學術評論.

特邀評論人: 戚文峰, 《密碼學報》副主編, 中國人民解放軍戰(zhàn)略支援部隊信息工程大學教授

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

評《有限域上幾類置換和完全置換》

置換在密碼學中有著非常廣泛的應用, 許多密碼算法的加解密變換就是密鑰控制下的置換, 而具有良好密碼性質的置換常常被用于構造重要密碼組件—非線性S 盒. 置換多項式是代數(shù)和密碼領域的重要研究問題, 在組合、編碼、密碼等領域都有著廣泛的應用, 目前對Dickson 多項式、二項式等特殊形式置換多項式的研究已有很多好的研究成果. 如果f(x) 和f(x)+x 均為置換, 則稱f(x) 為完全置換. 完全置換的提出源于正交拉丁方的構造, 因其好的密碼性質被應用于增強IDEA、SM4 等算法的安全性. 具有良好密碼性質的置換多項式和完全置換多項式的有效構造是密碼領域廣泛關注的熱點問題, 其研究具有重要的理論意義和實用價值. 《密碼學報》2019 年刊登的這篇論文研究了有限域上特殊類型的置換和完全置換多項式的構造問題, 運用跡函數(shù)、線性置換和Dickson 置換構造了有限域Fqn上六類形如γx+(h(x)) 的置換多項式, 證明了其中三類為完全置換; 考慮了xh(xs) 型置換, 基于已有的置換多項式的判定法則, 給出了Fqn上二項式γx+xs+1是置換的幾個充分條件, 得到了有限域上幾類新的完全置換, 也為完全置換多項式的構造提供新思路.

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.