perfect Nonlinear functions
Lecture Notes in Computer Science, 2009Co-Authors: Yves Edel, Alexander PottAbstract:Let $f:{\mathbb{F}_2^{n}}\to {\mathbb{F}_2^{n}}$ be an almost Perfect Nonlinear function (APN). The set $D_f:=\{(a,b)\: :\: f(x+a)-f(x)=b\mbox{\ has two solutions}\}$ can be used to distinguish APN functions up to equivalence. We investigate the multiplier groups of theses sets D f . This extends earlier work done by the authors [1].
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IMA Int. Conf. - On Designs and Multiplier Groups Constructed from Almost Perfect Nonlinear Functions
Cryptography and Coding, 2009Co-Authors: Yves Edel, Alexander PottAbstract:Let $f:{\mathbb{F}_2^{n}}\to {\mathbb{F}_2^{n}}$ be an almost Perfect Nonlinear function (APN). The set $D_f:=\{(a,b)\: :\: f(x+a)-f(x)=b\mbox{\ has two solutions}\}$ can be used to distinguish APN functions up to equivalence. We investigate the multiplier groups of theses sets D f . This extends earlier work done by the authors [1].
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a new almost Perfect Nonlinear function which is not quadratic
Advances in Mathematics of Communications, 2009Co-Authors: Yves Edel, Alexander PottAbstract:Following an example in [12], we show how to change one coordinate function of an almost Perfect Nonlinear (APN) function in order to obtain new examples. It turns out that this is a very powerful method to construct new APN functions. In particular, we show that our approach can be used to construct a ''non-quadratic'' APN function. This new example is in remarkable contrast to all recently constructed functions which have all been quadratic. An equivalent function has been found independently by Brinkmann and Leander [8]. However, they claimed that their function is CCZ equivalent to a quadratic one. In this paper we give several reasons why this new function is not equivalent to a quadratic one.
Tor Helleseth - One of the best experts on this subject based on the ideXlab platform.
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Optimal Ternary Cyclic Codes From Monomials
IEEE Transactions on Information Theory, 2013Co-Authors: Cunsheng Ding, Tor HellesethAbstract:Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Perfect Nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters [3m-1, 3m-1-2m, 4] by Carlet, Ding, and Yuan in 2005. In this paper, almost Perfect Nonlinear monomials, and a number of other monomials over GF(3m) are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented.
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new Perfect Nonlinear multinomials over f _ p 2k for any odd prime p
SETA '08 Proceedings of the 5th international conference on Sequences and Their Applications, 2008Co-Authors: Lilya Budaghyan, Tor HellesethAbstract:We introduce two infinite families of Perfect Nonlinear Dembowski-Ostrom multinomials over $\textbf{F}_{p^{2k}}$ where pis any odd prime. We prove that in general these functions are CCZ-inequivalent to previously known PN mappings. One of these families has been constructed by extension of a known family of APN functions over $\textbf{F}_{2^{2k}}$. This shows that known classes of APN functions over fields of even characteristic can serve as a source for further constructions of PN mappings over fields of odd characteristics. Besides, we supply results indicating that these PN functions define new commutative semifields. After the works of Dickson (1906) and Albert (1952), these are the firstly found infinite families of commutative semifields which are defined for all odd primes p.
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A New Family of Ternary Almost Perfect Nonlinear Mappings
IEEE Transactions on Information Theory, 2007Co-Authors: G.j. Ness, Tor HellesethAbstract:A mapping f(x) from GF(pn) to GF(pn) is differentially k-uniform if k is the maximum number of solutions x isin GF(pn) of f(x+a) - f(x) = b, where a, b isin GF(pn) and a ne 0. A 2-uniform mapping is called almost Perfect Nonlinear (APN). This correspondence describes new families of ternary APN mappings over GF(3n), n>3 odd, of the form f(x) = uxd + xd 2 where d1 = (3n-1)/2 - 1 and d2 = 3n - 2.
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On P-Ary Bent Functions Defined on Finite Fields
Mathematical Properties of Sequences and Other Combinatorial Structures, 2003Co-Authors: Young-sik Kim, Ji-woong Jang, Tor HellesethAbstract:It is known that a bent function corresponds to a Perfect Nonlinear function, which makes it difficult to do the differential cryptanalysis in DES and in many other block ciphers. In this paper, for an odd prime p, quadratic p-ary bent functions defined on finite fields are given from the families of p-ary sequences with optimal correlation property. And quadratic p-ary bent functions, that is, Perfect Nonlinear functions from the finite field Fp m to its prime field F p are constructed by using the trace functions.
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New families of almost Perfect Nonlinear power mappings
IEEE Transactions on Information Theory, 1999Co-Authors: Tor Helleseth, Chunming Rong, Daniel SandbergAbstract:A power mapping f(x)=x/sup d/ over GF(p/sup n/) is said to be differentially k-uniform if k is the maximum number of solutions x/spl isin/GF(p/sup n/) of f(x+a)-f(x)=b where a, b/spl isin/GF(p/sup n/) and a/spl ne/0. A 2-uniform mapping is called almost Perfect Nonlinear (APN). We construct several new infinite families of nonbinary APN power mappings.
Cunsheng Ding - One of the best experts on this subject based on the ideXlab platform.
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Shortened linear codes from APN and PN functions.
arXiv: Information Theory, 2020Co-Authors: Can Xiang, Chunming Tang, Cunsheng DingAbstract:Linear codes generated by component functions of Perfect Nonlinear (PN) and almost Perfect Nonlinear (APN) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$-ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$-ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes.
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Optimal Ternary Cyclic Codes From Monomials
IEEE Transactions on Information Theory, 2013Co-Authors: Cunsheng Ding, Tor HellesethAbstract:Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Perfect Nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters [3m-1, 3m-1-2m, 4] by Carlet, Ding, and Yuan in 2005. In this paper, almost Perfect Nonlinear monomials, and a number of other monomials over GF(3m) are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented.
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the weight distribution of a class of linear codes from Perfect Nonlinear functions
IEEE Transactions on Information Theory, 2006Co-Authors: Jin Yuan, Claude Carlet, Cunsheng DingAbstract:In this correspondence, the weight distribution of a class of linear codes based on Perfect Nonlinear functions (also called planar functions) is determined. The class of linear codes under study are either optimal or among the best codes known, and have nice applications in cryptography.
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A family of skew Hadamard difference sets
Journal of Combinatorial Theory Series A, 2006Co-Authors: Cunsheng Ding, Jin YuanAbstract:In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley-Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new Perfect Nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these Perfect Nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new Perfect Nonlinear functions has applications in cryptography, coding theory, and combinatorics.
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Note A family of skew Hadamard difference sets
2006Co-Authors: Cunsheng Ding, Jin YuanAbstract:In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley–Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new Perfect Nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these Perfect Nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley–Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new Perfect Nonlinear functions has applications in cryptography, coding theory, and combinatorics.
Xiangyong Zeng - One of the best experts on this subject based on the ideXlab platform.
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Linear Codes From Perfect Nonlinear Functions Over Finite Fields
IEEE Transactions on Communications, 2020Co-Authors: Xiangyong ZengAbstract:In this paper, a class of $p$ -ary 3-weight linear codes and a class of binary 2-weight linear codes are proposed respectively by virtue of the properties of the Perfect Nonlinear functions over $\mathbb {F}_{p^{m}}$ and $(m,s)$ -bent functions from $\mathbb {F}_{2^{m}}$ to $\mathbb {F}_{2^{s}}$ , where $p$ is an odd prime and $m, s$ are positive integers. The weight distributions are completely determined by the sign of the Walsh transform of weakly regular bent functions and the size of the preimage of the employed $(m,s)$ -bent functions at the zero point, respectively. As a special case, a class of optimal linear codes meeting Griesmer bound is obtained from our construction.
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SETA - Partially Perfect Nonlinear functions and a construction of cryptographic boolean functions
Sequences and Their Applications – SETA 2006, 2006Co-Authors: Xiangyong ZengAbstract:In this paper the concept of partially Perfect Nonlinear (PPN) function is introduced as an extension of binary partially Bent function and is used to construct a new class of Boolean functions with good cryptographic properties. The construction is a composition of a PPN function and a Boolean function. The Nonlinearity, correlation immunity, propagation criterion, and other cryptographic properties of the constructed functions are analyzed. In particular, new plateaued functions can be obtained by the proposed method and the construction of Khoo and Gong in [1] is improved.
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Partially Perfect Nonlinear Functions and a Construction of Cryptographic Boolean Functions
Lecture Notes in Computer Science, 2006Co-Authors: Xiangyong ZengAbstract:In this paper the concept of partially Perfect Nonlinear (PPN) function is introduced as an extension of binary partially Bent function and is used to construct a new class of Boolean functions with good cryptographic properties. The construction is a composition of a PPN function and a Boolean function. The Nonlinearity, correlation immunity, propagation criterion, and other cryptographic properties of the constructed functions are analyzed. In particular, new plateaued functions can be obtained by the proposed method and the construction of Khoo and Gong in [1] is improved.
Jin Yuan - One of the best experts on this subject based on the ideXlab platform.
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the weight distribution of a class of linear codes from Perfect Nonlinear functions
IEEE Transactions on Information Theory, 2006Co-Authors: Jin Yuan, Claude Carlet, Cunsheng DingAbstract:In this correspondence, the weight distribution of a class of linear codes based on Perfect Nonlinear functions (also called planar functions) is determined. The class of linear codes under study are either optimal or among the best codes known, and have nice applications in cryptography.
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A family of skew Hadamard difference sets
Journal of Combinatorial Theory Series A, 2006Co-Authors: Cunsheng Ding, Jin YuanAbstract:In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley-Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new Perfect Nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these Perfect Nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new Perfect Nonlinear functions has applications in cryptography, coding theory, and combinatorics.
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Note A family of skew Hadamard difference sets
2006Co-Authors: Cunsheng Ding, Jin YuanAbstract:In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley–Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new Perfect Nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these Perfect Nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley–Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new Perfect Nonlinear functions has applications in cryptography, coding theory, and combinatorics.
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Linear codes from Perfect Nonlinear mappings and their secret sharing schemes
IEEE Transactions on Information Theory, 2005Co-Authors: Claude Carlet, Cunsheng Ding, Jin YuanAbstract:In this paper, error-correcting codes from Perfect Nonlinear mappings are constructed, and then employed to construct secret sharing schemes. The error-correcting codes obtained in this paper are very good in general, and many of them are optimal or almost optimal. The secret sharing schemes obtained in this paper have two types of access structures. The first type is democratic in the sense that every participant is involved in the same number of minimal-access sets. In the second type of access structures, there are a few dictators who are in every minimal access set, while each of the remaining participants is in the same number of minimal-access sets.
