Almost Perfect Nonlinear Research Articles

A further study on the Ness-Helleseth function

Let Fpn be a finite field with pn elements. Ness and Helleseth in [29] first studied a class of functions over Fpn with the form f(x)=uxpn−32+xpn−2,u∈Fpn⁎, which is called the Ness-Helleseth function. The f(x) has been proved to be an almost perfect nonlinear (APN) function by Ness and Helleseth for p=3 in [29] and by Zeng et al. for any odd prime p in [43] under the condition pn≡3(mod4) and η(1+u)=η(u−1). In this paper, we continue to study the Ness-Helleseth functions under the condition that pn≡3(mod4) and η(1+u)≠η(u−1). Firstly, we prove that f(x) is a permutation polynomial with differential uniformity not more than 4 if η(1+u)=η(1−u). Moreover, for some more special u, f is an involution with differential uniformity at most 3. Secondly, we show that f(x) is a locally-APN function for u=±1. In addition, the differential spectrum and boomerang spectrum of f(x) are obtained via judging the number of solutions of some special equations. We obtain the first non-PN function that its boomerang uniformity can attain 0 or 1.

Study of cyclic codes from low differentially uniform functions and its consequences

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7].

Note on Budaghyan and Carlet's almost perfect nonlinear functions

Almost perfect nonlinear (APN) functions have good properties and are widely applied in sequence design and coding theory. Budaghyan and Carlet (2008) [5] constructed a family of APN hexanomials F3 over F22m with a certain technical condition. In this article, we give the number of APN hexanomials F3 and support a determination theorem for APN hexanomials F3 if i=1. Moreover, we construct a family of APN functions in bivariate form and show it is CCZ-equivalent to F3.

Revisiting some results on APN and algebraic immune functions

<p style='text-indent:20px;'>We push a little further the study of two recent characterizations of almost perfect nonlinear (APN) functions. We state open problems about them, and we revisit in their perspective a well-known result from Dobbertin on APN exponents. This leads us to a new result about APN power functions and more general APN polynomials with coefficients in a subfield <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{2^k} $\end{document}</tex-math></inline-formula>, which eases the research of such functions. It also allows to construct automatically many differentially uniform functions from them (this avoids calculations for proving their differential uniformity as done in a recent paper, which are tedious and specific to each APN function). In a second part, we give simple proofs of two important results on Boolean functions, one of which deserves to be better known but needed clarification, while the other needed correction.</p>

Some Binary Quasi-perfect Linear Codes Defined by APN Functions

In 2021, Tutdere proved that the covering radii R of a class of primitive binary cyclic codes with minimum distance strictly greater than an odd integer l satisfy r≤R≤l, where l, r are some integers depending on the given code. We here first discuss some equivalences of linear codes defined by Gold functions, which are quadratic APN (almost perfect nonlinear) functions. We then show that by applying the result of Tutdere one can find the covering radii of these quasi-perfect codes. In 2016, Li and Helleseth proved that the linear codes defined by the quadratic APN functions are quasi-perfect and they asked whether the linear codes defined by the non-quadratic APN functions are quasi-perfect or not. We here prove that the linear codes defined by some non-quadratic APN functions over the finite field〖 F〗_(2^m ) , for 1≤m≤8, are quasi-perfect, by computing the covering radii of these codes.

Biprojective Almost Perfect Nonlinear Functions

In this paper, we introduce the concept of biprojectivity in studying the cryptographically important almost perfect nonlinear (APN) functions. Although several known families of biprojective APN functions exist in the literature, the concept itself has not been explicitly observed and studied in detail. We give a survey of known APN families and functions that fall into the biprojective setting, then provide a method for finding such families and functions. We finally give two new infinite families of biprojective APN functions <inline-formula> <tex-math notation="LaTeX">${\mathsf {CCZ}}$ </tex-math></inline-formula>-inequivalent to any known APN function and study their properties.

On CCZ-inequivalence of some families of almost perfect nonlinear functions to permutations

Browning et al. (2010) exhibited almost perfect nonlinear (APN) permutations on $\mathbb {F}_{2^{6}}$ . This was the first example of an APN permutation on an even degree extension of $\mathbb {F}_{2}$ . In their approach of finding an APN permutation, Browning et al. made use of a necessary and sufficient condition based on the Walsh transform. In this paper, we give an algorithm based on a related necessary condition which checks whether a vectorial Boolean function is CCZ-inequivalent to a permutation. Using this algorithm, we are able to show that no function belonging to a known family of APN functions is equivalent to a permutation on $\mathbb {F}_{2^{2m}}$ , where m ≤ 6 (except for the known case on $\mathbb {F}_{2^{6}}$ ). We also give an EA-invariant based on the condition. Finally, we give a theoretical proof of the fact that no member of a specific family of APN functions is equivalent to a permutation on doubly-even degree extensions of $\mathbb {F}_{2}$ .

The Number of Almost Perfect Nonlinear Functions Grows Exponentially

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least $$\frac{\varphi (m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil $$ inequivalent APN functions on the finite field with $${2^{2m}}$$ elements, where $$\varphi $$ denotes Euler’s totient function. This is a great improvement of previous results: for even m, the best known lower bound has been $$\frac{\varphi (m)}{2}\left( \lfloor \frac{m}{4}\rfloor +1\right) $$ ; for odd m, there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi’s APN functions.

New generalized almost perfect nonlinear functions

APN (almost perfect non-linear) functions over finite fields of even characteristic are widely studied due to their applications to the design of symmetric ciphers resistant to differential attacks. This notion was recently generalized to GAPN (generalized APN) functions by Kuroda and Tsujie to odd characteristic p. They presented some constructions of GAPN functions, and other constructions were given by Zha et al. We present new constructions of GAPN functions both in the case of monomial and multinomial functions. Our monomial GAPN functions can be viewed as a further generalization of the Gold APN functions. We show that a certain technique used by Hou to construct permutations over finite fields also yields monomial GAPN functions. We also present several new constructions of GAPN functions which are sums of monomial GAPN functions, as well as new GAPN functions of degree p which can be written as the product of two powers of linearized polynomials. For this latter construction we describe some interesting differences between even and odd characteristic and also obtain a classification in certain cases.

A New Family of APN Quadrinomials

The binomial $B(x) = x^{3} + \beta x^{36}$ (where $\beta $ is primitive in $\mathbb {F}_{2^{2}}$ ) over $\mathbb {F}_{2^{10}}$ is the first known example of an Almost Perfect Nonlinear (APN) function that is not CCZ-equivalent to a power function, and has remained unclassified into any infinite family of APN functions since its discovery in 2006. We generalize this binomial to an infinite family of APN quadrinomials of the form $x^{3} + a (x^{2^{i}+1})^{2^{k}} + b x^{3 \cdot 2^{m}} + c (x^{2^{i+m}+2^{m}})^{2^{k}}$ from which $B(x)$ can be obtained by setting $a = \beta $ , $b = c = 0$ , $i = 3$ , $k = 2$ . We show that for any dimension $n = 2m$ with $m$ odd and $3 \nmid m$ , setting $(a,b,c) = (\beta, \beta ^{2}, 1)$ and $i = m-2$ or $i = (m-2)^{-1} \mod n$ yields an APN function, and verify that for $n = 10$ the quadrinomials obtained in this way for $i = m-2$ and $i = (m-2)^{-1} \mod n$ are CCZ-inequivalent to each other, to $B(x)$ , and to any other known APN function over $\mathbb {F}_{2^{10}}$ .

Classification of quadratic APN functions with coefficients in [formula omitted] for dimensions up to 9

Almost perfect nonlinear (APN) and almost bent (AB) functions are integral components of modern block ciphers and play a fundamental role in symmetric cryptography. In this paper, we describe a procedure for searching for quadratic APN functions with coefficients in F2 over the finite field F2n and apply this procedure to classify all such functions over F2n with n≤9. We discover two new APN functions (which are also AB) over F29 that are CCZ-inequivalent to any known APN function over this field. We also verify that there are no quadratic APN functions with coefficients in F2 over F2n with 6≤n≤8 other than the currently known ones.

Constructing APN Functions Through Isotopic Shifts

Almost perfect nonlinear (APN) functions over fields of characteristic 2 play an important role in cryptography, coding theory and, more generally, mathematics and information theory. In this paper we deduce a new method for constructing APN functions by studying the isotopic equivalence, concept defined for quadratic planar functions in fields of odd characteristic. In particular, we construct a family of quadratic APN functions which provides a new example of an APN mapping over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{2^{9}}$ </tex-math></inline-formula> and includes an example of another APN function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{9}+ \mathop {\mathrm {Tr}}\nolimits (x^{3})$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{2^{8}}$ </tex-math></inline-formula> , known since 2006 and not classified up to now. We conjecture that the conditions for this family are satisfied by infinitely many APN functions.

Almost perfect nonlinear families which are not equivalent to permutations

An important problem on almost perfect nonlinear (APN) functions is the existence of APN permutations on even-degree extensions of F2 larger than 6. Browning et al. (2010) gave the first known example of an APN permutation on the degree-6 extension of F2. The APN permutation is CCZ-equivalent to the previously known quadratic Kim κ-function (Browning et al. (2009)). Aside from the computer based CCZ-inequivalence results on known APN functions on even-degree extensions of F2 with extension degrees less than 12, no theoretical CCZ-inequivalence result on infinite families is known. In this paper, we show that Gold and Kasami APN functions are not CCZ-equivalent to permutations on infinitely many even-degree extensions of F2. In the Gold case, we show that Gold APN functions are not equivalent to permutations on any even-degree extension of F2, whereas in the Kasami case we are able to prove inequivalence results for every doubly-even-degree extension of F2.

Inverse function is not component-wise uniform

Recently, Carlet introduced (Carlet Finite. Fields Appl. 53, 226–253 (2018)) the concept of component-wise uniform (CWU) functions which is a stronger notion than being almost perfect nonlinear (APN). Carlet showed that crooked functions (in particular quadratic functions, including Gold functions) and the compositional inverse of a specific Gold function are CWU. Carlet also compiled a table on the component-wise uniformity of APN power functions, where for the extension degrees less than 12, the APN power functions which are always CWU are Gold, the compositional inverse of the specific Gold function and Kasami. In this paper, we show that the Inverse function is not CWU if n > 5. We also show that any additive shift of a derivative of the Inverse function is not a cyclic difference set.

Monomial Generalized Almost Perfect Nonlinear Functions

Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In particular, on finite fields of even characteristic, GAPN functions coincide with APN functions. In this paper, we study monomial GAPN functions for odd characteristic. We give monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic. Moreover, we define a generalization of exceptional APN functions and give typical examples.

Discrete antiderivatives for functions over mathop {{mathbb {F}}}_p^n

In the design of cryptographic functions, the properties of their discrete derivatives have to be carefully considered, as many cryptographic attacks exploit these properties. One can therefore attempt to first construct derivatives with the desired properties and then recover the function itself. Recently Suder developed an algorithm for reconstructing a function (also called antiderivative) over the finite field mathop {{mathbb {F}}}_{2^n} given its discrete derivatives in up to n linearly independent directions. Pasalic et al. also presented an algorithm for determining a function over mathop {{mathbb {F}}}_{p^n} given one of its derivatives. Both algorithms involve solving a p^n times p^n system of linear equations; the functions are represented as univariate polynomials over mathop {{mathbb {F}}}_{p^n}. We show that this apparently high computational complexity is not intrinsic to the problem, but rather a consequence of the representation used. We describe a simpler algorithm, with quasilinear complexity, provided we work with a different representation of the functions. Namely they are polynomials in n variables over mathop {{mathbb {F}}}_{p} in algebraic normal form (for p>2, additionally, we need to use the falling factorial polynomial basis) and the directions of the derivatives are the canonical basis of mathop {{mathbb {F}}}_{p}^n. Algorithms for other representations (the directions of the derivatives not being the canonical basis vectors or the univariate polynomials over mathop {{mathbb {F}}}_{p^n} mentioned above) can be obtained by combining our algorithm with converting between representations. However, the complexity of these conversions is, in the worst case, exponential. As an application, we develop a method for constructing new quadratic PN (Perfect Nonlinear) functions. We use an approach similar to the one of Suder, who used antiderivatives to give an alternative formulation of the methods of Weng et al. and Yu et al. for searching for new quadratic APN (Almost Perfect Nonlinear) functions.

Algebraic Construction of Near-Bent and APN Functions

Boolean functions play an important role in symmetric cryptosystems. In this paper, we have constructed near-bent Boolean functions algebraically with the help of Niho power function exponent in the trace form over a finite field. Specific cryptographic properties which may gain attention with suitable modifications are observed using these functions. If exponents are either Mersenne numbers or Fermat numbers, then the resulting functions are found to be APN (almost perfect nonlinear).

Differential uniformity and the associated codes of cryptographic functions

The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper, we consider more generally the codes associated with functions that have differential uniformity at least \begin{document}$ 4 $\end{document} . We emphasize, for such a function \begin{document}$ F $\end{document} , the role of codewords of weight \begin{document}$ 3 $\end{document} and \begin{document}$ 4 $\end{document} and of some cosets of its associated code \begin{document}$ C_F $\end{document} . We give some properties on codes associated with differential uniformity exactly \begin{document}$ 4 $\end{document} . We obtain lower bounds and upper bounds for the numbers of codewords of weight less than \begin{document}$ 5 $\end{document} of the codes \begin{document}$ C_F $\end{document} . We show that the nonlinearity of \begin{document}$ F $\end{document} decreases when these numbers increase. We obtain a precise expression to compute these numbers, when \begin{document}$ F $\end{document} is a plateaued or a differentially two-valued function. As an application, we propose a method to construct differentially \begin{document}$ 4 $\end{document} -uniform functions, with a large number of \begin{document}$ 2 $\end{document} -to- \begin{document}$ 1 $\end{document} derivatives, from APN functions.

Construction of CCZ transform for quadratic APN functions

Almost perfect nonlinear (APN) function is an important type of function in cryptography, especially quadratic APN function. Since the notion of CCZ-equivalence developed, the construction of CCZ transform for APN functions to obtain new APN functions became a critical issue in cryptography. Inspired by the result of Budaghyan who used Gold functions, this article gives the construction of CCZ transform for all quadratic vectorial Boolean functions and proves that for quadratic APN functions, the functions transformed have algebraic degree 3, thus EA-inequivalent to all quadratic functions, and have minimum algebraic degree 2, thus EA-inequivalent to all power functions.

On the differential equivalence of APN functions

Carlet et al. (Des. Codes Cryptogr. 15, 125–156, 1998) defined the associated Boolean function γF(a,b) in 2n variables for a given vectorial Boolean function F from $\mathbb {F}_{2}^{n}$ to itself. It takes value 1 if a≠0 and equation F(x) + F(x + a) = b has solutions. This article defines the differentially equivalent functions as vectorial functions having equal associated Boolean functions. It is an open problem of great interest to describe the differential equivalence class for a given Almost Perfect Nonlinear (APN) function. We determined that each quadratic APN function G in n variables, n ≤ 6, that is differentially equivalent to a given quadratic APN function F, can be represented as G = F + A, where A is affine. For the APN Gold function F, we completely described all affine functions A such that F and F + A are differentially equivalent. This result implies that the class of APN Gold functions up to EA-equivalence contains the first infinite family of functions, whose differential equivalence class is non-trivial.