特邀評(píng)論人: 戚文峰, 《密碼學(xué)報(bào)》副主編, 中國(guó)人民解放軍戰(zhàn)略支援部隊(duì)信息工程大學(xué)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(píng)《有限域上幾類置換和完全置換》

置換在密碼學(xué)中有著非常廣泛的應(yīng)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(zhì)的置換常常被用于構(gòu)造重要密碼組件—非線性S 盒. 置換多項(xiàng)式是代數(shù)和密碼領(lǐng)域的重要研究問題, 在組合、編碼、密碼等領(lǐng)域都有著廣泛的應(yīng)用, 目前對(duì)Dickson 多項(xiàng)式、二項(xiàng)式等特殊形式置換多項(xiàng)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(gòu)造, 因其好的密碼性質(zhì)被應(yīng)用于增強(qiáng)IDEA、SM4 等算法的安全性. 具有良好密碼性質(zhì)的置換多項(xiàng)式和完全置換多項(xiàng)式的有效構(gòu)造是密碼領(lǐng)域廣泛關(guān)注的熱點(diǎn)問題, 其研究具有重要的理論意義和實(shí)用價(jià)值. 《密碼學(xué)報(bào)》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(xiàng)式的構(gòu)造問題, 運(yùn)用跡函數(shù)、線性置換和Dickson 置換構(gòu)造了有限域Fqn上六類形如γx+(h(x)) 的置換多項(xiàng)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(xiàng)式的判定法則, 給出了Fqn上二項(xiàng)式γx+xs+1是置換的幾個(gè)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(xiàng)式的構(gòu)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.