学术评论.

特邀评论人: 戚文峰, 《密码学报》副主编, 中国人民解放军战略支援部队信息工程大学教授

Invited Reviewer: QI Wen-Feng, Associate Editor-in-Chief of Journal of Cryptologic Research, Professor of PLA Strategic Support Force Information Engineering University

评《有限域上几类置换和完全置换》

置换在密码学中有着非常广泛的应用, 许多密码算法的加解密变换就是密钥控制下的置换, 而具有良好密码性质的置换常常被用于构造重要密码组件—非线性S 盒. 置换多项式是代数和密码领域的重要研究问题, 在组合、编码、密码等领域都有着广泛的应用, 目前对Dickson 多项式、二项式等特殊形式置换多项式的研究已有很多好的研究成果. 如果f(x) 和f(x)+x 均为置换, 则称f(x) 为完全置换. 完全置换的提出源于正交拉丁方的构造, 因其好的密码性质被应用于增强IDEA、SM4 等算法的安全性. 具有良好密码性质的置换多项式和完全置换多项式的有效构造是密码领域广泛关注的热点问题, 其研究具有重要的理论意义和实用价值. 《密码学报》2019 年刊登的这篇论文研究了有限域上特殊类型的置换和完全置换多项式的构造问题, 运用迹函数、线性置换和Dickson 置换构造了有限域Fqn上六类形如γx+(h(x)) 的置换多项式, 证明了其中三类为完全置换; 考虑了xh(xs) 型置换, 基于已有的置换多项式的判定法则, 给出了Fqn上二项式γx+xs+1是置换的几个充分条件, 得到了有限域上几类新的完全置换, 也为完全置换多项式的构造提供新思路.

Review on “A Few Classes of Permutations and Complete Permutations over Finite Fields”

Permutation is widely used in Cryptography. The encryption and decryption transformation of many cryptographic algorithms is the permutation under key control. And permutation with good cryptographic properties is often used to construct nonlinear S-box, the important cryptographic component. Permutation polynomial is an important research problem in both Algebra and Cryptography, which is widely used in combination, coding,cryptography and other fields. At present, there are many good research results on Dickson polynomial, binomial and other special forms of permutation polynomial. If f(x) and f(x)+x are both permutations, then f(x) is called complete permutation. The concept of complete permutation, derived from the construction of orthogonal Latin squares, is used to enhance the security of IDEA, SM4 and other algorithms for its wonderful cryptographic properties. Because of this,the eきcient construction of permutation polynomials and complete permutation polynomials with good cryptographic properties is the focus of attention in the field of cryptography, with important theoretical significance and practical value. This paper, published in the Journal of Cryptologic Research in 2019,studies the construction of special types of permutation and complete permutation polynomials over finite fields,constructs six types of form γx+(h(x)) under finite fields Fqn by using trace functions, linear permutations and Dickson permutations, and proofs three of these are complete permutations. Also, this paper, studies the permutation of form xh(xs), proposes some necessary and suきcient conditions of that the binomial γx+xs+1under finite fields Fqnbased on the existing criteria of permutation polynomials, obtains some new types of complete permutations over finite fields, and also provides a new idea for the construction of complete permutation polynomials.