Designs, Codes and Cryptography最新文献_第3页

Understanding the new distinguisher of alternant codes at degree 2 了解 2 级交替编码的新区分度

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-19 DOI: 10.1007/s10623-025-01626-8

Axel Lemoine, Rocco Mora, Jean-Pierre Tillich

{"title":"Understanding the new distinguisher of alternant codes at degree 2","authors":"Axel Lemoine, Rocco Mora, Jean-Pierre Tillich","doi":"10.1007/s10623-025-01626-8","DOIUrl":"https://doi.org/10.1007/s10623-025-01626-8","url":null,"abstract":"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 et al. (2011) is only able to distinguish Goppa codes or alternant codes of rate very close to 1, in Couvreur et al. (in: Guo and Steinfeld (eds) Advances in Cryptology—ASIACRYPT 2023—29th International Conference on the Theory and Application of Cryptology and Information Security, Guangzhou, China, December 4–8, 2023, Proceedings, Part IV, Volume 14441 of LNCS, pp. 3–38, Springer, 2023) a much more powerful (and more general) distinguisher was proposed. It is based on computing the Hilbert series ({{{,textrm{HF},}}(d),;d in mathbb {N}}) of a Pfaffian modeling. The distinguisher of Faugère et al. (2011) can be interpreted as computing ({{,textrm{HF},}}(1)). Computing ({{,textrm{HF},}}(2)) still gives a polynomial time distinguisher for alternant or Goppa codes and is apparently able to distinguish Goppa or alternant codes in a much broader regime of rates as the one of Faugère et al. (2011). However, the scope of this distinguisher was unclear. We give here a formula for ({{,textrm{HF},}}(2)) corresponding to generic alternant codes when the field size q satisfies (q geqslant r), where r is the degree of the alternant code. We also show that this expression for ({{,textrm{HF},}}(2)) provides a lower bound in general. The value of ({{,textrm{HF},}}(2)) corresponding to random linear codes is known and this yields a precise description of the new regime of rates that can be distinguished by this new method. This shows that the new distinguisher improves significantly upon the one given in Faugère et al. (2011).","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"17 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143849751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

New upper bounds for wide-sense frameproof codes 宽义防帧码的新上界

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-18 DOI: 10.1007/s10623-025-01631-x

Chengyu Sun, Xin Wang

引用次数: 0

Knot theory and error-correcting codes 结理论与纠错码

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-18 DOI: 10.1007/s10623-025-01615-x

Altan B. Kılıç, Anne Nijsten, Ruud Pellikaan, Alberto Ravagnani

引用次数: 0

Utilizing two subfields to accelerate individual logarithm computation in extended tower number field sieve 利用两个子域加速扩展塔数场筛中单个对数的计算

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-10 DOI: 10.1007/s10623-025-01590-3

Yuqing Zhu, Chang Lv, Jiqiang Liu

{"title":"Utilizing two subfields to accelerate individual logarithm computation in extended tower number field sieve","authors":"Yuqing Zhu, Chang Lv, Jiqiang Liu","doi":"10.1007/s10623-025-01590-3","DOIUrl":"https://doi.org/10.1007/s10623-025-01590-3","url":null,"abstract":"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 is calculated by establishing a relation with the factor base. Although computing the factor-base elements is the most time-consuming part of NFS, it can be performed only once and treated as pre-computation for a fixed finite field when multiple logarithms need to be computed. In this paper, we present a method for accelerating individual logarithm computation by utilizing two subfields. We focus on the case where the extension degree of the finite field is a multiple of 6 within the extended tower number field sieve framework. Our method allows for the construction of an element with a lower degree, while maintaining the same coefficient bound compared to Guillevic’s method, which uses only one subfield. Consequently, the element derived from our approach enjoys a smaller norm, which will improve the efficiency in individual logarithm computation.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"26 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143819557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The geometry of covering codes in the sum–rank metric 和秩度量中覆盖码的几何形状

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-09 DOI: 10.1007/s10623-025-01628-6

Matteo Bonini, Martino Borello, Eimear Byrne

引用次数: 0

Fast multiplication and the PLWE–RLWE equivalence for an infinite family of maximal real subfields of cyclotomic fields 分环场无穷一族极大实子域的快速乘法和PLWE-RLWE等价

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-07 DOI: 10.1007/s10623-025-01601-3

Joonas Ahola, Iván Blanco-Chacón, Wilmar Bolaños, Antti Haavikko, Camilla Hollanti, Rodrigo M. Sánchez-Ledesma

{"title":"Fast multiplication and the PLWE–RLWE equivalence for an infinite family of maximal real subfields of cyclotomic fields","authors":"Joonas Ahola, Iván Blanco-Chacón, Wilmar Bolaños, Antti Haavikko, Camilla Hollanti, Rodrigo M. Sánchez-Ledesma","doi":"10.1007/s10623-025-01601-3","DOIUrl":"https://doi.org/10.1007/s10623-025-01601-3","url":null,"abstract":"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 complexity in the dimension of the field, as it makes use of the fast Discrete Cosine Transform (DCT). Our approach assumes that the two input polynomials are given in a basis of Chebyshev-like polynomials, in contrast to the customary power basis. To validate this assumption, we prove that the change of basis from the power basis to the Chebyshev-like basis can be computed with ({mathcal {O}}(n log n)) arithmetic operations, where n is the problem dimension. Finally, we provide a heuristic and theoretical comparison of the vulnerability to some attacks for the pth cyclotomic field versus the maximal totally real subextension of the 4pth cyclotomic field for a reasonable set of parameters of cryptographic size.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"74 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Constructions of binary cyclic codes with minimum weights exceeding the square-root lower bound 最小权值超过平方根下界的二进制循环码的构造

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-07 DOI: 10.1007/s10623-025-01621-z

Hai Liu, Chunyu Gan, Chengju Li, Xueying Shi

{"title":"Constructions of binary cyclic codes with minimum weights exceeding the square-root lower bound","authors":"Hai Liu, Chunyu Gan, Chengju Li, Xueying Shi","doi":"10.1007/s10623-025-01621-z","DOIUrl":"https://doi.org/10.1007/s10623-025-01621-z","url":null,"abstract":"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 three forms: (p^2, p_1p_2, 2p_2) for odd primes (p, p_1, p_2). In this paper, we mainly construct several classes of binary cyclic codes with parameters ([2^lambda -1, k ge 2^{lambda -1}, d ge sqrt{n}]). Specifically, the binary cyclic codes ({mathcal {C}}_{(1, p^2)}), ({mathcal {C}}_{(1, 2p_2)}), ({mathcal {C}}_{(2, 2p_2)}), and ({mathcal {C}}_{(1, p_1p_2)}) have minimum distance (d ge sqrt{n}) though their dimensions satisfy (k > frac{n+1}{2}). Moreover, two classes of binary cyclic codes ({mathcal {C}}_{(2, p^2)}) and ({mathcal {C}}_{(2, p_1p_2)}) with dimension (k= frac{n+1}{2}) and minimum distance d much exceeding the square-root bound are presented, which extends the results given by Sun, Li, and Ding [30]. In fact, the rate of these two classes of binary cyclic codes are around (frac{1}{2}) and the lower bounds on their minimum distances are close to (frac{n}{log _2 n}). In addition, their extended codes are also investigated.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"21 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Quantum codes and irreducible products of characters 量子码与字符的不可约积

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-05 DOI: 10.1007/s10623-025-01599-8

Eric Kubischta, Ian Teixeira

引用次数: 0

Constructions of locally recoverable codes with large availability 具有大可用性的局部可恢复代码的构造

IF 1.6 2区数学

Designs, Codes and Cryptography Pub Date : 2025-04-05 DOI: 10.1007/s10623-025-01624-w

Giacomo Micheli, Vincenzo Pallozzi Lavorante, Abhi Shukul, Noah Smith

引用次数: 0

A new method for erasure decoding of convolutional codes 一种新的卷积码擦除译码方法