非线性密码函数若干关键问题研究

The nonlinearity profile and differential uniformity of cryptographic functions are the two most important properties in the design of symmetric cryptography algorithms. The main objective of this project is to carry out in-depth study on the nonlinearity profile and differential uniformity of cryptographic functions, based on some well-understood theories and powerful tools such as the finite field theory, the combinatorial design theory, the fourth moment of the Walsh transform and the concatenation of multiple functions. We will first treat the nonlinearity profile of Boolean functions, in which we mainly focus on the Dobbertin conjecture and determining the exact values of higher-order nonlinearities of several classes of bent and semi-bent functions. Secondly, we will concentrate on constructing balanced Boolean functions with very low maximum absolute values, particularity, obtaining counterexamples against the Zhang-Zheng conjecture in the case of the number of variables is congruent to 0 modulo 4. Thirdly, will we consider the construction of balanced vectorial Boolean functions with low differential uniformity, especially the construction of differentially 4-uniform permutations. Finally, we will study the cryptographic properties of cryptographic functions and some applications of cryptographic functions in the constructions of binary linear codes, for examples, the relation between the fast algebraic immunity and the higher-order nonlinearities of Boolean functions, the relation between the differential uniformity and the Walsh spectrum of vectorial Boolean functions, and the construction of distance-optimal binary linear codes derived from cryptographic functions. Research on these subjects has both important theoretical significance and great practical value.

非线性轮廓和差分均匀度是密码函数在对称密码算法设计中最重要的两个密码学性质。本项目拟基于有限域、组合设计、布尔函数多重级联、高阶Walsh变换等强有力的理论和工具，对密码函数的非线性轮廓和差分均匀度展开深入研究：1.研究布尔函数的非线性轮廓，重点研究Dobbertin猜想问题和给出若干类Bent及Semi-bent函数的高阶非线性度确切值；2.研究低最大自相关绝对值平衡布尔函数的设计，特别是给出Zhang-Zheng猜想在变元个数模4余0时的一类反例；3.研究低差分均匀度平衡向量值布尔函数的设计，尤其是具有已知最大非线性度的4-差分置换的构造；4.研究密码函数的密码学性质及相关编码，包括布尔函数快速代数免疫度与高阶非线度之间的关系、向量值布尔函数的差分分布与谱值分布之间的关系、距离最优二元线性码的构造及其应用等相关问题。这些课题的研究具有重要的理论意义和应用价值。

对称密码体制算法设计中所使用的密码函数应同时兼具多种密码学性质。本项目对非线性密码函数的非线性轮廓和差分均匀度开展了系统深入的研究工作，并对密码函数不同密码性质之间的相互关系和距离最优二元线性码的构造及应用开展了研究工作。主要取得以下研究进展：（1）基于Maiorana-McFarland型Bent函数，构造了一大类具有极低最大自相关绝对值的轻量级平衡布尔函数，该类函数亦为Zhang-Zheng猜想反例；（2）基于组合理论和代数曲线理论，构造了第一类具有极低最大自相关绝对值的 (2k,k)-密码函数，该类函数亦为第一类具有超低差分-线性均匀度的多输出布尔函数；（3）基于加性高斯噪声和极大似然估计，我们首次理论建立了S-盒的分量组合布尔函数的最低自相关值与其抵抗侧信道密码分析能力之间的一个关系，并构造了若干类具有低自相关值的平衡密码函数；（4）构造了一类Almost perfect nonlinear function函数，提高了若干类密码函数的高阶非线性度下界，构造了几类满足Griesmer界的距离最优二元线性码。基于以上研究工作，项目负责人在国内外重要学术期刊和学术会议上发表标注基金资助的高水平研究论文19篇，其中SCI源期刊论文17篇、国际会议EI检索论文1篇，含信息安全领域顶级期刊《IEEE Transactions on Information Forensics and Security》和密码与信息理论旗舰期刊《IEEE Transactions on Information Theory》各1篇，密码与编码领域著名期刊《Designs, Codes and Cryptography》论文2篇，组合数学著名期刊《SIAM Journal on Discrete Mathematics》论文1篇，国际会议印度密码学年会论文1篇，有限域领域著名期刊《Finite Fields and Their Applications》论文3篇，离散数学著名期刊《Discrete Applied Mathematics》和《Discrete Mathematics》各1篇。项目负责人获评2019年密码算法学术会议最佳论文，参与提交至中国密码学会主办的“全国密码算法设计竞赛”的分组密码算法FBC获评三等奖。该项目的研究进展具有重要的理论意义和潜在的应用价值。