Perfect Nonlinear Functions Research Articles

A quantum query algorithm for computing the degree of a perfect nonlinear Boolean function

The degree of a Boolean function is a basic primitive that has applications in coding theory and cryptography. This paper considers a problem of computing the degree of a perfect nonlinear Boolean function in a quantum system. The details are as follows: Given a promise that the function f is either linear or perfect nonlinear in $$F^{n}_{d}$$ , we propose a quantum algorithm 1 to distinguish which case it is with a high probability, where d is an even number. Furtherly, for computing the degree of a perfect nonlinear Boolean function f, we present a quantum Algorithm 2 to solve it by calling quantum Algorithm 1 when $$d=2$$ . The quantum query complexity of the proposed quantum Algorithm 2 is O(s), and the space complexity (the number of quantum logic gate) is $$O(2^{s})$$ , where $$s+1=\text {deg}(f)$$ . The analysis shows that the quantum Algorithm 2 proposed in this paper is more efficient than any classical algorithm for solving this problem.

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.

Three new classes of generalized almost perfect nonlinear power functions

Generalized almost perfect nonlinear (GAPN) functions are algebraic generalization of almost perfect nonlinear functions and have important applications in cryptography and finite geometry. Based on the well known binomial theorem and some combinatory formulas, the conjecture proposed by Kuroda is partially disproved and three new classes of GAPN power functions for odd characteristic are presented.

On APN exponents, characterizations of differentially uniform functions by the Walsh transform, and related cyclic-difference-set-like structures

In this paper, we summarize the results obtained recently in three papers on differentially uniform functions in characteristic 2, and presented at the workshop WCC 2017 in Saint-Petersburg, and we give new results on these functions. Firstly, we recall the recent connection between almost perfect nonlinear (APN) power functions and the two notions in additive combinatorics of Sidon sets and sum-free sets; we also recall a characterization of APN exponents which leads to a property of Dickson polynomials in characteristic 2 previously unobserved, which is generalizable to all finite fields. We also give a new characterization of APN exponents in odd dimension by Singer sets. Secondly, after recalling the recent multiple generalization to differentially $$\delta $$ -uniform functions of the Chabaud–Vaudenay characterization of APN functions by their Walsh transforms, we generalize the method to all criteria on vectorial functions dealing with the numbers of solutions of equations of the form $$\sum _{i\in I}F(x+u_{i,a})+L_a(x)+u_a=0$$ , with $$L_a$$ linear; we give the examples of injective functions and of o-polynomials; we also deduce a generalization to differentially $$\delta $$ -uniform functions of the Nyberg characterization of APN functions by means of the Walsh transforms of their derivatives. Thirdly, we recall the two notions of componentwise APNness (CAPNness) and componentwise Walsh uniformity (CWU). We recall why CAPN functions can exist only if n is odd and why crooked functions (in particular, quadratic APN functions) are CWU. We also recall that the inverse of one of the Gold permutations is CWU and not the others. Another potential class of CWU functions is that of Kasami functions. We consider the difference sets with Singer parameters equal to the complement of $$\varDelta _F=\{F(x)+F(x+1)+1; x\in \mathbb {F}_{2^n}\}$$ where F is a Kasami function. These sets have another potential property, called the cyclic-additive difference set property, which is related to the CWU property in the case of power permutations (n odd). We study cyclic-additive difference sets among Singer sets. We recall the main properties of Kasami functions and of the related set $$\varDelta _F$$ shown by Dillon and Dobbertin and we observe and prove new expressions for $$\varDelta _F$$ .

On an algorithm generating 2-to-1 APN functions and its applications to “the big APN problem”

Almost perfect nonlinear (APN) functions are of great interest to many researchers since they have the optimal resistance to the differential attack. The existence of bijective APN functions in even number of variables is an important open problem, and there is only one known example of such a function at present. In this paper we consider a special subclass of 2-to-1 vectorial Boolean functions that can allow us to search and construct APN permutations. We proved that each 2-to-1 function is potentially EA-equivalent to a permutation and proposed an algorithm that generates special symbol sequences for constructing 2-to-1 APN functions. Also, we described two methods for searching APN permutations, that are based on sequences generated by this algorithm.

A Construction of Multiple Optimal ZCZ Sequence Sets With Good Cross Correlation

Zero correlation zone (ZCZ) sequences are a class of spreading sequences having ideal auto-correlation and cross correlation in a zone around the origin. They have been extensively studied in recent years due to their important applications in quasi-synchronous code division multiple access systems. In this paper, a construction of ZCZ sequence sets is proposed based on perfect nonlinear functions. It generates multiple ZCZ sequence sets with the properties: 1) each sequence is perfect in the sense that its out-of-phase auto-correlation is always zero; 2) each ZCZ sequence set is optimal with respect to the Tang–Fan–Matsufuji bound in which all the sequences are pairwise cyclically distinct; and 3) the maximum inter-set cross correlation of multiple sequence sets achieves the well-known Sarwate bound.

Algebraic manipulation detection codes with perfect nonlinear functions under non-uniform distribution

Algebraic manipulation detection codes with perfect nonlinear functions under non-uniform distribution

Fourier transforms and bent functions on finite groups

Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis $${\widehat{G}}$$ , consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on $${\widehat{G}}$$ , and obtain characterizations of a bent function by its Fourier transforms (as functions on $${\widehat{G}}$$ ). For a function f from G to another finite group, we define a dual function $${\widetilde{f}}$$ on $${\widehat{G}}$$ , and characterize the nonlinearity of f by its dual function $${\widetilde{f}}$$ . Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.

Binary linear codes from vectorial boolean functions and their weight distribution

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding et al. recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contain the all-one codeword, and determine their weight distribution. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.’s generic construction.

Zero-Difference Balanced Functions With New Parameters and Their Applications

As an optimal combinatorial object, zero-difference balanced (ZDB) functions introduced by Ding in 2008, are a generalization of the well-known perfect nonlinear functions. ZDB functions have received much attention in recent years due to its important applications in coding theory and sequence design. One objective of this paper is to present a construction of ZDB functions based on a kind of generalized cyclotomy. It generates ZDB functions over cyclic group with new parameters which can not be produced by earlier constructions. Another objective of this paper is to employ these ZDB functions to obtain at the same time: 1) optimal constant-composition codes; 2) perfect difference systems of sets; and 3) optimal frequency-hopping sequences, all with new parameters.

Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: The gold case

An almost perfect nonlinear (APN) function \begin{document}$f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$\end{document} (necessarily polynomial) is called exceptional APN if it is APN on infinitely many extensions of \begin{document}$\mathbb{F}_{2^n}$\end{document} . Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number \begin{document}$(2^k+1)$\end{document} or a Kasami-Welch number \begin{document}$(2^{2k}-2^k+1)$\end{document} . When the degree of the polynomial function is a Gold number or a Kasami-Welch number, several partial results have been obtained by several authors including us. In this article we address these exceptions. We almost prove the exceptional APN conjecture in the Gold degree case when \begin{document}$\deg{(h(x))}$\end{document} is odd. We also show exactly when the corresponding multivariate polynomial \begin{document}$φ(x, y, z)$\end{document} is absolutely irreducible. Also, there is only one result known when \begin{document}$f(x)=x^{2^{k}+1} + h(x)$\end{document} , and \begin{document}$\deg(h(x))$\end{document} is even. Here, we extend this result as well, thus making progress in this case that seems more difficult.

Nonlinear functions and difference sets on group actions

Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G, H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat X$. We will also investigate the existence and constructions of $G$-perfect nonlinear functions and $G$-bent functions. Several known results in [2,6,10,17] are direct consequences of our results.

Characterization of almost perfect nonlinear functions in terms of subfunctions

Abstract The paper is concerned with combinatorial description of almost perfect nonlinear functions (APN-functions). A complete characterization of n-place APN-functions in terms of (n − 1)-place subfunctions is obtained. An n-place function is shown to be an APN-function if and only if each of its (n − 1)-place subfunctions is either an APN-function or has the differential uniformity 4 and the admissibility conditions hold. A detailed characterization of 2, 3 or 4-place APN-functions is presented.

On the conjecture on APN functions and absolute irreducibility of polynomials

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field \(\mathbb {F}\) is called exceptional APN, if it is also APN on infinitely many extensions of \(\mathbb {F}\). In this article we consider the most studied case of \(\mathbb {F}=\mathbb {F}_{2^n}\). A conjecture of Janwa–Wilson and McGuire–Janwa–Wilson (1993/1996), settled in 2011, was that the only monomial exceptional APN functions are the monomials \(x^n\), where \(n=2^k+1\) or \(n={2^{2k}-2^k+1} \) (the Gold or the Kasami exponents, respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our results is that all functions of the form \(f(x)=x^{2^k+1}+h(x)\) (for any odd degree h(x), with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture. We also show absolute irreducibility of a class of multivariate polynomials over finite fields (by repeated hyperplane sections, linear transformations, and reductions) and discuss their applications.

Two new message authentication codes based on APN functions and stream ciphers

AbstractAfter the concept of the active wiretapper was proposed, integrity protection became more important than ever before. Therefore, message authentication code, a method that protects the message from being modified in an undetectable way, attracts more attention nowadays. In this paper, we propose two new message authentication codes based on almost perfect nonlinear functions and stream ciphers. The security of both new constructions is proved by giving upper bounds of the probability of the successful substitution forgery attacks against our new message authentication codes, and these upper bounds are negligible. We implement our algorithms and compare their time consumption with the time consumption of EIA1, the message authentication code used in the 4G LTE system. The results show that our algorithms are overwhelmingly faster than EIA1. Moreover, our new constructions are resistant to cycling and linear forgery attacks, which can be applied to EIA1. Copyright © 2016 John Wiley & Sons, Ltd.

Dembowski-Ostrom polynomials from reversed Dickson polynomials

This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2. The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form \({x^{1 + {2^\alpha }}}\) is almost perfect nonlinear. It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions.

The arithmetic of Carmichael quotients

Carmichael quotients for an integer $m\ge 2$ are introduced analogous to Fermat quotients, by using Carmichael function $\lambda(m)$. Various properties of these new quotients are investigated, such as basic arithmetic properties, sequences derived from Carmichael quotients, Carmichael-Wieferich numbers, and so on. Finally, we link Carmichael quotients to perfect nonlinear functions.

Perfect nonlinear functions and cryptography

In the late 1980s the importance of highly nonlinear functions in cryptography was first discovered by Meier and Staffelbach from the point of view of correlation attacks on stream ciphers, and later by Nyberg in the early 1990s after the introduction of the differential cryptanalysis method. Perfect nonlinear (PN) and almost perfect nonlinear (APN) functions, which have the optimal properties for offering resistance against differential cryptanalysis, have since then been an object of intensive study by many mathematicians. In this paper, we survey some of the theoretical results obtained on these functions in the last 25 years. We recall how the links with other mathematical concepts have accelerated the search on PN and APN functions. To illustrate the use of PN and APN functions in practice, we discuss examples of ciphers and their resistance to differential attacks. In particular, we recall that in cryptographic applications suboptimal functions are often used.