摘要
为了衡量S盒(S-boxes)抵抗差分分析的能力,Nyberg于1993年提出了差分均匀度的概念.在实际应用中,常要求密码算法中所使用的S盒是低差分函数.除了在密码学中有着重要的应用,低差分函数在序列设计、编码理论以及组合设计等领域也有着广泛应用.此外,差分谱是衡量密码函数抵抗差分分析变体的重要指标,并且也可用于设计理论、编码理论等领域,因此吸引了大量的关注.但目前已确定差分谱的幂函数并不多.本文的主要工作是确定了一类幂函数的差分谱.1997年,Helleseth和Sandberg证明了幂函数x p^(n)-1/2+2在Fp^(n)上的差分均匀度不超过4,其中p为奇素数.特别地,当p≠3且p^(n)≡1(mod 4)时,幂函数xp^(n)-1/2+2的差分均匀度不超过3.本文利用有限域上指数和及椭圆曲线相关知识,完全确定了当p≠3且p^(n)≡1(mod 4)时,幂函数x p^(n)-1/2+2在Fp^(n)上的差分谱,丰富了已有结果.
For characterizing the resistance of S-boxes against differential cryptanalysis,the differential uniformity of an S-box was presented by Nyberg in 1993.Practically,cryptographic functions are desirable to be with low differential uniformity.It is worth noting that besides the applications in cryptography,functions with low differential uniformity are also widely used in sequence designs,coding theory and combinatorial designs.In addition,the differential spectrum is an important measure on the resistance of cryptographic functions against some variants of the differential cryptanalysis,and it also has wide applications in design theory,coding theory and other areas.Thus the differential spectrums of S-boxes are shown to be of great interest.However,there are few results regarding the differential spectrums of power functions.The work of this paper is to determine the differential spectrum of a class of power functions.In 1997,Helleseth and Sandberg proved that the differential uniformity of x p^(n)-1/2+2 over Fp^(n) is less than or equal to 4,where p is an odd prime.Especially,the differential uniformity of power function x p^(n)-1/2+2 over Fp^(n) is less than or equal to 3 when p≠3 and p^(n)≡1(mod 4).Based on some knowledge of elliptic curves and exponential sums over finite fields,this paper gives a complete determination on the differential spectrum of x p^(n)-1/2+2 over Fp^(n) with p≠3 and p^(n)≡1(mod 4).
作者
姜沙
李康荃
李宇玻
屈龙江
JIANG Sha;LI Kang-Quan;LI Yu-Bo;QU Long-Jiang(College of Liberal Arts and Sciences,National University of Defense Technology,Changsha 410073,China;Hunan Engineering Research Center of Commercial Cryptography Theory and Technology Innovation,Changsha 410073,China)
出处
《密码学报》
CSCD
2022年第3期484-495,共12页
Journal of Cryptologic Research
基金
国家自然科学基金(62032009)
国防科技大学科研计划项目(ZK22-14)。
关键词
幂函数
差分谱
椭圆曲线
指数和
分圆数
power function
differential spectrum
elliptic curve
exponential sum
cyclotomic number
