This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing through a series of results how the properties of the knot translate into code parameters. We show that knots can be used to obtain error-correcting codes with prescribed

Frameproof codes are used to fingerprint digital data. It can prevent copyrighted materials from unauthorized use. To determine the maximum size of the frameproof codes is a crucial problem in this research area. In this paper, we study the upper bounds for frameproof codes under Boneh-Shaw descendant (wide-sense descendant). First, we give new upper bounds for wide-sense 2-frameproof codes to improve

Distinguishing Goppa codes or alternant codes from generic linear codes (Faugère et al. in Proceedings of the IEEE Information Theory Workshop—ITW 2011, Paraty, Brasil, October 2011, pp. 282–286, 2011) has been shown to be a first step before being able to attack McEliece cryptosystem based on those codes (Bardet et al. in IEEE Trans Inf Theory 70(6):4492–4511, 2024). Whereas the distinguisher of Faugère

Twisted generalized Reed–Solomon (TGRS) codes as a generalization of generalized Reed–Solomon (GRS) codes have attracted a lot of attention from many researchers in recent years. In this paper, we investigate the conditions for the equality of two classes of TGRS codes with different parameters. Moreover, we construct the permutation automorphism groups of two classes of TGRS codes and show they are

The hardness of discrete logarithm problem (DLP) over finite fields forms the security foundation of many cryptographic schemes. When the characteristic is not small, the state-of-the-art algorithms for solving the DLP are the number field sieve (NFS) and its variants. NFS first computes the logarithms of the factor base, which consists of elements of small norms. Then, for a target element, its logarithm

We introduce the concept of a sum–rank saturating system and outline its correspondence to covering properties of a sum–rank metric code. We consider the problem of determining the shortest length of a sum–rank-\(\rho \)-saturating system of a fixed dimension, which is equivalent to the covering problem in the sum–rank metric. We obtain upper and lower bounds on this quantity. We also give constructions

We prove the equivalence between the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems for the maximal totally real subfield of the \(2^r 3^s\)th cyclotomic field for \(r \ge 3\) and \(s \ge 1\). Moreover, we describe a fast algorithm for computing the product of two elements in the ring of integers of these subfields. This multiplication algorithm has quasilinear

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Constructing binary cyclic codes with parameters \([n, \frac{n+1}{2}, d \ge \sqrt{n}]\) is an interesting topic in coding theory, as their minimum distances have a square-root bound. Let \(n=2^\lambda -1\), where \(\lambda \) has

In a recent paper, we defined a type of weighted unitary design called a twisted unitary 1-group and showed that such a design automatically induced error-detecting quantum codes. We also showed that twisted unitary 1-groups correspond to irreducible products of characters thereby reducing the problem of code-finding to a computation in the character theory of finite groups. Using a combination of

Let p be a prime number, m be a positive integer, and \(q=p^m\). For any fixed locality r such that \(p\not \mid r(r+1)\), we construct infinite families of locally recoverable codes with availabilty of nodes lower bounded by \(q/r!+O(\sqrt{q})\) and number of locality sets equal to \(q^2/(r+1)!+O(q^{3/2})\).

In this paper, we propose a new erasure decoding algorithm for convolutional codes using the generator matrix. This implies that our decoding method also applies to catastrophic convolutional codes in opposite to the classic approach using the parity-check matrix. We compare the performance of both decoding algorithms. Moreover, we enlarge the family of optimal convolutional codes (complete-MDP) based

The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, t-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined

Nonlinear feedback shift registers (NFSRs) are widely used in the design of stream ciphers and the cycle structure of an NFSR is a fundamental problem still open. In this paper, a new configuration of Galois NFSRs, called F-Ring NFSRs, is proposed. It is shown that an n-bit F-Ring NFSR generates n sequences with the same period simultaneously, that is, sequences from all bit registers have the same

Nonlinear feedback shift registers (NFSRs) are used in many recent stream ciphers as their main building blocks. Two NFSRs are said to be isomorphic if their state diagrams are isomorphic, and to be equivalent if their sets of output sequences are equal. So far, numerous work has been done on the equivalence of NFSRs with same bit number, but much less has been done on their isomorphism. Actually,

A De Bruijn cycle is a cyclic sequence in which every word of length n over an alphabet \(\mathcal {A}\) appears exactly once. De Bruijn tori are a two-dimensional analogue. Motivated by recent progress on universal partial cycles and words, which shorten De Bruijn cycles using a wildcard character, we introduce universal partial tori and matrices. We find them computationally and construct infinitely

In this paper, for an odd prime power q and an integer \(m\ge 2\), let \(\mathcal {C}(q,m)\) be a one-weight irreducible cyclic code with parameters \([q^m-1,m,(q-1)q^{m-1}]\), we consider the complete weight enumerator and the weight distribution of the square \(\big (\mathcal {C}(q,m)\big )^2\), whose dual has \(\lfloor \frac{m}{2}\rfloor +1\) zeros. Using the character sums method and the known

The learning parity with noise (LPN) problem underlines several classic cryptographic primitives. Researchers have attempted to show the algorithmic difficulty of this problem by finding a reduction from the decoding problem of linear codes, for which several hardness results exist. Earlier studies used code smoothing as a technical tool to achieve such reductions for codes with vanishing rate. This

In this paper, we introduce and study the problem of binary stretch embedding of edge-weighted graphs in both integer and fractional settings. Roughly speaking, the binary stretch embedding problem for a weighted graph G is to find a mapping from the vertex set of G, to the vertices of a hypercube graph such that the distance between every pair of the vertices is not reduced under the mapping, hence

A \(2-(v, k, \lambda )\) design is additive if, up to isomorphism, the point set is a subset of an abelian group G and every block is zero-sum. This definition was introduced in Caggegi et al. (J Algebr Comb 45:271-294, 2017) and was the starting point of an interesting new theory. Although many additive designs have been constructed and known designs have been shown to be additive, these structures

Lattices have many significant applications in cryptography. In 2021, the p-adic signature scheme and public-key encryption cryptosystem were introduced. They are based on the Longest Vector Problem (LVP) and the Closest Vector Problem (CVP) in p-adic lattices. These problems are considered to be challenging and there are no known deterministic polynomial time algorithms to solve them. In this paper

Homomorphic encryption (HE) is one of the mainstream cryptographic tools used to enable secure outsourced computation. A typical task is secure matrix computation, which is a fundamental operation used in various outsourced computing applications such as statistical analysis and machine learning. In this paper, we present a new framework for secure multiplication of two matrices with size \(r \times

A function \(f: {\mathbb {F}}_q \rightarrow {\mathbb {F}}_q\), is called an almost perfect nonlinear (APN) if \(f(X+a)-f(X) =b\) has at most 2 solutions for every \(b,a \in {\mathbb {F}}_q\), with a nonzero. Furthermore, it is called an exceptional APN if it is an APN on infinitely many extensions of \({\mathbb {F}}_q\). These problems are equivalent to finding rational points on the corresponding

Impossible differential cryptanalysis is a crucial cryptanalytical method for symmetric ciphers. Given an impossible differential, the key recovery attack typically proceeds in two steps: generating pairs of data and then identifying wrong keys using the guess-and-filtering method. At CRYPTO 2023, Boura et al. first proposed a new key recovery technique—the differential meet-in-the-middle attack, which

A problem of current interest, also motivated by applications to Coding theory, is to find explicit equations for maximal curves, that are projective, geometrically irreducible, non-singular curves defined over a finite field \(\mathbb {F}_{q^2}\) whose number of \(\mathbb {F}_{q^2}\)-rational points attains the Hasse-Weil upper bound \(q^2+2\mathfrak {g}q+1\) where \(\mathfrak {g}\) is the genus of

Multivariate public key cryptography (MPKC) is one of the most promising alternatives to build quantum-resistant signature schemes, as evidenced in NIST’s call for additional post-quantum signature schemes. The main assumption in MPKC is the hardness of the Multivariate Quadratic (MQ) problem, which seeks for a common root to a system of quadratic polynomials over a finite field. Although the Crossbred

The Internet of Things (IoT) has become a necessary part of modern technology, enabling devices to connect and interact with each other. Unless applicable cryptographic components have adequate security protection, the IoT could easily leak private data. Impossible differential cryptanalysis (IDC) is one of the best-known techniques for cryptanalysis of block ciphers. Several papers are aimed at formalizing

We present a new result on the nonexistence of generalized bent functions (GBFs) from \((\mathbb {Z}/t\mathbb {Z})^n\) to \(\mathbb {Z}/t\mathbb {Z}\) (called type [n, t]) for a large class. Assume p is an odd prime number. By showing certain quadratic norm form equations having no integral points, we obtain a universal result on the nonexistence of GBFs with type \([n, 2p^e]\) when p and n satisfy

This paper provides new bounds on the size of spheres in any coordinate-additive metric with a particular focus on improving existing bounds in the sum-rank metric. We derive improved upper and lower bounds based on the entropy of a distribution related to the Boltzmann distribution, which work for any coordinate-additive metric. Additionally, we derive new closed-form upper and lower bounds specifically

Let \(\ell ^m\) be a power with \(\ell \) a prime greater than 3 and \(m\) a positive integer such that 3 is a primitive root modulo \(2\ell ^m\). Let \(\mathbb {F}_3\) be the finite field of order 3, and let \(\mathbb {F}\) be the \(\ell ^{m-1}(\ell -1)\)-th extension field of \(\mathbb {F}_3\). Denote by \(\text {Tr}\) the absolute trace map from \(\mathbb {F}\) to \(\mathbb {F}_3\). For any \(\alpha

We study the hardness of the Syndrome Decoding problem, the base of most code-based cryptographic schemes, such as Classic McEliece, in the presence of side-channel information. We use ChipWhisperer equipment to perform a template attack on Classic McEliece running on an ARM Cortex-M4, and accurately classify the Hamming weights of consecutive 32-bit blocks of the secret error vector \(\textbf{e}\in

We compute the weight distribution of the binary Reed–Muller code \({\mathcal {R}} (4,9)\) by combining the methodology described in D. V. Sarwate’s Ph.D. thesis from 1973 with newer results on the affine equivalence classification of Boolean functions. More specifically, to address this problem posed, e.g., in the book of MacWilliams and Sloane, we apply an enhanced approach based on the classification

Given two linear codes, the code equivalence problem asks to find an isometry mapping one code into the other. The problem can be described in terms of group actions and, as such, finds a natural application in signatures derived from a Zero-Knowledge Proof system. A recent paper, presented at Asiacrypt 2023, showed how a proof of equivalence can be significantly compressed by describing how the isometry

This paper focuses on the study of triple-twisted generalized Reed–Solomon (TTGRS) codes over a finite field \({\mathbb {F}}_q\), having twists \(\varvec{t} = (1, 2, 3)\) and hooks \(\varvec{h} = (0, 1, 2)\). We have obtained the necessary and sufficient conditions for such TTGRS codes to be MDS, AMDS, and AAMDS via algebraic techniques. We have also enumerated these codes for some particular values

An orientable sequence of order n over an alphabet\(\{0,1,\ldots , k{-}1\}\) is a cyclic sequence such that each length-n substring appears at most once in either direction. When \(k= 2\), efficient algorithms are known to construct binary orientable sequences, with asymptotically optimal length, by applying the classic cycle-joining technique. The key to the construction is the definition of a parent

SCARF, an ultra low-latency tweakable block cipher, is the first cipher designed for cache randomization. The block cipher design is significantly different from other common tweakable block ciphers; with a block size of only 10 bits, and yet the input key size is a whopping 240 bits. Notably, the majority of the round key in its round function is absorbed into the data path through AND operations

Let p be any prime number such that \(p\equiv 1 \pmod 4\), and let \({\mathbb {F}}_p\) be the finite field of p elements. In this paper, we first construct a new AMDS symbol-pair cyclic code of length 4p and of symbol-pair distance 9 by examining its generator polynomial. We then use the generator polynomial to obtain a family of \((p-1)/2\) AMDS symbol-pair constacyclic codes of the same length and

In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes (GAG codes) using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter r in such codes by using only non-rational places of degree at most r. This is, up to the author’s knowledge, a new way to construct locally recoverable codes (LRCs). We give an example

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against pair errors in symbol-pair read channels. This special channel structure is motivated by the limitations of the reading process in high density data storage systems, where it is no longer possible to read individual symbols. In this work, we study bounds and constructions of codes in symbol-pair metric. By using

Two-dimensional multilength optical orthogonal codes (2D MLOOCs) were proposed as a means of simultaneously reducing the chip rate and accommodating multimedia services with multiple bit rates and quality of service (QoS) requirements in OCDMA networks. This paper considers two-dimensional multilength optical orthogonal codes with inter-cross-correlation of \(\lambda =2\). New upper bounds on the size

Non-overlapping codes over a given alphabet are defined as a set of words satisfying the property that no prefix of any length of any word is a suffix of any word in the set, including itself. When the word lengths are variable, it is additionally required that no word is contained as a subword within any other word. In this paper, we present a new construction of variable-length non-overlapping codes

Anemoi is a family of compression and hash functions over finite fields \(\mathbb {F}_q\) for efficient Zero-Knowledge applications. Its round function is based on a novel permutation \(\mathcal {H}: \mathbb {F}_q^2 \rightarrow \mathbb {F}_q^2\), called the open Flystel, which is parametrized by a permutation \(E: \mathbb {F}_q \rightarrow \mathbb {F}_q\) and two functions \(Q_\gamma , Q_\delta : \mathbb

This work is devoted to solving some closely related open problems on the average and asymptotic behavior of the 2-adic complexity of binary sequences. First, for fixed N, we prove that the expected value \(E^{\text {2-adic}}_N\) of the 2-adic complexity over all binary sequences of length N is close to \(\frac{N}{2}\) and the deviation from \(\frac{N}{2}\) is at most of order of magnitude \(\log (N)\)

Complete permutations in addition over finite fields have attracted many scholars’ attention due to their wide applications in combinatorics, cryptography, sequences, and so on. In 2020, Tu et al. introduced the concept of the complete permutation in the sense of multiplication (CPM for short). In this paper, we further study the constructions and applications of CPMs. We mainly construct many classes

In a previous paper, the authors determined the parameters of all strongly regular graphs that can be decomposed into a divisible design graph and a Hoffman coclique. As a counterpart of this result, in the present paper we determine the parameters of all strongly regular graphs that can be decomposed into a divisible design graph and a Delsarte clique. In particular, an infinite family of strongly

Soft Analytical Side Channel Attacks (SASCA) are a powerful family of Side Channel Attacks (SCA) that allows the recovery of secret values with only a small number of traces. Their effectiveness lies in the Belief Propagation (BP) algorithm, which enables efficient computation of the marginal distributions of intermediate values. Post-quantum schemes such as Kyber, and more recently, Hamming Quasi-Cyclic

A code is said to be equidistant if the distance between any two distinct codewords of the code is the same. In this paper, we have studied equidistant single-orbit cyclic and quasi-cyclic subspace codes. The orbit code generated by a subspace U in \({\mathbb {F}}_{q^n}\) such that the dimension of U over \({\mathbb {F}}_q\) is t or \(n-t\), \(\text{ where }~t=\dim _{{\mathbb {F}}_q}(\text{ Stab }(U)\cup

Flag codes have received a lot of attention due to its application in random network coding. In 2021, Alonso-González et al. constructed optimal \((n,{\mathcal {A}})_{q}\)-Optimum distance flag codes (ODFC) for \({\mathcal {A}}\subseteq \{1,2,\ldots ,k,n-k,\ldots ,n-1\}\) with \(k\in {\mathcal {A}}\) and \(k\mid n\). In this paper, we introduce a new construction of \((n,{\mathcal {A}})_q\)-ODFCs by

This paper focuses on the cryptanalysis of the ASCON family using automatic tools. We analyze two different problems with the goal to obtain new modelings, both simpler and less computationally heavy than previous works (all our models require only a small amount of code and run on regular desktop computers). The first problem is the search for Meet-in-the-middle attacks on reduced-round ASCON–XOF

Recent improvements to garbled circuits are mainly focused on reducing their size. The state-of-the-art construction of Rosulek and Roy (Crypto 2021) requires \(1.5\kappa \) bits for garbling AND gates in the free-XOR setting. This is below the previously proven lower bound \(2\kappa \) in the linear garbling model of Zahur, Rosulek, and Evans (Eurocrypt 2015). Whether their construction is optimal

A \((v,k,\lambda )\)-BIBD \((X,\mathcal {B})\) has a nesting if there is a mapping \(\phi :\mathcal {B}\rightarrow X\) such that \((X,\{B\cup \{\phi (B)\}\mid B\in \mathcal {B}\})\) is a \((v,k+1,\lambda +1)\)-packing. If the \((v,k+1,\lambda +1)\)-packing is a \((v,k+1,\lambda +1)\)-BIBD, then this nesting is said to be perfect. We show that given any positive integers k and \(\lambda \), if \(k\ge

Classical strong external difference families (SEDFs) are much-studied combinatorial structures motivated by information security applications; it is conjectured that only one classical abelian SEDF exists with more than two sets. Recently, non-disjoint SEDFs were introduced; it was shown that families of these exist with arbitrarily many sets. We present constructions for both classical and non-disjoint

In this paper, a security evaluation framework for ARX ciphers, using modular addition as non-linear component, against rotational-XOR differential cryptanalysis is proposed. We first model all the possible propagations for rotational-XOR difference and rotational-XOR differential probability by some conjunctive normal form clauses. Then, acceleration techniques of automatic search are presented to

It is well-known that functions over finite fields play a crucial role in designing substitution boxes (S-boxes) in modern block ciphers. In order to analyze the security of an S-box, recently, three new tables have been introduced: the Extended Boomerang Connectivity Table (EBCT), the Lower Boomerang Connectivity Table (LBCT), and the Upper Boomerang Connectivity Table (UBCT). In fact, these tables

Side Channel Attacks (SCA) exploit physical information leakage from devices performing cryptographic operations, posing significant security threats. While SCA has been extensively studied in the context of block ciphers, similar analyses on stream ciphers and constructions like authenticated encryption are less explored. In this paper, we present a novel enhancement to existing SCA techniques based

A combinatorial trade is a pair of sets of blocks of elements that can be exchanged while preserving relevant subset intersection constraints. The class of balanced and swap-robust minimal trades was proposed in Pan et al. (in: 2022 IEEE International Symposium on Information Theory (ISIT), IEEE, pp 2385–2390, 2022) for exchanging blocks of data chunks stored on distributed storage systems in an access-

Experimental results show that, when the order n is odd, there are de Bruijn sequences such that the corresponding complement sequence and the reverse sequence are the same. In this paper, we propose one efficient method to generate such de Bruijn sequences. This solves an open problem asked by Fredricksen forty years ago for showing the existence of such de Bruijn sequences when the odd order \(n

Due to the operational efficiency and lower computational costs of the Chebyshev polynomial compared to ECC, this chaotic system has attracted widespread attention in public key cryptography. However, the single recurrence coefficient limitation and inherent short-period flaw, often render the Chebyshev polynomials cryptosystem ineffective against various attacks, such as Exhaustive Attacks and Ciphertext-Only

Linear complementary pairs (LCPs) of codes have been studied since they were introduced in the context of discussing mitigation measures against possible hardware attacks to integrated circuits. In this situation, the security parameters for LCPs of codes are defined as the (Hamming) distance and the dual distance of the codes in the pair. We study the properties of LCPs of skew constacyclic codes

We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of \(F(x)=x^{2^e}h(\textrm{Tr}_{2^{2m}/2^m}(x))\), where \(e\ge 0\) and h(x) is a permutation over \({\mathbb {F}}_{2^m}\). If h(x) is monomial, the nonlinearity of F(x) is shown to be at most \( 2^{2\,m-1}-2^{\lfloor \frac{3\,m}{2}\rfloor }\) and some non-plateaued

Let \(\mathbb {F}_q\) be the finite field with q elements, where q is a power of a prime p. Given a monic polynomial \(f \in \mathbb {F}_q[x]\) that is not divisible by x, there exists a positive integer \(e=e(f)\) such that f(x) divides the binomial \(x^e-1\) and e is minimal with this property. The integer e is commonly known as the order of f and we write \(\textrm{ord}(f)=e\). Motivated by a recent