In this paper, we study twisted group codes over finite groups by introducing a novel symmetric and non-degenerate bilinear form to define the α -dual of a code. Because the Euclidean inner product is not generally G-invariant for twisted group algebras unless α = α ^-1 , our proposed twisted bilinear form provides a natural framework for studying these codes. Moreover, we prove that this duality preserves important properties such as the maximum distance separable (MDS) property. We further investigate α -linear complementary dual ( α -LCD) codes, establishing that they are generated by idempotents and providing necessary conditions for such codes to be MDS. Additionally, we study the behavior of twisted duals under cohomologous 2-cocycles, showing that the resulting structures are equivalent. Finally, we present a decoding algorithm for α -LCD codes based on this new framework.
We investigate cyclic 𝔽_q -linear codes over the alphabet 𝔽_q𝔽_q^2 , where q is a prime power. First, its generator polynomials and minimal spanning set are determined. Then, examples of cyclic 𝔽_q -linear 𝔽_q^2 -codes that satisfy the well-known Singleton bound are constructed. Using a Gray map, we produce certain optimal linear codes over 𝔽_3 . Finally, we obtain a few optimal ternary linear complementary dual (LCD) codes from certain 𝔽_3 -linear 𝔽_3𝔽_9 -codes.
This article focuses specifically on the study of self-dual double cyclic codes over a finite field 𝔽_q . A self-dual double cyclic code is a double cyclic code that is equal to its dual. Structurally, a double cyclic code of length (r, s) over 𝔽_q is a 𝔽_q[x] -submodule of 𝔽_q,r,s:=𝔽_q[x]/⟨ x^r-1⟩×𝔽_q[x]/⟨ x^s-1⟩ . Moreover, any double cyclic code of length (r, s) over 𝔽_q is generated by two pairs of polynomials in 𝔽_q,r,s . From the properties of the generating elements, we provide the necessary and sufficient conditions such that two pairs of polynomials in 𝔽_q,r,s generate a self-dual code. Furthermore, we examine the existence of self-dual double cyclic codes for some specific lengths: (r, r); (r, 2r) and (2r, r); and (r, s), where (r,s)=1 . For each case, we provide a construction method with some explicit examples over various finite fields. We also observe some connections between self-dual double cyclic codes and other classes of self-dual codes.
Time-lock puzzles are unique cryptographic primitives that use computational complexity to keep information secret for some period of time, after which security expires. This topic, while over 25 years old, is still in a state where foundations are not well understood: For example, current analysis techniques of time-lock primitives provide no sound mechanism to build composed multi-party cryptographic protocols which use expiring security as a building block. Further, there are analyses that employ idealizations and simulators of unrealistic computational power to be an acceptable sound security argument. Our goal with this short paper is to advocate for understanding what approaches may lead to sound modeling beyond idealization, and what approaches may, in fact, be hopeless at this task of sound modeling. We explain in this paper how existing attempts at this subtle problem lack either composability, a fully consistent analysis, or functionality. The subtle flaws in the existing frameworks reduce to an impossibility result by Mahmoody et al., who showed that time-lock puzzles with super-polynomial gaps (between committer and solver) cannot be constructed from random oracles alone (or any repetitive computation where the next state is completely random given the prior state); yet still the analyses of algebraic puzzles today treat the solving process as if each step is a generic or random oracle. We point out that if the generation process relies on a trapdoor function that cannot be treated as a random oracle (to allow efficient generation while avoiding this impossibility result), then, to be consistent, the analysis of the solving process should also not treat such a trapdoor function (and its intermediate states) as a random oracle. We further prove that there are no time-lock puzzles schemes with efficient generation that are based only on random oracle or generic group operations. We also delineate additional issues with the proof techniques used for time-lock puzzles. Specifically, when a time-lock puzzle must retain privacy for some amount of time, the reduction should bound the running time of the simulator. It is not a valid security argument to give the simulator more running time than the adversary would require to solve a puzzle. We survey the adherence of various attempts to this principle, as well as the properties that different attempts achieve toward composition.
Spoofing-robust automatic speaker verification (SASV) technology is used to safeguard voice-based authentication systems from fraudulent attempts. Such a system should be able to verify that the speech was spoken by the target speaker, as in standard automatic speaker verification, and should also be robust against spoofing attacks. This research employs an understandable and explicable embedding derived from the probability mass function (PMF) of waveform amplitudes in the time-domain. Using the logical access (LA) database from the ASVspoof2019 challenge for evaluation, we show that the performance of the countermeasure (CM) system is enhanced when it is gender-dependent. The CM system demonstrated an equal error rate (EER) of 8.6
The Restricted Syndrome Decoding Problem (RSDP) is a variant of the well-known syndrome decoding problem. It has recently been turned into a post-quantum signature scheme named CROSS by Baldi et al.. It is a scheme highlighted for being computationally friendly and providing a compact signature and public key size. This paper investigates an Oracle-based definition of the RSDP that has already proved useful in constructing other cryptographic primitives. We propose a new solving algorithm for this novel and interesting problem. Our approach is to first introduce a new weight definition for vectors over 𝔽_p and then develop a solving algorithm similar to the BKW algorithm that includes finding many such low-weight vectors in a dual space. We make use of several advanced techniques, such as Covering codes and Subspace Hypothesis Testing. We show that when there are many samples, our algorithm can be more advantageous than prior work based on information-set decoding adaptations for RSDP or algebraic approaches.
Quantum computing poses a significant threat to the security of many current cryptographic algorithms. This imminent risk has accelerated the development and standardization of new cryptosystems, known as post-quantum cryptosystems. These cryptosystems are designed to protect digital information in the quantum era using classical computing resources. CRYSTALS-Kyber is a lattice-based post-quantum encryption scheme that has been selected as a standard by the National Institute of Standards and Technology. This work presents a novel kleptographic attack against the CRYSTALS-Kyber scheme, providing both theoretical insights and practical results that demonstrate its feasibility. One conclusion is that this post-quantum cryptosystem may be vulnerable to kleptographic backdoors. As a result, this study offers a mathematical analysis, software implementations, and practical methodologies to detect such backdoors, thereby contributing to the development of more transparent and verifiable cryptographic standards.
Since their introduction, many results on bent partitions appeared in various directions and in several papers. The objective of this survey is to provide an introduction to this relatively young topic and a comprehensive overview of the main results. We present the known constructions of bent partitions, including those derived from generalized semifield spreads, their modifications, and secondary construction methods. We summarize the relations to special classes of vectorial bent functions and connections with combinatorial objects such as partial difference sets, strongly regular graphs, amorphic association schemes, and LP-packings. The article concludes with a discussion of several open problems.
Recent studies have shown that the preimage set partitions of weakly regular bent functions, particularly those that are vectorial dual–bent, can give rise to association schemes. The first construction of association schemes from non-weakly regular bent functions, namely from ternary generalized Maiorana–McFarland bent functions from 𝔽_3^n×𝔽_3^k×𝔽_3^k to 𝔽_3 for n = 1 and n = 2 , is presented in Özbudak and Pelen (J. Algebraic Combin. 56 (2022), 635–658). This construction was substantially generalized to arbitrary odd primes p and positive integers n in a recent work of Anbar et al. (Finite Fields Appl. 103 (2025), Paper No. 102568), using a variant of generalized Maiorana–McFarland bent functions. In this paper, we refine the construction of Anbar et al. to obtain association schemes on 𝔽_p^n×𝔽_p^k×𝔽_p^k with ( p^k - 1/p - 1(p + 1) + p ) , ( p^k - 1/p - 1(p + 1) + p - 1 ) and ( p^k - 1/p - 1(p + 1) + p - 1/2) association classes, depending on n and on the choice of bent functions employed in the construction. We further emphasize that the association schemes previously obtained in the works of Özbudak–Pelen and Anbar et al. arise as fusion schemes of those constructed in this paper.
Minimal linear codes play a central role in applications such as secret sharing and secure communication, and their construction often reveals deep connections between coding theory and other areas of mathematics. Using a representation theoretic approach we construct six new minimal binary linear codes. Our approach exploits the action of the adjoint Chevalley group F_4(2) on its natural 26-dimensional module, yielding codes of very large length, small dimension, and large minimum distance. The resulting codes are projective, divisible, and invariant under F_4(2) acting transitively on coordinates. We provide a geometric description of minimum weight codewords and show that their stabilizers are maximal or second maximal subgroups of F_4(2) . Furthermore, we demonstrate that these codes provide examples of both narrow and wide minimal codes. Our work links the structure of an exceptional finite group to the construction of highly symmetric minimal codes, thereby enriching the interplay between group theory and coding theory.
BCH codes are an important class of cyclic codes with wide applications in communication and storage systems. However, compared to primitive LCD BCH codes, LCD properties of BCH codes of length n=q^m+1/r , r ∤ (q+1) are seldom studied. In this paper, we investigate the parameters of LCD BCH codes with length 2^m+1/5, 2^m+1/9 and 3^m+1/5. A new method is proposed to calculate the coset leaders modulo n , and the dimensions of LCD BCH codes 𝒞_(q,n,δ ,b) with some given designed distances are determined. Furthermore, we compute the Bose distance of 𝒞_(q,n,δ ,0) . These results may be helpful to construct other families of LCD BCH codes. Some LCD BCH codes of length n=q^m+1/r with good minimum weights are constructed.
A multi-coupon is a batch of coupons (or tokens) a customer can redeem to a merchant in order to obtain some good or service. When all the tokens are intended to be redeemed by the same customer, their sharing is a fraud. Completely preventing token sharing is not possible when the system is endowed with privacy-preserving features such as anonymity and unlinkability. So, several options to make multi-coupon splitting difficult have been proposed in the research literature. One of them is the so-called strong unsplittability approach. All the existing unsplittable multi-coupon systems proposed so far have been designed using cryptography whose security assumes the existence of a hard to factor RSA modulus. Since the integer factorization problem will cease to be hard if a powerful enough quantum computer is eventually built, all these systems are constructed over quantum-vulnerable technology. In this paper, we prove strongly unsplittable multi-coupons can be designed using quantum-resistant cryptography. The proposal employs lattice-based cryptography whose security holds on the assumed hardness of variants of the short integer solution (SIS) and learning with errors (LWE) problems. To the best of authors’ knowledge, no such system has been proposed so far.
The main contribution of this paper is to introduce Subspace Differential Uniformity (SDU) for S-boxes and block ciphers. The SDU is essentially a measure of how well any function spreads input differences clustered in affine subspaces away from affine clusters in output differences. We provide some lower bounds for the SDU and describe an efficient algorithm for computing the SDU. Moreover, we provide results for some popular classes of S-boxes up to n=8 .
