4-uniform Permutations Research Articles

Ambiguity and deficiency for differentially [formula omitted]-uniform monomial permutations on [formula omitted

For a function f:Fq→Fq on a finite field Fq, new measures—ambiguity A(f) and deficiency D(f) of f were raised by Panario et al. (2010, 2011) to judge non-injectivity and non-surjectivity of the difference functions Δaf:Fq→Fq(a∈Fq∗). In this paper, we determine the values of A(f) and D(f) for three differentially 4-uniform monomial permutations, i.e., inverse function, Gold function and Dobbertin function on F2n (2∣n).

On the construction of differentially 4-uniform involutions

An involution is a permutation whose compositional inverse is itself. Differentially 4-uniform involutions with high algebraic degree and high nonlinearity are important for the design of block ciphers because they possess good cryptographic properties and the same component can be used in both encryption circuit and decryption circuit. A well known differentially 4-uniform involution is the multiplicative inverse function, which is used as the S-boxes in AES and Camellia. Although many differential 4-uniform permutations have been constructed, there are only a few classes of differentially 4-uniform involutions. Recently, Charpin et al. presented an idea of constructing involutions, which is called piece by piece construction. With this method, we construct a family of differentially 4-uniform involutions with optimal algebraic degree and high nonlinearity. It is also shown with the help of computer that such involutions which are CCZ-inequivalent to the known differentially 4-uniform permutations in small number of even dimensions are constructed. Furthermore, the number of CCZ-inequivalent classes of our involutions over F2n increases exponentially when n increases.

Differentially 4-Uniform Permutations with the Best Known Nonlinearity from Butterflies

Many block ciphers use permutations defined over the finite field F22k with low differential uniformity, high nonlinearity, and high algebraic degree to provide confusion. Due to the lack of knowledge about the existence of almost perfect nonlinear (APN) permutations over F22k, which have lowest possible differential uniformity, when k > 3, constructions of differentially 4-uniform permutations are usually considered. However, it is also very difficult to construct such permutations together with high nonlinearity; there are very few known families of such functions, which can have the best known nonlinearity and a high algebraic degree. At Crypto’16, Perrin et al. introduced a structure named butterfly, which leads to permutations over F22k with differential uniformity at most 4 and very high algebraic degree when k is odd. It is posed as an open problem in Perrin et al.’s paper and solved by Canteaut et al. that the nonlinearity is equal to 22k−1−2k. In this paper, we extend Perrin et al.’s work and study the functions constructed from butterflies with exponent e = 2i + 1. It turns out that these functions over F22k with odd k have differential uniformity at most 4 and algebraic degree k +1. Moreover, we prove that for any integer i and odd k such that gcd(i, k) = 1, the nonlinearity equality holds, which also gives another solution to the open problem proposed by Perrin et al. This greatly expands the list of differentially 4-uniform permutations with good nonlinearity and hence provides more candidates for the design of block ciphers.

New secondary constructions of differentially 4-uniform permutations over

ABSTRACTIn this paper, we generalize the switching method and present a method to construct new differentially 4-uniform permutations from two known ones by determining the corresponding cycle sets. As for applications, by determining all the cycle sets of and related to the inverse functions, respectively, we present two efficient constructions of differentially 4-uniform permutations. Moreover, it has been checked by the Magma software for small n that, both constructions give many new Carlet–Charpin–Zinoviev (CCZ)-inequivalent classes of functions that are not CCZ-equivalent to the known functions.

A new method to investigate the CCZ-equivalence between functions with low differential uniformity

Recently, many new classes of differentially 4-uniform permutations have been constructed. However, it is difficult to decide whether they are CCZ-inequivalent or not. In this paper, we propose a new notion called “Projected Differential Spectrum”. By considering the properties of the projected differential spectrum, we find several relations that should be satisfied by CCZ-equivalent functions. Based on these results, we mathematically prove that any differentially 4-uniform permutation constructed in [11] by C. Carlet, D. Tang, X. Tang, et al., is CCZ-inequivalent to the inverse function. We also get two interesting results with the help of computer experiments. The first one is a proof that any permutation constructed in [11] is CCZ-inequivalent to a function which is the summation of the inverse function and any Boolean function on F22k when 4≤k≤7. The second one is a differentially 4-uniform permutation on F26 which is CCZ-inequivalent to any function in the aforementioned two classes.

New explicit constructions of differentially 4-uniform permutations via special partitions of [formula omitted

In this paper, we further study the switching constructions of differentially 4-uniform permutations over F22k from the inverse function and propose several new explicit constructions. In our constructions, we first partition the finite field F22k into some minimal subsets that are closed under both mappings x↦1x−1+1 and x↦ωx, where ω∈F22k⁎ is of order 3. Then, by utilizing some properties of such subsets to extend differentially 4-uniform permutations over the subfield F4 or F24 to that over F22k, we give new constructions of differentially 4-uniform permutations over F22k for the cases k odd, k/2 odd and k/2 even respectively. As compared to previous constructions, our new constructions explicitly give large numbers (at least 222k−2−1) of functions.

A new family of differentially 4-uniform permutations over $$\mathbb{F}_{2^{2k} }$$ for odd k

We study the differential uniformity of a class of permutations over $\mathbb{F}_{2^{n}}$ with $n$ even. These permutations are different from the inverse function as the values $x^{-1}$ are modified to be $(\gamma x)^{-1}$ on some cosets of a fixed subgroup $\langle\gamma\rangle$ of $\mathbb{F}_{2^{n}}^*$. We obtain some sufficient conditions for this kind of permutations to be differentially 4-uniform, which enable us to construct a new family of differentially 4-uniform permutations that contains many new Carlet-Charpin-Zinoviev equivalent (CCZ-equivalent) classes as checked by Magma for small numbers $n$. Moreover, all of the newly constructed functions are proved to possess optimal algebraic degree and relatively high nonlinearity.

New differentially 4-uniform permutations by modifying the inverse function on subfields

Permutations over ??22k$\mathbb {F}_{2^{2k}}$ with low differential uniformity, high algebraic degree and high nonlinearity are of great cryptographic importance since they can be chosen as the substitution boxes (S-boxes) for many block ciphers with SPN (Substitution Permutation Network) structure. A well known example is that the S-box of the famous Advanced Encryption Standard (AES) is derived from the inverse function on ??28$\mathbb {F}_{2^{8}}$, which has been proved to be a differentially 4-uniform permutation with the optimal algebraic degree and known best nonlinearity. Recently, Zha et al. proposed two constructions of differentially 4-uniform permutations over ??22k$\mathbb {F}_{2^{2k}}$, say Gt and Gs, t with Tr(s?1) = 1, by applying affine transformations to the inverse function on some subfields of ??22k$\mathbb {F}_{2^{2k}}$ (Zha et al. Finite Fields Appl. 25, 64---78, 2014). In this paper, we generalize their method by applying other types of EA (extended affine) equivalent transformations to the inverse function on some subfields of ??22k$\mathbb {F}_{2^{2k}}$ and present two new constructions of differentially 4-uniform permutations, say F? and Fs, ? with Tr(s?1) = 1. Furthermore, we prove that all the functions Gt with different t are CCZ (Carlet-Charpin-Zinoviev) equivalent to our subclass F0, while all the functions Gs, t with different t are CCZ-equivalent to our subclass Fs,0. In addition, both our two constructions give many new CCZ-inequivalent classes of such functions, as checked by computer in small numbers of variables. Moreover, all these newly constructed permutations are proved to have the optimal algebraic degree and high nonlinearity.

Constructing New Piecewise Differentially 4-Uniform Permutations from Known APN Functions

A method for constructing piecewise differentially 4-uniform permutations has been recently introduced by Zha, Hu and Sun. By using this method, we provide two new infinite families of differentially 4-uniform permutations from the known APN functions. The CCZ inequivalence between these constructed functions and the known differentially 4-uniform permutations are also investigated by computation.

Constructing new APN functions and bent functions over finite fields of odd characteristic via the switching method

The switching method is a very powerful method to construct new APN functions and differentially 4-uniform permutations over finite fields of even characteristic. In this paper, using this method, we present several new constructions of infinite classes of nonpower APN functions and two new classes of bent functions in finite fields of odd characteristic.

Some Binomial and Trinomial Differentially 4-Uniform Permutation Polynomials

Permutation polynomials with low differential uniformity are important candidate functions to design substitution boxes of block ciphers. In this paper, we investigate several classes of differential 4-uniform binomial and trinomial permutation polynomials over the finite field [Formula: see text] of [Formula: see text] elements.

Further results on differentially 4-uniform permutations over $\mathbb{F}_{2^{2m} } $

We present several new constructions of differentially 4-uniform permutations over \(\mathbb{F}_{2^{2m} } \) by modifying the values of the inverse function on some subsets of \(\mathbb{F}_{2^{2m} } \). The resulted differentially 4-uniform permutations have high nonlinearities and algebraic degrees, which provide more choices for the design of crytographic substitution boxes.

More constructions of differentially 4-uniform permutations on $${\mathbb {F}}_{2^{2k}}$$ F 2 2 k

Differentially $$4$$4-uniform permutations on $${\mathbb {F}}_{2^{2k}}$$F22k with high nonlinearity are chosen as Substitution boxes in many block ciphers and some stream ciphers. Recently, Qu et al. (IEEE Trans Inf Theory, 59(7), 4675---4686, 2013) introduced a class of functions, which are called preferred functions, to construct a lot of infinite families of such permutations. In this paper, we propose a particular type of Boolean functions to characterize the preferred functions. On the one hand, such Boolean functions can be determined by solving linear equations, and they give rise to a huge number of differentially $$4$$4-uniform permutations over $${\mathbb {F}}_{2^{2k}}$$F22k. Hence they may provide more choices for the design of Substitution boxes. On the other hand, by investigating the number of these Boolean functions, we show that the number of CCZ-inequivalent differentially $$4$$4-uniform permutations over $${\mathbb {F}}_{2^{2k}}$$F22k grows exponentially when $$k$$k increases, which gives a positive answer to an open problem proposed in Qu et al.(IEEE Trans Inf Theory, 59(7), 4675---4686, 2013).

Constructing new differentially 4-uniform permutations from the inverse function

Two new families of differentially 4-uniform permutations over F22m are constructed by modifying the values of the inverse function on some subfield of F22m and by applying affine transformations on the function. The resulted 4-uniform permutations have high nonlinearity and algebraic degree. A family of differentially 6-uniform permutations with high nonlinearity and algebraic degree is also constructed by making the modification on an affine subspace of F22m.

Constructing Differentially 4-Uniform Permutations Over <formula formulatype="inline"><tex Notation="TeX">${\BBF}_{2^{2k}}$</tex> </formula> via the Switching Method

Many block ciphers use permutations defined on F(22k ) with low differential uniformity, high nonlinearity, and high algebraic degree as their S-boxes to provide confusion. It is well known that, for a function on F(2n), the lowest differential uniformity is 2 and the functions achieving this lower bound are called almost perfect nonlinear (APN) functions. However, due to the lack of knowledge on APN permutations on F(22k ), differentially 4-uniform permutations are usually chosen as S-boxes. For example, the currently endorsed Advanced Encryption Standard chooses one such function, the multiplicative inverse function, as its S-box. By a recent survey on differentially 4-uniform permutations over F(22k ), there are only five known infinite families of such functions, and most of them have small algebraic degrees. In this paper, we apply the powerful switching method to discover many CCZ-inequivalent infinite families of such functions on F(22k ) with optimal algebraic degree, where k is an arbitrary positive integer. This greatly expands the list of differentially 4-uniform permutations and hence provide more choices for the S-boxes. Furthermore, lower bounds for the nonlinearity of the functions obtained in this paper are presented and they imply that some infinite families have high nonlinearity.

A negative answer to Bracken–Tan–Tan's problem on differentially 4-uniform permutations over [formula omitted

Permutations with low differential uniformity are widely used in cipher design. Recently, Bracken, Tan and Tan (2012) [5] presented a method to construct differentially 4-uniform permutations by changing certain conditions of known APN functions. They guessed that only two classes of existing quadratic APN functions have this property. They succeeded in proving one class and left the other one as an open problem. In this paper, with the help of a computer, those polynomials are proved to be differentially 4-uniform but may not be permutation polynomials, which give a negative answer to this problem.

Constructing differentially 4-uniform permutations over GF(22m ) from quadratic APN permutations over GF(22m+1)

In this paper, by means of the idea proposed by Carlet (ACISP 1-15, 2011), differentially 4-uniform permutations with the best known nonlinearity over $${\mathbb{F}_{2^{2m}}}$$ are constructed using quadratic APN permutations over $${\mathbb{F}_{2^{2m+1}}}$$ . Special constructions are given using the Gold functions. The algebraic degree of the constructions and their compositional inverses is also investigated. One construction and its compositional inverse both have algebraic degree m + 1 over $${\mathbb{F}_2^{2m}}$$ .