In the thesis a new way of checking whether a function is CCZ-equivalent to a permutation is given. The results for known families of almost perfect nonlinear (APN) functions are presented for functions defined over G...
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In the thesis a new way of checking whether a function is CCZ-equivalent to a permutation is given. The results for known families of almost perfect nonlinear (APN) functions are presented for functions defined over GF(2n ), for even n ≤ 12. The ways how to reduce the number of polynomials from each family are studied. For functions of the form x3 + a-1 tr1(a3 x9 ) it is shown, that they cannot be CCZ-equivalent to a permutation on fields GF(24n ) for n ∈ ℕ .
We propose a new set of benchmark problems that come from the field of cryptography. These problems are often easy to define, they are relevant in practice, and some of them have a known optimal value. The solutions f...
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ISBN:
(纸本)9781450349390
We propose a new set of benchmark problems that come from the field of cryptography. These problems are often easy to define, they are relevant in practice, and some of them have a known optimal value. The solutions from these problems can be used as cryptographic primitives and consequently function as components of cryptographic algorithms. They can be compared not only with other evolutionary computation techniques but with a broad spectrum of techniques. Finally, due to the fact that many of the optimal solutions are not known (although we know they must exist), finding such solutions would also have a significant practical impact.
We show that there is no vectorialboolean function of degree , with some conditions on , which is APN over infinitely many extensions of its field of definition. It is a new step in the proof of the conjecture of Aub...
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We show that there is no vectorialboolean function of degree , with some conditions on , which is APN over infinitely many extensions of its field of definition. It is a new step in the proof of the conjecture of Aubry, McGuire and Rodier.
It is known that the set of rotation symmetric booleanfunctions has many functions with various useful properties for cryptography. This study shows how to construct some families of rotation symmetric functions whic...
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It is known that the set of rotation symmetric booleanfunctions has many functions with various useful properties for cryptography. This study shows how to construct some families of rotation symmetric functions which are balanced or plateaued. The authors also consider vectorial boolean functions [that is, maps from GF(2)(n) to GF(2)(m)] which are k-rotation symmetric and they give two infinite families of such functions which are permutations with the maximum possible algebraic degree. The families of functions that they give provide a source, which can be searched for functions with other useful cryptographic properties.
Budaghyan and Carlet (2008) [4] constructefi a family of almost perfect nonlinear (APN) hexanomials over a field with r(2) elements, and with terms of degrees r + 1, s + 1, rs + 1, rs + r, rs + s, and r + s, where r =...
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Budaghyan and Carlet (2008) [4] constructefi a family of almost perfect nonlinear (APN) hexanomials over a field with r(2) elements, and with terms of degrees r + 1, s + 1, rs + 1, rs + r, rs + s, and r + s, where r = 2(m) and s = 2(n) with GCD(m, n) = 1. The construction requires a certain technical condition, which was verified empirically in a finite number of examples. Bracken. Tan, and Tan (2011) [1] proved that the condition holds when m 2 or 4 (mod 6). In this article, we prove that the construction of Budaghyan and Carlet produces APN polynomials for all relatively prime values of m and n. More generally, if GCD(m, n) = k >= 1, Budaghyan and Carlet showed that the nonzero derivatives of the hexanomials are 2k-toone maps from F-r2 to F-r2, provided the same technical condition holds. We prove that their construction produces polynomials with this property for all m and n. Published by Elsevier Inc.
作者：
Wang, QichunKan, HaibinFudan Univ
Sch Comp Sci Shanghai Key Lab Intelligent Informat Proc Shanghai 200433 Peoples R China Fudan Univ
Dept Comp Sci Shanghai 200433 Peoples R China
Constructing APN or 4-differentially uniform permutations achieving all the necessary criteria is an open problem, and the research on it progresses slowly. In ACISP 2011, Car let put forth an idea for constructing di...
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Constructing APN or 4-differentially uniform permutations achieving all the necessary criteria is an open problem, and the research on it progresses slowly. In ACISP 2011, Car let put forth an idea for constructing differentially uniform permutations using extension fields, which was illustrated with a construction of a 4-differentially uniform (n, n)-permutation. The permutation has optimum algebraic degree and very good nonlinearity. However, it was proved to be a permutation only for n odd. In this note, we investigate further the construction of differentially uniform permutations using extension fields, and construct a 4-differentially uniform (n, n)-permutation for any n. These permutations also have optimum algebraic degree and very good nonlinearity. Moreover, we consider a more general type of construction, and illustrate it with an example of a 4-differentially uniform (n, n)-permutation with good cryptographic properties.
A general mathematical framework behind algebraic cryptanalytic attacks is developed. The framework relates to finding algebraic equations induced by vectorial boolean functions and, in particular, equations of low al...
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A general mathematical framework behind algebraic cryptanalytic attacks is developed. The framework relates to finding algebraic equations induced by vectorial boolean functions and, in particular, equations of low algebraic degree. The equations may involve only a subset of input variables and may or may not be conditioned on the values of output variables. In addition, the equations may have a constrained form interesting for the so-called fast algebraic attacks. A possible divide-and-conquer effect is pointed out and the notion of algebraic immunity order, naturally extending the notion of correlation immunity order, is defined. An application of general results to stream ciphers known as combiners with or without memory, with possibly multiple outputs, is studied in particular detail and the concept of divide-and-conquer algebraic attacks is introduced. Special properties of combiners with finite input memory, such as nonlinear filter generators, are also established. It is also pointed out that Grijbner basis algorithms may be used for finding low-degree induced algebraic equations.
We construct two classes of balanced S-boxes with high nonlinearity 2(n-1) - 2((n-1)/2) for n odd. From known results, it can be deduced that for any S-box which has nonlinearity 2(n-1) -2((n-1)/2) , the unrestricted ...
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We construct two classes of balanced S-boxes with high nonlinearity 2(n-1) - 2((n-1)/2) for n odd. From known results, it can be deduced that for any S-box which has nonlinearity 2(n-1) -2((n-1)/2) , the unrestricted nonlinearity is lower bounded by 2(n-1) -2((m+n-1)/2) while the generalized nonlinearity is lower bounded by 2(n-1) -(2(m) -1) 2((n-1)/2). We prove that the lower bound on the unrestricted nonlinearity of both our S-box constructions can be increased to 2(n-1) -2((m+n)/2-1). For the first class of S-box, the lower bound on generalized nonlinearity can be increased to 2(n-1) -2((n-1)/2+m-1). For the second class, the generalized nonlinearity is proven to be exactly 2(n-1) -2((m+n)/2-1), which is much higher than the lower bound for our first construction. The first class of S-boxes have low maximum differential while the second class corresponds to GMW sequences, whose algebraic structure allows us to construct a larger family of S-boxes. Moreover, both classes of S-boxes can attain high algebraic degree. We also compare our constructions with some known functions with high unrestricted and/or generalized nonlinearity.
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