Search

In coding theory, the well-known class of block codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), form a class of cyclic error-correcting codes constructed using polynomials over a finite field. They are used for various critical practical applications in communication and storage due to their efficient encoding and decoding algorithms. In the past sixty years, significant progress has been made in understanding BCH codes’ dimensions and minimum distances. However, there has been limited research on the minimum distances of the dual codes of BCH codes, making it challenging to determine their actual minimum distances. Therefore, developing accurate lower bounds on the minimum distances of the dual codes of BCH codes is crucial and exciting. In this paper, we primarily use the multiplier technique proposed by Huffman and Pless to investigate the lower bounds on minimum distances of the dual codes $\mathcal {C}_{(q,q^{m}-1,\delta)}^{\perp } $ of the primitive narrow-sense BCH codes with designed distance $\delta $ . When $q = p^{e}$ with $e \ge 2$ , we improve the lower bounds on minimum distances of the dual codes $\mathcal {C}_{(q,q^{m}-1,\delta)}^{\perp } $ in the ranges $p^{ei}-p^{e-1}+2 \le \delta \le p^{ei+e-1}-p^{e-1}+1$ , where $m \ge 2$ and $1 \le i \le m-1$ . These new lower bounds are much tighter than the previously known bounds in the literature. This technique also applies to the study of binary dual codes $\mathcal {C}_{(2,2^{m}-1,\delta)}^{\perp } $ , for which we obtain tight lower bounds for $\delta = 2^{t}$ , where $m \ge 5$ is odd and $2 \le t \le m-3$ is even.

Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codes ${\mathcal{ C}}_{f,g}$ over $\mathbb {F}_{q^{m}}$ parameterized by any two functions $f, g$ over $\mathbb {F}_{q^{m}}$ , and then study the properties of six types of ${\mathcal{ C}}_{f,g}$ , its punctured code ${\mathcal{ C}}^{\ast}_{f,g}$ and their corresponding subfield codes over $\mathbb {F}_{q}$ . The classification of ${\mathcal{ C}}_{f,g}$ is based on special choices of $f,g$ as trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types of ${\mathcal{ C}}_{f,g}$ , we explicitly determine the weight distributions and dualities of ${\mathcal{ C}}_{f,g}, {\mathcal{ C}}_{f,g}^{\ast}$ and their subfield codes over $\mathbb {F}_{q}$ . The remaining four types of ${\mathcal{ C}}_{f,g}$ are restricted to $q=2$ , and the weight distributions and dualities of the subfields code ${\mathcal{ C}}_{f,g}^{(q)}$ and ${\cal C_{f,g}^{\ast}}^{(q)}$ are completely determined. Most of the resultant linear codes (over $\mathbb {F}_{q^{m}}$ or over $\mathbb {F}_{q}$ ) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at https://www.codetables.de. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of $[2^{4m-2},2m+1,2^{4m-3}]$ quaternary Hermitian self-orthogonal codes are obtained with $m \geq 2$ . As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.

A one-third century ago, as a means to speed up the elliptic curve method (ECM) for integer factoring, Montgomery suggested using a special elliptic curve form over prime fields and developed an addition chain to compute scalar multiplication on them, which nowadays are famous as Montgomery curves and Montgomery ladder. Kim et al. (http://eprint.iacr.org/2017/669. 2017) and Kim et al. (Adv Math Commun https://doi.org/10.3934/amc.2020090. 2020) found the Montgomery ladder very efficient on every short Weierstrass curve, leading to the most efficient regular scalar multiplication algorithms, which was further improved by Hamburg (https://ches.2017.rump.cr.yp.to/. 2020) and Hamburg (http://eprint.iacr.org/2020/437. 2020). However, the efficiency of the Montgomery ladder in general Montgomery curves remained not improved at all since firstly presented by Montgomery. This paper addresses the long-standing Elliptic Curve Cryptography (ECC) problem. The topic of this article is considered one of the topics that have attracted much attention from the cryptographic community following the launch of a multi-year project called “Post-Quantum Cryptography Standardization" by the National Institute of Standards and Technology (NIST) and also thanks partly to featuring one of the smallest keys of any algorithm known in the literature that is conjectured to be quantum resistant. To the best of our knowledge, this article provides, for the first time after Peter L. Montgomery’s, an improvement of arithmetic in general Montgomery curves, including point doubling and differential addition, which are the most fundamental operations in the context of ECC and supersingular isogeny-based primitives such as Supersingular Isogeny Diffie–Hellman (SIDH) or Supersingular Isogeny Key Encapsulation (SIKE), as well as ECM.

Permutation polynomials over finite fields are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. This family of polynomials constitutes an active research area in which advances are being made constantly. In particular, constructing infinite classes of permutation polynomials over finite fields with good differential properties (namely, low) remains an exciting problem despite much research in this direction for many years. This article exhibits low differentially uniform power permutations over finite fields of odd characteristics. Specifically, its objective is twofold concerning the power functions $F(x)=x^{\frac {p^{n}+3}{2}}$ defined over the finite field ${\mathbb {F}}_{p^{n}}$ of order $p^{n}$ , where $p$ is an odd prime, and $n$ is a positive integer. The first is to complement some former results initiated by Helleseth and Sandberg in 1997 by solving the open problem left open for more than twenty years concerning the determination of the differential spectrum of $F$ when $p^{n}\equiv 3\pmod 4$ and $p\neq 3$ . The second is to determine the exact value of its differential uniformity. Our achievements are obtained firstly by evaluating some character sums over ${\mathbb {F}}_{p^{n}}$ (which amounts to evaluating the number of ${\mathbb {F}}_{p^{n}}$ -rational points on some related curves and secondly by computing the number of solutions in $({\mathbb {F}}_{p^{n}})^{4}$ of a system of equations presented by Helleseth, Rong, and Sandberg, naturally appears while determining the differential spectrum of $F$ . We show that in the considered case ( $p^{n}\equiv 3\pmod 4$ and $p\neq 3$ ), $F$ is an APN power permutation when $p^{n}=11$ , and a differentially 4-uniform power permutation otherwise.

Complete complementary codes (CCCs) have important applications in communication, radar, and information security. In modern communication and radar systems, certain spectrum is reserved or prohibited from transmission, which leads to the so-called spectrally null constrained (SNC) problem. Compared with vast works on conventional CCCs, relatively little is known about SNC-CCCs. One objective of this paper is to derive several constructions of CCCs with more flexible settings from the graph of extended Boolean functions. This generalizes earlier achievements on CCCs from recent literature. Another objective of this paper is to employ the graphs of extended Boolean functions and polynomial representation of sequences to construct SNC-CCCs.

Studying the Schur square of a linear code is an important research topic in coding theory. Schur squares have important applications in cryptography and private information retrieval schemes, notably in secure multiparty computing or designing bilinear multiplication algorithms in finite extensions of finite fields through the notion of supercodes. Thanks to their exciting applications in cryptography, squares and powers of several linear codes have been investigated. In this paper, we will focus on the Schur square of a relevant well-known subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), which have wide applications in communication and storage systems and benefit from explicit defining sets that include consecutive integers, which gives the advantage of analyzing the parameters of BCH codes and their complements. Our main objective is to investigate the parameters of the squares of primitive narrow-sense BCH codes $\mathcal C(\delta)$ and their complements $\mathcal C(\delta)^{c}$ . We will present two sufficient and necessary conditions to guarantee that $\mathcal C^{2}(\delta) \ne \Bbb F_{q}^{n}$ and $\mathcal C^{2}(\delta)^{c} \ne \Bbb F_{q}^{n}$ by giving restrictions on designed distance $\delta $ , where $2 \le \delta \le n$ . Based on these two characterizations, the dimensions and minimum distances of $\mathcal C^{2}(\delta)$ and $\mathcal C^{2}(\delta)^{c}$ are investigated in some cases. The dimensions of these squares are determined explicitly, and lower bounds on the minimum distance are given.

It has been forty years since the TCP/IP protocol blueprint, which is the core of the modern worldwide Internet, was published. Over this long period, technology has made rapid progress. These advancements are slowly putting pressure and new demands on the underlying network architecture design. Therefore, there was a need for innovations that could handle the increasing demands of new technologies like IoT while ensuring secrecy and privacy. It is how named data networking (NDN) came into the picture. NDN enables robust data distribution with interest-based content retrieval and a leave-copy-everywhere caching policy. Even though NDN has surfaced as a future envisioned and decisive machinery for data distribution in IoT, it suffers from new data security challenges like content poisoning attacks. In this attack, an attacker attempts to introduce poisoned content with an invalid signature into the network. Given the circumstances, there is a need for a cost-effective signature scheme, requiring inexpensive computing resources and fast when implemented. An identity-based signature scheme (IBS) is the natural choice to address this problem. Herein, we present an IBS, namely Mul-IBS relying on multivariate public key cryptography (MPKC), which leads the race among the post-quantum cryptography contenders. A 5-pass identification scheme accompanying a safe and secure signature scheme based on MPKC works as key ingredients of our design. Our Mul-IBS attains optimal master public key size, master secret key size, and user’s secret key size in the context of multivariate identity-based signatures. The proposed scheme Mul-IBS is proven to be secure in the model, existential unforgeability under chosen-message and chosen identity attack (uf-cma), contingent upon the fact that Multivariate Quadratic (MQ) problem is NP-hard. The proposed design Mul-IBS can be utilized as a crucial cryptographic building block to build a robust and resilient IoT-based NDN architecture.

Permutation polynomials have been extensively studied for their applications in cryptography, coding theory, combinatorial design, etc. An important subfamily of permutations is the class of involutions (those permutations are equal to their compositional inverse). Elements of this class have been used frequently for block cipher designs and coding theory. In this article, we further investigate this corpus using new approaches, specifically from two-to-one (2-to-1) functions and (in some cases) using the graph indicators introduced by Carlet in 2020. In our constructions of involutions over the finite field $\mathbb {F}_{2^{n}}$ of order $2^{n}$ , we shall intensively use 2-to-1 mappings over $\mathbb {F}_{2^{n}}$ . More specifically, we present a new constructive method to design involutions from 2-to-1 mappings through their graph indicator and derive new involutions from known 2-to-1 mappings. Besides, we also propose several new classes of 2-to-1 mappings, including 2-to-1 hexanomials, 2-to-1 mappings of the form $(x^{2^{k}}+x+\delta)^{s_{1}}+(x^{2^{k}}+x+\delta)^{s_{2}}+cx$ , and 2-to-1 mappings from linear 2-to-1 mappings. We also exhibit the corresponding involutions of the constructed 2-to-1 mappings. Furthermore, an infinite family of involutions with differential uniformity at most 4 (EA-inequivalent to the inverse function) is obtained. Finally, we highlight that all our derived families of involutions have no fixed point, further accentuating their cryptographic interest.

Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase and end-shift positions and zero cross-correlation sums around the end-shift positions. CZCPs can be crucial in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency-selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio (CZCR). A construction framework of CZCPs with a large ZCZ ratio is proposed using Turyn’s method on some seed CZCPs and GCPs. By choosing suitably the seed CZCPs, we obtain 24 classes of new CZCPs with large CZCR. Especially, if the GCP is strengthened, our resultant CZCPs have the maximum $\mathrm {\mathbf{CZCR}}\approx \frac {M-1}{M}$ for $M\in \{6,12,24,28,48,56\}$ . We also obtain optimal CZCPs with new parameters (28, 13), (48, 23), (56, 27), (96, 47) and (112, 55), which can be extended to $(96N,47N)$ -CZCPs and $(112N,55N)$ -CZCPs respectively for any Golay number $N$ .

This article focuses on the so-called locally-APN power functions introduced by Blondeau, Canteaut and Charpin, which generalize the well-known notion of APN functions and possibly more suitable candidates against differential attacks. Specifically, given two coprime positive integers $m$ and $k$ such that $\gcd (2^{m}+1,2^{k}+1)=1$ , we investigate the locally-APN-ness property of the Niho type power function $F(x)=x^{s(2^{m}-1)+1}$ over the finite field $\mathbb {F}_{2^{2m}}$ for $s=(2^{k}+1)^{-1}$ , where $(2^{k}+1)^{-1}$ denotes the multiplicative inverse modulo $2^{m}+1$ . By employing finer studies of the number of solutions of certain equations over finite fields, we prove that $F(x)$ is locally-APN and determine its differential spectrum. We emphasize that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for $2\leq m\leq 10$ . In addition, we also determine the boomerang spectrum of $F(x)$ by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang and Li.

This article was initially motivated by an interesting work of Kyureghyan and Suder, in 2014, whereas an open problem it was stated to express the Hamming weights of the inverses of Kasami exponent $2^{2k}-2^{k}+1$ modulo $2^{n}-1$ in terms of $k$ and $n$ when it exists. In 2020, Kölsch solved the open problem by developing the modular add-with-carry approach first formally introduced in 2001. By these two nice works, the Hamming weights of inverses of all known APN and 4-differentially uniform exponents over ${\mathbb F}_{2^{n}}$ have been determined. However, their common difficulty is guessing the inverse related structures by performing some numerical experiments. As also mentioned by Fu in [Discrete Math. 345, 112658, 2022], where inversions of some exponents are revisited, it is unclear how their approaches can be generalized to others except that they studied. In this paper, we first present a new way to find the Hamming weights of inverses of Kasami exponents, which might be generalized to other exponents which are independent of the field size. Although this provides an alternative simpler solution to the aforementioned open problem and the found representations for inverses are fundamentally different from ones found by Kölsch, the representations still do not express the least positive residues of inverses in a closed form. Then, we present a method to find explicitly the least positive residue of the modular inverse without a prior guess and experiment when the exponent depends on the field size. With this method, we determine the inverses and their Hamming weights of all known APN, 4-differentially uniform and complete permutation polynomial (CPP) exponents over ${\mathbb F}_{2^{n}}$ , which solves a research problem raised in [Des. Codes, Cryptogr. 88, 2597-2621, 2020].

Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. Li et al. (2021) have recently identified some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li et al. by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field $\mathbb {F}_{2^{n}}$ of order $2^{n}$ . We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li et al. (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).

The arithmetic in a finite field constitutes the core of public key cryptography like RSA, ECC or pairing-based cryptography. This paper discusses an efficient hardware implementation of the Coarsely Integrated Operand Scanning (CIOS) method of Montgomery modular multiplication combined with an effective systolic architecture designed with a two-dimensional array of processing elements. The systolic architecture increases the speed of calculation by combining the concepts of pipelining and the parallel processing into a single concept. We propose the CIOS method for the Montgomery multiplication using a systolic architecture. As far as we know, this is the first implementation of such design. The proposed architectures are designed for field programmable gate array platforms. They targeted to reduce the number of clock cycles of the modular multiplication. The presented implementation results of the CIOS algorithms focus on different security levels useful in cryptography. This architecture has been designed in order to use the flexible DSP48 on Xilinx Field-Programmable Gate Array’s. Our architecture is scalable and depends only on the number and size of words. For instance, we provide results of implementation for 8-, 16-, 32- and 64-bit-long words in 33, 66, 132 and 264 clock cycles. We highlight the fact that for a given number of word, the number of clock cycles is constant. We propose a general version of our systolic architecture presented in SPACE2016.

This paper is devoted to the constructions of bent vectorial functions, that is, maximally non-linear multi-output Boolean functions. Such functions contribute to an optimal resistance to both linear and differential attacks of those cryptosystems in which they are involved as substitution boxes (S-boxes). We survey, study more in details and generalise the known primary and secondary constructions of bent functions, and we introduce new ones.