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Abstract: We define the class of triplicate functions as a generalization of 3-to-1 functions over \(\mathbb {F}_{2^{n}}\) for even values of n. We investigate the properties and behavior of triplicate functions, and of 3-to-1 among triplicate functions, with particular attention to the conditions under which such functions can be APN. We compute the exact number of distinct differential sets of power APN functions and quadratic 3-to-1 functions; we show that, in this sense, quadratic 3-to-1 functions are a generalization of quadratic power APN functions for even dimensions, in the same way that quadratic APN permutations are generalizations of quadratic power APN functions for odd dimensions. We show that quadratic 3-to-1 APN functions cannot be CCZ-equivalent to permutations in the case of doubly-even dimensions. We compute a lower bound on the Hamming distance between any two quadratic 3-to-1 APN functions, and give an upper bound on the number of such functions over \(\mathbb {F}_{2^{n}}\) for any even n. We survey all known infinite families of APN functions with respect to the presence of 3-to-1 functions among them, and conclude that for even n almost all of the known infinite families contain functions that are quadratic 3-to-1 or are EA-equivalent to quadratic 3-to-1 functions. We also give a simpler univariate representation in the case of singly-even dimensions of the family recently introduced by Göloglu than the ones currently available in the literature. We conduct a computational search for quadratic 3-to-1 functions in even dimensions n ≤ 12. We find six new APN instances for n = 10, and the first sporadic APN instance for n = 12 since 2006. We provide a list of all known 3-to-1 APN functions for n ≤ 12. PubDate: 2022-05-20

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Abstract: Multi-twisted (MT) additive codes over finite fields constitute a special class of additive codes and are generalizations of cyclic additive codes. In this paper, we study a class of 1-generator MT additive codes over finite fields. We further study the asymptotic behaviour of the rates and relative Hamming distances of a certain special subclass of 1-generator MT additive codes by using probabilistic methods. We show that there exists an asymptotically good infinite sequence of MT additive codes of length pγℓ with block length \(p^{\gamma } \to \infty\) over \(\mathbb {F}_{q^{t}}\) of rate \(0<\frac {f}{\ell t p^{d}}<1\) and relative Hamming distance at least δ, where ℓ ≥ 1 and t ≥ 2 are integers, q is a prime power, \(\mathbb {F}_{q^{t}}\) is the finite field of order qt, p is an odd prime satisfying \(\gcd (p,q)=\gcd (p,q-1)=1,\) f is the multiplicative order of q modulo p, d is the largest positive integer such that pd divides qf − 1, γ is a positive integer satisfying γ ≥ d + 1, and δ is a positive real number satisfying \(h_{q^{t}}(\delta ) < 1-\frac {1}{\ell t},\) (here \(h_{q^{t}}\) denotes the qt-ary entropy function). This shows that the family of MT additive codes over finite fields is asymptotically good. PubDate: 2022-05-19

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Abstract: Constant dimension codes (CDCs) have drawn extensive attention due to their applications in random network coding. A fundamental problem for CDCs is to explore the maximum possible cardinality Aq(n,d,k) of a set of k-dimensional subspaces in \(\mathbb {F}^{n}_{q}\) such that the subspace distance statisfies dis(U,V ) = 2k − 2 dim(U ∩ V ) ≥ d for all pairs of distinct subspaces U and V in this set. In this paper, by means of an appropriate combination of the matrix blocks from rank metric codes and small CDCs, we present three constructions of CDCs based on the generalized block inserting construction by Niu et al. in 2021. According to our constructions, we obtain 28 new lower bounds for CDCs which are better than the previously known lower bounds. PubDate: 2022-05-19

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Abstract: Two linear codes are said to be a linear ℓ-intersection pair if their intersection has dimension ℓ. Guenda et al. (Des Codes Cryptogr. 88, 133–152, 2020) constructed most of the linear ℓ-intersection pairs of MDS codes and we complement their results by constructing some linear ℓ-intersection pairs of MDS codes over \(\mathbb {F}_{q}\) of lengths n = q,q + 1. Furthermore, we construct all the possible linear ℓ-intersection pairs of MDS codes over \(\mathbb {F}_{2^{m}}\) of length n = 2m + 2 ≥ 6. As a consequence, linear ℓ-intersection pairs of MDS codes for all possible parameters are given. Moveover, we can apply our results to asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) and obtain all the possible pure MDS CSS-type AEAQECCs. PubDate: 2022-05-18

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Abstract: In this paper, we present quantum codes via algebraic geometry codes on norm-trace curves. We provide a lower bound of minimum Hamming distance for q-ary quantum code, where q = 2e (e ≥ 3). In order to get this, we determine Feng-Rao function values for the elements of Weierstrass semigroups on norm-trace curves. We present the order-bound on the minimum Hamming distance of one-point dual codes. Furthermore, we give a certain increasing sequence of one-point codes on norm-trace curves. We construct quantum codes from the sequence of one-point codes via the CSS construction. These give a better lower bound on the minimum Hamming distance of q-ary quantum code than some previous results. PubDate: 2022-05-14

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Abstract: At CRYPTO 2019, Chen et al. have shown a beyond the birthday bound secure n-bit to n-bit PRF based on public random permutations. Followed by the work, Dutta and Nandi have proposed a beyond the birthday bound secure nonce based MAC nEHtMp based on public random permutation. In particular, the authors have shown that nEHtMp achieves tight 2n/3-bit security (with respect to the state size of the permutation) in the single-user setting, and their proven bound gracefully degrades with the repetition of the nonces. However, we have pointed out that their security proof is not complete (albeit it does not invalidate their security claim). In this paper, we propose a minor variant of nEHtMp construction, called \(\textsf {nEHtM}^{*}_{p}\) and show that it achieves a tight 2n/3 bit security in the multi-user setting. Moreover, the security bound of our construction also degrades gracefully with the repetition of nonces. Finally, we have instantiated our construction with the PolyHash function to realize a concrete beyond the birthday bound secure public permutation-based MAC, \(\textsf {nEHtM}_{p}^{+}\) in the multi-user setting. PubDate: 2022-05-11

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Abstract: Abstract The inverse, the Gold, and the Bracken-Leander functions are crucial for building S-boxes of block ciphers with good cryptographic properties in symmetric cryptography. These functions have been intensively studied, and various properties related to standard attacks have been investigated. Thanks to novel advances in symmetric cryptography and, more precisely, those pertaining to boomerang cryptanalysis, this article continues to follow this momentum and further examine these functions. More specifically, we revisit and bring new results about their Difference Distribution Table (DDT), their Boomerang Connectivity Table (BCT), their Feistel Boomerang Connectivity Table (FBCT), and their Feistel Boomerang Difference Table (FBDT). For each table, we give explicit values of all entries by solving specific systems of equations over the finite field \(\mathbb {F}_{2^n}\) of cardinality \(2^n\) and compute the cardinalities of their corresponding sets of such values. The explicit values of the entries of these tables and their cardinalities are crucial tools to test the resistance of block ciphers based on variants of the inverse, the Gold, and the Bracken-Leander functions against cryptanalytic attacks such as differential and boomerang attacks. The computation of these entries and the cardinalities in each table aimed to facilitate the analysis of differential and boomerang cryptanalysis of S-boxes when studying distinguishers and trails. PubDate: 2022-05-03

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Abstract: Abstract We study LCD (linear complementary dual) and ACD (additive complementary dual) codes over a noncommutative non-unital ring E with four elements. This is the first attempt to construct LCD codes over a noncommutative non-unital ring. We show that free LCD codes over E are directly related to binary LCD codes. We introduce ACD codes over E. They include free LCD codes over E as a special case. These facts imply that LCD and ACD codes over E are worth studying. In particular, we characterize a free LCD E-code C in terms of a binary generator matrix G. We also define an ACD code over E, called a left-ACD code. We give several conditions for the existence of left-ACD codes. PubDate: 2022-05-01

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Abstract: Abstract In this paper we have studied repeated root cyclic codes of length pk over \(R=\mathbb {Z}_{p^{2}}+u\mathbb {Z}_{p^{2}}\) , u2 = 0, where p is a prime and k is a positive integer. We have determined a unique set of generators for these codes and obtained some results on their Lee distances. A minimal spanning set for them has been obtained and their ranks are determined. Further, we have determined the complete algebraic structure of principally generated cyclic codes in this class. An upper bound for the Lee distance of linear codes over R is presented. We have considered two Gray maps \(\psi :R \rightarrow \mathbb {Z}_{p}^{4}\) and \(\phi _{1}:R \rightarrow \mathbb {Z}_{p^{2}}^{2}\) , and using them, we have obtained some optimal binary linear codes as well as some quaternary linear codes from cyclic codes of length 4 over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\) . Three of the quaternary linear codes obtained are new, and the remaining of them have the best known parameters for their lengths and types. We have also obtained some optimal ternary codes of length 12 as Gray images of repeated root cyclic codes of length 3 over \(\mathbb {Z}_{9}+u\mathbb {Z}_{9}\) . PubDate: 2022-05-01

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Abstract: Abstract For positive integers k ≥ 2 and t, let m = 2kt and α be a primitive element of the finite field \(\mathbb {F}_{2^{m}}\) . In this paper, we study the parameters of a class of cyclic codes \(\mathcal {C}_{(1,v)}\) which has two zeros α and αv with \(v=\frac {2^{m}-1}{2^{t}+1}\) . It is shown that \(\mathcal {C}_{(1,v)}\) is optimal or almost optimal with respect to the sphere packing bound. Based on some results of Kloosterman sums and Gaussian periods, the weight distribution of the dual code of \(\mathcal {C}_{(1,v)}\) is completely determined when t = 5. PubDate: 2022-05-01

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Abstract: Abstract Let q be a prime power. In this paper, we investigate the maximum designed distances of LCD BCH codes over \(\mathbb {F}_{q^{2}}\) such that they contain their Hermitian dual codes, and also calculate their dimensions. As an application, we construct some quantum codes with good parameters from LCD BCH codes. PubDate: 2022-05-01

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Abstract: Abstract Combinatorial t-designs are closely related to coding theory and finite geometries and have nice applications in some engineering areas. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. In this paper, this approach is further investigated with a class of linear codes related to the extended BCH codes and their duals over finite fields. By determining the parameters of these codes, an infinite family of 3-designs is presented and their parameters are also explicitly determined. In addition, the parameters of some combinatorial 2-designs from the extended codes of certain primitive BCH codes are also determined. Moreover, Research Problem 9.4 proposed by Ding in [Designs from Linear Codes, World Scientific, Singapore, pp: 257, 2018)] is also solved. PubDate: 2022-05-01

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Abstract: Abstract A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are then obtained. Moreover, by the Möbius action over finite fields, for any known self-dual GRS codes, we give a systematic way to construct new self-dual GRS codes with flexible evaluation points. PubDate: 2022-05-01

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Abstract: Abstract Several constructions of Mutually Unbiased Bases (MUBs) borrow tools from combinatorial objects. In this paper we focus on how one can construct Approximate Real MUBs (ARMUBs) with improved parameters using results from the domain of Resolvable Block Designs (RBDs). We first explain the generic idea of our strategy in relating the RBDs with MUBs/ARMUBs, which are sparse (the basis vectors have small number of non-zero co-ordinates). Then specific parameters are presented, for which we can obtain new classes and improve the existing results. To be specific, we present an infinite family of \(\lceil \sqrt {d}\rceil \) many ARMUBs for dimension d = q(q + 1), where q ≡ 3 mod 4 and it is a prime power, such that for any two vectors v1,v2 belonging to different bases, \( \langle {v_{1} v_{2}}\rangle < \frac {2}{\sqrt {d}}\) . We also demonstrate certain cases, such as d = sq2, where q is a prime power and sq ≡ 0 mod 4. These findings subsume and improve our earlier results in [Cryptogr. Commun. 13, 321-329, January 2021]. This present construction idea provides several infinite families of such objects, not known in the literature, which can find efficient applications in quantum information processing for the sparsity, apart from suggesting that parallel classes of RBDs are intimately linked with MUBs/ARMUBs. PubDate: 2022-05-01

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Abstract: Abstract In a prior paper (Boukerrou et al. IACR Trans. Symmetric Cryptol. 2020(1), 331–362 2020), Boukerrou et al. introduced the Feistel Boomerang Connectivity Table (FBCT). FBCT is an important cryptanalytic technique on Feistel ciphers. In fact, the coefficients of FBCT are actually related to the second-order zero differential spectra of functions in even characteristic. In this paper, we push further the study initiated in Boukerrou et al. (IACR Trans. Symmetric Cryptol. 2020(1), 331–362 2020). Almost perfect nonlinear (APN) functions and the inverse function are interesting in cryptography and coding theory. In Boukerrou et al. (IACR Trans. Symmetric Cryptol. 2020(1), 331–362 2020), Boukerrou et al. determined the second-order zero differential spectra of APN functions and the inverse function in even characteristic. In order to derive further cryptographic properties of APN functions and the inverse function in odd characteristic, we calculate the second-order zero differential spectra of some APN functions and the inverse function in odd characteristic. In addition, these APN functions and the inverse function have low second-order zero differential uniformity. PubDate: 2022-05-01

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Abstract: Abstract In this paper we study the linear congruential generator on elliptic curves from the cryptographic point of view. We show that if sufficiently many of the most significant bits of the composer and of three consecutive values of the sequence are given, then one can recover the seed and the composer (even in the case where the elliptic curve is private). The results are based on lattice reduction techniques and improve some recent approaches of the same security problem. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for nonlinear congruential generators. Several examples are tested using implementations of ours algorithms. PubDate: 2022-05-01

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Abstract: Abstract The DryGASCON scheme was one of authenticated encryption (AE) algorithms submitted to the ongoing NIST Lightweight Cryptography Standardization Process. Such a competition aims to standardize lightweight cryptographic algorithms and determine lightweight AE schemes that are suitable for use in constrained environments suitable for several emerging areas in which highly-constrained devices are interconnected. This article deals with DryGASCON and aims to evaluate its security. The reasons for this choice are twofold. The first reason is related to its unique design. The second one is that no forgery cryptanalysis has been performed on it in the literature to the best of our knowledge. Specifically, this paper presents practical forgery attacks on DryGASCON by exploiting internal collisions of the underlying permutation. During our cryptanalysis, we investigate collisions and discuss how to find them. Next, we mounted forgery attacks on DryGASCON128 with an optimal probability 2− 13 for a class of weak keys and with certainty (optimal probability 1) for pairs of related keys. We show that the number of elements from the weak-key class depends on the size of the secret keys. In addition, we also find forgeries of DryGASCON256 in the related-key scenario. Forgery attacks succeeded without the reuse of nonce. Our results threaten the authenticity and robustness of DryGASCON. However, thanks to our analysis, we shed light on the causes of its weaknesses, and we manage to draw constructive conclusions and recommendations for future AE designs schemes, notably similar to DryGASCON. PubDate: 2022-05-01

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Abstract: Abstract This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to ⌊(d − 1)/2⌋ in polynomial-time, where d is the minimum distance of the code. PubDate: 2022-04-26

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Abstract: Abstract Self-dual codes over the ring \(\mathbb {Z}_{4}\) are related to combinatorial designs and unimodular lattices. First, we discuss briefly how to construct self-dual cyclic codes over \(\mathbb {Z}_{4}\) of arbitrary even length. Then we focus on solving one key problem of this subject: for any positive integers k and m such that m is even, we give a direct and effective method to construct all distinct Hermitian self-dual cyclic codes of length 2k over the Galois ring GR(4,m). This then allows us to provide explicit expressions to accurately represent all these Hermitian self-dual cyclic codes in terms of binomial coefficients. In particular, several numerical examples are presented to illustrate our applications. PubDate: 2022-04-21

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