Sihem Mesnager

Vikas Srivastava, Debasish Roy, Sihem Mesnager et al.

Voting is a primary democratic activity through which voters select representatives or approve policies. Conventional paper ballot elections have several drawbacks that might compromise the fairness, effectiveness, and accessibility of the voting process. Therefore, there is an increasing need to design safer, effective, and easily accessible alternatives. E-Voting is one such solution that uses digital tools to simplify voting. Existing state-of-the-art designs for secure E-Voting are based on number-theoretic hardness assumptions. These designs are no longer secure due to quantum algorithms such as Shor's algorithm. We present the design and analysis of \textit{first} post-quantum secure end-to-end verifiable E-Voting protocol based on multivariate polynomials to address this issue. The security of our proposed design depends on the hardness of the MQ problem, which is an NP-hard problem. We present a simple yet efficient design involving only standard cryptographic primitives as building blocks.

Biplab Chatterjee, Sihem Mesnager, Ratnesh Kumar Mishra et al.

In this paper, we extend a necessary and sufficient condition for a linear code over a Galois ring to be minimal and establish new bounds on the length of an $m$-dimensional minimal linear code. Building upon this structural characterization, we further generalize the function-based minimality criteria introduced by Wu \emph{et al.} (Cryptogr. Commun. 14, 875-895, 2022) from the finite field setting to the framework of Galois rings. The transition from fields to rings introduces substantial algebraic challenges due to the presence of zero divisors and the richer module structure of $\mathrm{GR}(p^{n},\ell)$. By exploiting Frobenius duality and the chain structure of Galois rings, we derive refined necessary and sufficient conditions ensuring that linear codes arising from functions over $\mathrm{GR}(p^{n},\ell)$ are minimal. As an application of these criteria, we construct several infinite families of minimal linear codes over Galois rings, thereby significantly generalizing the constructions of Wu \emph{et al.} to the ring setting. Our results provide a unified framework that connects minimality theory, module duality over Frobenius rings, and function-based code constructions.

Hengfeng Liu, Sihem Mesnager, Chunming Tang et al.

Locally recoverable codes (LRCs) have emerged as fundamental objects in modern coding theory, primarily due to their pivotal role in distributed and cloud storage systems. A major breakthrough in their construction was achieved by Tamo and Barg, who introduced the notion of \emph{good polynomials} as a key structural ingredient. In this article, we propose a natural generalization of this paradigm by introducing the concept of \emph{good rational functions}. Building upon this extension, we develop a unified and flexible framework for constructing optimal LRCs. To quantify the quality of a rational function, we embed the problem into the rich context of algebraic function field theory and Galois theory. This perspective allows us to extend the Galois-theoretic framework originally developed by Micheli for good polynomials. In particular, we derive structural and quantitative results on the number of totally split rational places associated with rational functions. Furthermore, we construct explicit families of good rational functions that outperform all good polynomials of the same degree. As a consequence, we obtain infinite families of optimal LRCs with improved parameters compared to those arising from the classical Tamo-Barg construction. These results highlight the intrinsic strength of our approach.

Kwang Ho Kim, Sihem Mesnager, Jong Hyok Choe et al.

Linearized polynomials over finite fields have been intensively studied over the last several decades. Interesting new applications of linearized polynomials to coding theory and finite geometry have been also highlighted in recent years. Let $p$ be any prime. Recently, preimages of the $p-$linearized polynomials $\sum_{i=0}^{\frac kl-1} X^{p^{li}}$ and $\sum_{i=0}^{\frac kl-1} (-1)^i X^{p^{li}}$ were explicitly computed over $\GF{p^n}$ for any $n$. This paper extends that study to $p-$linearized polynomials over $\GF{p}$, i.e., polynomials of the shape $$L(X)=\sum_{i=0}^t α_i X^{p^i}, α_i\in\GF{p}.$$ Given a $k$ such that $L(X)$ divides $X-X^{p^k}$, the preimages of $L(X)$ can be explicitly computed over $\GF{p^n}$ for any $n$.

Claude Carlet, Kwang Ho Kim, Sihem Mesnager

Using recent results on solving the equation $X^{2^k+1}+X+a=0$ over a finite field $\mathbb{F}_{2^n}$, we address an open question raised by the first author in WAIFI 2014 concerning the APN-ness of the Kasami functions $x\mapsto x^{2^{2k}-2^k+1}$ with $gcd(k,n)=1$, $x\in\mathbb{F}_{2^n}$.

Sihem Mesnager, Longjiang Qu

Two-to-one ($2$-to-$1$) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of $x^rh(x^{(q-1)/d})$, those from permutation polynomials, from linear translators and from APN functions. Then we present $2$-to-$1$ polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of $2$-to-$1$ mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results.

Sihem Mesnager, Chunming Tang, Maosheng Xiong

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.

Sihem Mesnager, Kwang Ho Kim, Dujin Jo et al.

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.

Sihem Mesnager, Fengrong Zhang, Chunming Tang et al.

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.

Jian Liu, Sihem Mesnager

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.

Tor Helleseth, Alexander Kholosha, Sihem Mesnager

In this paper, the relation between binomial Niho bent functions discovered by Dobbertin et al. and o-polynomials that give rise to the Subiaco and Adelaide classes of hyperovals is found. This allows to expand the class of bent functions that corresponds to Subiaco hyperovals, in the case when $m\equiv 2 (\bmod 4)$.