Cartesian authentication codes from functions with optimal nonlinearity

In this paper, we present several classes of authentication codes using functions with perfect nonlinearity and optimum nonlinearity. Some of the authentication codes are optimal. On the other hand, these authentication codes are easy to implement due to their simple algebraic structures.     

低编码复杂度不规则准循环LDPC码的构造方法

针对不规则低密度奇偶校验码（LDPC码）误码性能好,但编码复杂度高的问题,利用重复积累码（RA码）能有效编码的特性和掩模技术,提出了一种不规则LDPC码的构造方法,该码具有线性复杂度的编码算法。该构造方法,首先对RA码的校验矩阵进行了改进,消除了RA码常产生的错误平层效应;然后基于范德蒙矩阵构造了一种新的校验矩阵,该校验矩阵具有代数结构,易于硬件实现。理论分析和实验结果表明,构造的不规则LDPC码的编码复杂度低于Mackay随机码,在加性高斯白噪声（AWGN）信道条件下,误码率为1.0×10^-4时,比Mackay随机码性能提高约0.4~0.6 dB。     

Asymmetric Code-Theoretical Schemes Constructed with the Use of Algebraic Geometric Codes

Asymmetric coding schemes are developed that are constructed with the use of algebraic geometric codes. Basic analytical expressions are obtained that connect the parameters of algebraic geometric codes with the parameters of asymmetric coding schemes based on these codes. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 47–57, May–June 2005.     

Non-split Toric Codes

Problems of Information Transmission - We introduce a new wide class of error-correcting codes, called non-split toric codes. These codes are a natural generalization of toric codes where non-split...     

Toric dynamical systems

Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded.     

环Fp uFp vFp uvFp上的二次剩余码

针对环Fp+ uFp+ vFp+ uvFp上的二次剩余码进行了研究,其中u2=u,v2=v,uv=vu,p是一个奇素数.首先引入了环Fp+ uFp+vFp+ uvFp上长为n的循环码的相关知识,用幂等元的形式定义了环Fp+ uFp+vFp+uvFp上的二次剩余码,给出了其定义和性质,并讨论了它们与其扩展码之间的关系和对偶性质.最后,给出了环F3+uF3+vF3+uvF3上长为11的二次剩余码的幂等生成元的具体形式.     

大围长结构化LDPC码构造方法

在LDPC码的构造中,校验矩阵拥有大围长对于改善码的性能有着重要的意义.结构化是提高码实用性的关键.提出一种低复杂度的基于列差搜索法(Column-Difference Search AIgorithm)的准循环LDPC码构造方法,用以设计大围长和任意码率的规则QC-LDPC码(以下称为CDS-LDPC码).该方法可线性编码,易于IC实现.仿真结果表明,该方法构造的各种码率CDS-LDPC码在BER性能上均优于对应的随机码,与同属QC-LDPC码的Tanner码和Array码相比明显提高了0.79～3.28dB,并且在码长、码率等参数设计上更为灵活.     

Multipartite quantum systems and symplectic toric manifolds

In this paper we study the geometrical structures of multi-qubit states based on symplectic toric manifolds. After a short review of symplectic toric manifolds, we discuss the space of a single quantum state in terms of these manifolds. We also investigate entangled multipartite states based on moment map and Delzant’s construction of toric manifolds and algebraic toric varieties.     

Design of two step deterministic interleaver for turbo codes

The performance and design of a deterministic interleaver for short frame turbo codes is considered in this paper. The main characteristic of this class of deterministic interleaver is that their algebraic design selects the best permutation generator such that the points in smaller subsets of the interleaved output are uniformly spread over the entire range of the information data frame. It is observed that the interleaver designed in this manner improves the minimum distance of first few spectral lines of minimum distance spectrum. Finally we introduce a circular shift in the permutation function to reduce the correlation between the parity bits corresponding to the original and interleaved data frames to improve the decoding capability of MAP decoder. The design is focused on combining good permutations with de-correlation property. Our solution to design a deterministic interleaver outperforms the semi-random interleavers and the deterministic interleavers reported in the literature.     

防合谋指纹印码的新构造

针对防盗版应用中防合谋码个数种类少的问题,提出利用基于有理正规曲线的一个组合设计——强部分平衡t-设计构造防合谋码。利用该方法构造新的防合谋指纹印码,新码多于已知个数,而且码的效率比以前所有的防合谋码更优。     

Initial ideals of unimodular integer programs

This article concerns the initial ideal computation for toric ideals, which arises in the computational algebraic approach for integer programs. It is known that the initial ideals of toric ideals contain many information about original toric ideals that are beneficial for solving the integer programs. Therefore, their analyses may bring further comprehensions for structure of integer programs and can help us to construct a new type of algorithm. There are two known algorithms for initial ideal computation. One is based on the theory of Gröbner bases and the other is proposed by Thomas and Hos¸ten. In this article, we propose another type of algorithm to perform the initial ideal computation for the toric ideals defined by unimodular matrices. The unimodular matrices is one of the important matrix classes, that are often looming as the coefficient matrices of some combinatorial optimization problems with the forms of integer programs. Ours are based on a combinatorial relation between the vector matroids of defining matrices and the minimal generators of initial ideals.     

New construction of highly nonlinear resilient S-boxes via linear codes

Highly nonlinear resilient functions play a crucial role in nonlinear combiners which are usual hardware oriented stream ciphers. During the past three decades, the main idea of construction of highly nonlinear resilient functions are benefited from concatenating a large number of affine subfunctions. However, these resilient functions as core component of ciphers usually suffered from the guess and determine attack or algebraic attack since the n-variable nonlinear Boolean functions can be easily given rise to partial linear relations by fixing at most n/2 variables of them. How to design highly nonlinear resilient functions (S-boxes) without concatenating a large number of n/2 variables affine subfunctions appears to be an important task. In this article, a new construction of highly nonlinear resilient functions is proposed. These functions consist of two classes subfunctions. More specially, the first class (nonlinear part) contains both the bent functions with 2 k variables and some affine subfunctions with n/2 − k variables which are attained by using [ n/2 − k, m, d] disjoint linear codes. The second class (linear part) includes some linear subfunctions with n/2 variables which are attained by using [ n/2, m, d] disjoint linear codes. It is illustrated that these resilient functions have high nonlinearity and high algebraic degree. In particular, It is different from previous well-known resilient S-boxes, these new S-boxes cannot be directly decomposed into some affine subfunctions with n/2 variables by fixing at most n/2 variables. It means that the S-boxes (vectorial Boolean functions) which use these resilient functions as component functions have more favourable cryptography properties against the guess and determine attack or algebraic attacks.     

Problems and solutions from the fourth International Students’ Olympiad in Cryptography (NSUCRYPTO)

Mathematical problems and their solutions from the fourth International Students’ Olympiad in cryptography (NSUCRYPTO-2017) are presented. We consider problems related to attacks on ciphers and hash functions, cryptographic Boolean functions, linear branch numbers, addition chains, and error correction codes, among others. We discuss several open problems involving the algebraic structures of cryptographic functions, useful proof-of-work algorithms, the Boolean hidden shift problem, and quantum computing.     

大围长结构化LDPC码线性编码器设计

校验矩阵拥有大围长对于改善LDPC码的性能有着重要的意义.结构化是提高码实用性的关键.提出一种基于列差搜索算法(Column-Difference Search Algorithm)和步进边增长算法(Progressive Edge Growth Algorithm,PEG)的准循环LDPC码构造方法,可以设计任意围长和码率的QC-LDPC码(称为CP-LDPC码).利用该码校验矩阵的近似下三角特性,推导出递推编码方法,使得该码编码复杂度与码长成线性关系.仿真结果表明,CP-LDPC码在BER性能上与随机码以及同属QC-LDPC码的Tanner码和Array码相比有明显提高,优于随机码、Tanner码和Array码1.35dB～3.95dB.     

对偶距离为5的极大自正交码及其子码

研究了自对偶码与其删截得到的极大自正交码的等价性问题。利用删截法构造出码长n满足21≤n≤29、对偶距离为5的二元极大自正交码。再用随机搜索算法研究了所得到的二元极大自正交码的子码,构造出它们的对偶距离为3和5的子码的生成矩阵。研究了这些子码构成的码链以及它们的对偶码构成的码链。利用所得到的码链,由Steane构造法构造出距离为5的具有很好参数的量子纠错码。     

Towards toric absolute factorization

This article presents an algorithmic approach to study and compute the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry.     

量子纠错码的一个统一构造方法

在量子通信和量子计算中,量子纠错码起着至关重要的作用。人们已经利用Hamming码、BCH码、Reed-Solomon码等各种循环码、常循环码、准循环码来构造量子纠错码。利用准缠绕码将这些构造方法统一起来,给出了准缠绕码包含其对偶码的充分必要条件及准缠绕码的一个新构造方法,并且利用准缠绕码构造了新的量子纠错码。     

距离为6的二元自对偶码的子码

用随机搜索算法研究了码长n满足22≤n≤30且距离为6的二元自对偶码的子码,构造出它们的对偶距离为3、4、5和6的子码的生成矩阵。研究了这些子码构成的码链以及它们的对偶码构成的码链。利用所得到的码链,由Steane构造法构造出距离为5和6的具有很好参数的量子纠错码,改进了前人得到的几个量子纠错码的参数。     

Remarks concerning the optimization of matrix codes

This paper describes the results of a general theory of matrix codes correcting a set of given types of multiple errors. A detailed study has been made of certain matrix classes of these systematic binary error correcting codes that will correct typical errors of some digital channels. These codes published by Elias,^(2,3) Hobb's,⁽⁵⁾ and Voukalis⁽¹¹⁾ account for this theory and other new families of binary systematic matrix codes of arbitrary size, correcting random, burst and clusters of errors are given here. Also presented here are the basic ideas of each of these codes. We can easily find practical decoding algorithms for each of these codes. The characteristic calculation of the parity check equations that the information matrix codebook has to satisfy are also shown. Further on we deal with the optimum construction of these codes showing their use in certain applications. We answer questions such as: “What is the optimum size of the code?” “What is the best structure of the code?” “What is the probability of error correction and the mean error correction performance?” Consequently, in this paper we also describe the results of an extensive search for optimum matrix codes designed to correct a given set of multiple errors as well as their implementation.