编码与密码学前沿研讨会：后量子密码及其相关问题会议安排

四川 成都 西南交通大学2024年12月22日-2024年12月24日

本次会议主题包括(但不限于)：为了促进学术交流，加强学术合作，由西南交通大学信息科学与技术学院、数学学院主办的“编码与密码学前沿研讨会：后量子密码及其相关问题”，将于2024年12月22日（周日）至12月24日（周二）在西南交通大学召开。本次研讨会将邀请编码、密码、组合数学与数论等领域的专家参加，共同研讨相关方向的最新研究进展和发展趋势，为相关学者提供一个学术平台，交流最新发展动态及学术成果，促进信息学科、数学学科理论等相关领域的交叉、融合与发展，为该领域的老师、学生提供一个相互学习和交流的场所。

联系人：

唐春明 (tangchunmingmath@163.com; 18582182739)

罗荣 (luorong@swjtu.edu.cn; 13882112127)

报告摘要

题目：超奇异椭圆曲线同源密码

报告人：徐茂智教授

摘要：椭圆曲线是构造密码算法的重要源泉，椭圆曲线密码包括基于点的标量乘的密码、基于双线性配对的密码和基于超奇异椭圆曲线同源的密码。前两者已经得到广泛应用，而超奇异椭圆曲线同源密码因为其抗量子攻击的性质，成为密码学界和应用数学界的一个研究热点。 本报告介绍椭圆曲线概念、加法，并给出超奇异椭圆曲线及同源的概念，进而介绍使用超奇异椭圆曲线同源构造的密码基础和进展情况。涉及代数、数论、代数曲线知识和密码学的概念。

报告题目：素数判定问题综述

报告人：邓映蒲教授

摘要：判定一个大整数是否是素数是计算数论的基本问题之－，在如密码中有重要应用。我们综述素数判定的一些重要算法，包括概率算法与确定性算法，如著名的AKS算法，还讲述一些特殊数的素数判定方法

报告题目：后量子密码学简介

报告人：潘彦斌副研究员

摘要：随着量子计算技术的快速发展，目前所广泛使用的基于传统数论问题的公钥密码体制受到了严重威胁。因此，抗量子密码体制的研制近年来备受关注。本报告将简要介绍主流后量子密码体制的相关数学理论、发展现状以及尚待解决的一些重要问题。

报告题目：ANALYSIS OF ROTH-LEMPEL CODES

报告人：周海燕教授

摘要：Near maximum distance separable (NMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes due to their algebraic properties and excellent error-correcting capabilities.This report focuses on Roth-Lempel codes and establishes necessary and sufficient conditions for them to be NMDS and further completely determine its weight distributions . Besides, we illustrate the linearly inequivalence of Roth-Lempel codes and NMDS codes of elliptic curve type when their corresponding code lenthts exceed $\frac{4(q+2\sqrt{q}+1)}{5}-1$. Finally we show that some special linear codes of elliptic-curve type are not equivalent to Roth-Lempel code C by Schur product.

报告题目：Weights of a class of projective geometry codes

报告人：麻常利教授

摘要：Linear codes are an important class of error-correcting codes and widely used in secret sharing schemes, combinational designs, authentication codes and so on. Let C(n-1,q) be the p-ary linear code generated by the rows of the incidence matrix of points and hyperplanes of PG(n-1,q), with q=ps, s\geq1 and p prime. This is a special class of projective geometry codes. The weights of C(n-1,q) have attracted a great many of research in recent years. Some previous results are as follows: the minimal weight of C(n-1,q) is θn-1, where θn=qn-1q-1; the second minimal weight of C(n-1,q) is 2qn-2; the third minimal weight of C(2,p) is 2p+1, with p prime and p≥11; the fourth minimum weight of C(2,p) is 3p-3, with p prime and p≥5. What are all the weights of C(n-1,q)? This is what we focus on in this paper. Our main results are as follows:

(i) We present that C(n-1,q) is a cyclic code, and give its generator polynomial and parity-check polynomial.

(ii) We give a formula to calculate some weights of C(n-1,q) by the weight distribution of CD, where CD is constructed from the defining set D.

(iii) When q>2 is even, we prove that the weight w of C(2,q) is w\equiv1(\mbox{or }0)~(\mbox{mod}~4). Furthermore, we give a method to calculate some weights of C(2,q) , and present a conjecture.

(iv) When q is even, we prove that each codeword of C2,q⊥ is a linear combination of the incidence vectors of some linear hyperovals, and every hyperoval is the sum of q+2 linear hyperovals, where 、C2,q⊥ is the dual code of C(2,q).

报告题目：Large-size families of Costas arrays with low cross-correlation

报告人：王琦教授

摘要：Costas arrays have been extensively investigated for decades due to their applications in Radar systems and their close connections to combinatorics. In this talk, I will introduce some new recent results on families of Costas arrays with low cross-correlation. More precisely, by employing some results on the number of roots of certain polynomials over finite fields, we are able to derive bounds on the cross-correlation of several large-size families of Costas arrays.

报告题目：子空间码的消息认证码的构造

报告人：张俊教授

摘要：子空间码是线性网络编码的一类重要纠错码。由于网络的复杂性，网络中的替代攻击/污染攻击是常见的安全问题，消息认证码是保证消息完整性、防止这两类攻击的有效手段。报告中，我们将利用经典纠错码对子空间码设计一类消息认证码。

报告题目：同源密码及其计算

报告人：胡志副教授

报告内容：同源密码是后量子密码领域的一类重要研究对象，其优势是密钥长度短，劣势是安全基础研究历史短、底层理论复杂且实现效率低。2022年SIDH被攻破，以及后续SQIsign被遴选到NIST后量子密码签名算法标准征集中，相关事件使得同源密码得到了众多关注。本报告将介绍同源密码的安全基础、同源密码方案以及相关同源计算，并探讨同源密码的发展前景。