arXiv subjects

Sihem Mesnager

Publications and source records attributed to Sihem Mesnager.

At least 73 records · Page 4Linked to original sources

arXiv2019

Solutions of $x^{q^k}+\cdots+x^{q}+x=a$ in $GF{2^n}$

Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field is fairly large. Thus, it may be of great interest to find an explicit representation of the solutions independently of the field base. This was previously done only for quadratic equations over a binary finite field. This paper gives an explicit representation of solutions for a much wider class of affine polynomials over a binary prime field.

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arXiv2019

Codebooks from generalized bent $\mathbb{Z}_4$-valued quadratic forms

Codebooks with small inner-product correlation have application in unitary space-time modulations, multiple description coding over erasure channels, direct spread code division multiple access communications, compressed sensing, and coding theory. It is interesting to construct codebooks (asymptotically) achieving the Welch bound or the Levenshtein bound. This paper presented a class of generalized bent $\mathbb{Z}_4$-valued quadratic forms, which contain functions of Heng and Yue (Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$. Cryptogr. Commun. 9(1), 41-53, 2017). By using these generalized bent $\mathbb{Z}_4$-valued quadratic forms, we constructs optimal codebooks achieving the Levenshtein bound. These codebooks have parameters $(2^{2m}+2^m,2^m)$ and alphabet size $6$.

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arXiv2019

Solving $x^{2^k+1}+x+a=0$ in $\mathbb{F}_{2^n}$ with $\gcd(n,k)=1$

Let $N_a$ be the number of solutions to the equation $x^{2^k+1}+x+a=0$ in $\GF {n}$ where $\gcd(k,n)=1$. In 2004, by Bluher \cite{BLUHER2004} it was known that possible values of $N_a$ are only 0, 1 and 3. In 2008, Helleseth and Kholosha \cite{HELLESETH2008} have got criteria for $N_a=1$ and an explicit expression of the unique solution when $\gcd(k,n)=1$. In 2014, Bracken, Tan and Tan \cite{BRACKEN2014} presented a criterion for $N_a=0$ when $n$ is even and $\gcd(k,n)=1$. This paper completely solves this equation $x^{2^k+1}+x+a=0$ with only condition $\gcd(n,k)=1$. We explicitly calculate all possible zeros in $\GF{n}$ of $P_a(x)$. New criterion for which $a$, $N_a$ is equal to $0$, $1$ or $3$ is a by-product of our result.

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arXiv2019

A class of narrow-sense BCH codes over $\mathbb{F}_q$ of length $\frac{q^m-1}{2}$

BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorics design. This paper studies a class of linear codes of length $ \frac{q^m-1}{2}$ over $\mathbb{F}_q$ with special trace representation, where $q$ is an odd prime power. With the help of the inner distributions of some subsets of association schemes from bilinear forms associated with quadratic forms, we determine the weight enumerators of these codes. From determining some cyclotomic coset leaders $δ_i$ of cyclotomic cosets modulo $ \frac{q^m-1}{2}$, we prove that narrow-sense BCH codes of length $ \frac{q^m-1}{2}$ with designed distance $δ_i=\frac{q^m-q^{m-1}}{2}-1-\frac{q^{ \lfloor \frac{m-3}{2} \rfloor+i}-1}{2}$ have the corresponding trace representation, and have the minimal distance $d=δ_i$ and the Bose distance $d_B=δ_i$, where $1\leq i\leq \lfloor \frac{m+3}{4} \rfloor$.

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arXiv2019

On the boomerang uniformity of (quadratic) permutations over $F_{2^n}$

At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool called Boomerang Connectivity Table (BCT) for measuring the resistance of a block cipher against the boomerang attack (which is an important cryptanalysis technique introduced by Wagner in 1999 against block ciphers). Next, Boura and Canteaut introduced an important parameter (related to the BCT) for cryptographic Sboxes called boomerang uniformity. In this context, we present a brief state-of-the-art on the notion of boomerang uniformity of vectorial functions (or Sboxes) and provide new results. More specifically, we present a slightly different (and more convenient) formulation of the boomerang uniformity and show that the row sum and the column sum of the boomerang connectivity table can be expressed in terms of the zeros of the second-order derivative of the permutation or its inverse. Most importantly, we specialize our study of boomerang uniformity to quadratic permutations in even dimension and generalize the previous results on quadratic permutation with optimal BCT (optimal means that the maximal value in the Boomerang Connectivity Table equals the lowest known differential uniformity). As a consequence of our general result, we prove that the boomerang uniformity of the binomial differentially $4$-uniform permutations presented by Bracken, Tan, and Tan equals $4$. This result gives rise to a new family of optimal Sboxes.

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arXiv2018

A Proof of the Beierle-Kranz-Leander Conjecture related to Lightweight Multiplication in $\mathds{F}_{2^n}$

Lightweight cryptography is a key tool for building strong security solutions for pervasive devices with limited resources. Due to the stringent cost constraints inherent in extremely large applications (ranging from RFIDs and smart cards to mobile devices), the efficient implementation of cryptographic hardware and software algorithms is of utmost importance to realize the vision of generalized computing. In CRYPTO 2016, Beierle, Kranz and Leander have considered lightweight multiplication in $\mathds{F}_{2^n}$. Specifically, they have considered the fundamental question of optimizing finite field multiplications with one fixed element and investigated which field representation, that is which choice of basis, allows for an optimal implementation. They have left open a conjecture related to two XOR-count. Using the theory of linear algebra, we prove in the present paper that their conjecture is correct. Consequently, this proved conjecture can be used as a reference for further developing and implementing cryptography algorithms in lightweight devices.

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arXiv2018

Improved upper bound on root number of linearized polynomials and its application to nonlinearity estimation of Boolean functions

To determine the dimension of null space of any given linearized polynomial is one of vital problems in finite field theory, with concern to design of modern symmetric cryptosystems. But, the known general theory for this task is much far from giving the exact dimension when applied to a specific linearized polynomial. The first contribution of this paper is to give a better general method to get more precise upper bound on the root number of any given linearized polynomial. We anticipate this result would be applied as a useful tool in many research branches of finite field and cryptography. Really we apply this result to get tighter estimations of the lower bounds on the second order nonlinearities of general cubic Boolean functions, which has been being an active research problem during the past decade, with many examples showing great improvements. Furthermore, this paper shows that by studying the distribution of radicals of derivatives of a given Boolean functions one can get a better lower bound of the second-order nonlinearity, through an example of the monomial Boolean function $g_μ=Tr(μx^{2^{2r}+2^r+1})$ over any finite field $\GF{n}$.

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arXiv2018

Cyclic bent functions and their applications in codes, codebooks, designs, MUBs and sequences

Let $m$ be an even positive integer. A Boolean bent function $f$ on $\GF{m-1} \times \GF {}$ is called a \emph{cyclic bent function} if for any $a\neq b\in \GF {m-1}$ and $ε\in \GF{}$, $f(ax_1,x_2)+f(bx_1,x_2+ε)$ is always bent, where $x_1\in \GF {m-1}, x_2 \in \GF {}$. Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on $\GF {m-1} \times \GF {}$ and their applications. The first objective of this paper is to construct a new class of cyclic bent functions, which includes all known constructions of cyclic bent functions as special cases. The second objective is to use cyclic bent functions to construct good mutually unbiased bases (MUBs), codebooks and sequence families. The third objective is to study cyclic semi-bent functions and their applications. The fourth objective is to present a family of binary codes containing the Kerdock code as a special case, and describe their support designs. The results of this paper show that cyclic bent functions and cyclic semi-bent functions have nice applications in several fields such as symmetric cryptography, quantum physics, compressed sensing and CDMA communication.

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arXiv2018

Two constructions of optimal pairs of linear codes for resisting side channel and fault injection attacks

Direct sum masking (DSM) has been proposed as a counter-measure against side-channel attacks (SCA) and fault injection attacks (FIA), which are nowadays important domains of cryptanalysis. DSM needs two linear codes whose sum is direct and equals a whole space $\Bbb F_q^n$. The minimum distance of the former code and the dual distance of the latter should be as large as possible, given their length and dimensions. But the implementation needs in practice to work with words obtained by appending, to each codeword $y$ of the latter code, the source word from which $y$ is the encoding. Let $\mathcal C_1$ be an $[n, k]$ linear code over the finite field $\Bbb F_q$ with generator matrix $G$ and let $\mathcal C_2$ be the linear code over the finite field $\Bbb F_q$ with generator matrix $[G, I_k]$. It is then highly desired to construct optimal pairs of linear codes satisfying that $d(\mathcal C_2^\perp)= d(\mathcal C_1^\perp)$. In this paper, we employ the primitive irreducible cyclic codes to derive two constructions of optimal pairs of linear codes for resisting SCA and FIA, where the security parameters are determined explicitly. To the best of our knowledge, it is the first time that primitive irreducible cyclic codes are used to construct (optimal) pairs of codes. As a byproduct, we obtain the weight enumerators of the codes $\mathcal C_1, \mathcal C_2, \mathcal C_1^\perp$, and $\mathcal C_2^\perp$ in our both constructions.

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arXiv2018

Strongly regular graphs from weakly regular plateaued functions

The paper provides the first constructions of strongly regular graphs and association schemes from weakly regular plateaued functions over finite fields of odd characteristic. We generalize the construction method of strongly regular graphs from weakly regular bent functions given by Chee et al. in [Journal of Algebraic Combinatorics, 34(2), 251-266, 2011] to weakly regular plateaued functions. In this framework, we construct strongly regular graphs with three types of parameters from weakly regular plateaued functions with some homogeneous conditions. We also construct a family of association schemes of class p from weakly regular p-ary plateaued functions.

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arXiv2018

Several classes of minimal linear codes with few weights from weakly regular plateaued functions

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. There are several methods to construct linear codes, one of which is based on functions over finite fields. Recently, many construction methods of linear codes based on functions have been proposed in the literature. In this paper, we generalize the recent construction methods given by Tang et al. in [IEEE Transactions on Information Theory, 62(3), 1166-1176, 2016] to weakly regular plateaued functions over finite fields of odd characteristic. We first construct three weight linear codes from weakly regular plateaued functions based on the second generic construction and determine their weight distributions. We next give a subcode with two or three weights of each constructed code as well as its parameter. We finally show that the constructed codes in this paper are minimal, which confirms that the secret sharing schemes based on their dual codes have the nice access structures.

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arXiv2018

On the Menezes-Teske-Weng's conjecture

In 2003, Alfred Menezes, Edlyn Teske and Annegret Weng presented a conjecture on properties of the solutions of a type of quadratic equation over the binary extension fields, which had been convinced by extensive experiments but the proof was unknown until now. We prove that this conjecture is correct. Furthermore, using this proved conjecture, we have completely determined the null space of a class of linear polynomials.

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arXiv2018

Further study on the maximum number of bent components of vectorial functions

In 2018, Pott, at al. have studied in [IEEE Transactions on Information Theory. Volume: 64, Issue: 1, 2018] the maximum number of bent components of vectorial function. They have presented serval nice results and suggested several open problems in this context. This paper is in the continuation of their study in which we solve two open problems raised by Pott et al. and partially solve an open problem raised by the same authors. Firstly, we prove that for a vectorial function, the property of having the maximum number of bent components is invariant under the so-called CCZ equivalence. Secondly, we prove the non-existence of APN plateaued having the maximum number of bent components. In particular, quadratic APN functions cannot have the maximum number of bent components. Finally, we present some sufficient conditions that the vectorial function defined from $\mathbb{F}_{2^{2k}}$ to $\mathbb{F}_{2^{2k}}$ by its univariate representation: $$ αx^{2^i}\left(x+x^{2^k}+\sum\limits_{j=1}^ργ^{(j)}x^{2^{t_j}} +\sum\limits_{j=1}^ργ^{(j)}x^{2^{t_j+k}}\right)$$ has the maximum number of {components bent functions, where $ρ\leq k$}. Further, we show that the differential spectrum of the function $ x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})$ (where $i,t_1,t_2$ satisfy some conditions) is different from the binomial function $F^i(x)= x^{2^i}(x+x^{2^k})$ presented in the article of Pott et al. Finally, we provide sufficient and necessary conditions so that the functions $$Tr_1^{2k}\left(αx^{2^i}\left(Tr^{2k}_{e}(x)+\sum\limits_{j=1}^ργ^{(j)}(Tr^{2k}_{e}(x))^{2^j} \right)\right) $$ are bent.

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arXiv2017

Vectorial Boolean functions and linear codes in the context of algebraic attacks

In this paper we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain $2^{n}$-ary cyclic codes (which we prove that they are LCD codes) extending results due to Rønjom and Helleseth. The knowledge of the minimum distance of those codes gives rise to a lower bound on the algebraic immunity of the associated vectorial function. Furthermore, we solve an open question raised by Mesnager and Cohen. We also present some properties of those cyclic codes (whose generator polynomials determined by vectorial functions) as well as their weight enumerator. In addition we generalize the so-called algebraic complement and study its properties.

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arXiv2017

Characterizations of o-polynomials by the Walsh transform

The notion of o-polynomial comes from finite projective geometry. In 2011 and later, it has been shown that those objects play an important role in symmetric cryptography and coding theory to design bent Boolean functions, bent vectorial Boolean functions, semi-bent functions and to construct good linear codes. In this note, we characterize o-polynomials by the Walsh transform of the associated vectorial functions.

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arXiv2017

New characterization and parametrization of LCD Codes

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their symplectic basis. Using such a characterization,we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results of the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of q-ary LCD codes have orthonormal basis, where q is a power of an odd prime.

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arXiv2017

Weightwise perfectly balanced functions with high weightwise nonlinearity profile

Boolean functions with good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the recent FLIP stream cipher. In this paper, we propose a large class of weightwise perfectly balanced (WPB) functions, which is not extended affinely (EA) equivalent to the known constructions. We also discuss the weightwise nonlinearity profile of these functions, and present general lower bounds on $k$-weightwise nonlinearity, where $k$ is a power of $2$. Moreover, we exhibit a subclass of the family. By a recursive lower bound, we show that these subclass of WPB functions have very high weightwise nonlinearity profile.

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arXiv2017

On σ-LCD codes

Linear complementary pairs (LCP) of codes play an important role in armoring implementations against side-channel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with $σ$ complementary dual ($σ$-LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. As Euclidean LCD codes, $σ$-LCD codes can also be used to construct LCP of codes. We show that, for $q > 2$, all q-ary linear codes are $σ$-LCD and that, for every binary linear code $\mathcal C$, the code $\{0\}\times \mathcal C$ is $σ$-LCD. Further, we study deeply $σ$-LCD generalized quasi-cyclic (GQC) codes. In particular, we provide characterizations of $σ$-LCD GQC codes, self-orthogonal GQC codes and self-dual GQC codes, respectively. Moreover, we provide constructions of asymptotically good $σ$-LCD GQC codes. Finally, we focus on $σ$-LCD Abelian codes and prove that all Abelian codes in a semi-simple group algebra are $σ$-LCD. The results derived in this paper extend those on the classical LCD codes and show that $σ$-LCD codes allow the construction of LCP of codes more easily and with more flexibility.

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