Functions with good differential-uniformity properties have important applications in coding theory and sequence design in addition to the applications in cryptography. The differential spectrum of a cryptographic function is useful for estimating its resistance to some variants of differential cryptanalysis. The objective of this paper is to determine the differential spectrum of the power function $ x^{p^{2k}-p^k+1} $ over $ \mathbb F_{p^n} $, where $ p $ is an odd prime, $ n, k, e $ are integers with $ \gcd(n,k) = e $ and $ \frac{n}{e} $ being odd. In particular, when $ n $ is odd and $ e = 1 $, our result includes a recent one (IEEE Trans. Inform. Theory 65(10): 6819-6826) as a special case.

中文翻译：

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一类幂函数在有限域上的微分谱

具有良好差分均匀性的函数除了在密码学中的应用外，在编码理论和序列设计中也具有重要的应用。密码函数的差分频谱可用于估计其对差分密码分析的某些变体的抵抗力。本文的目的是确定幂函数$ x ^ {p ^ {2k} -p ^ k + 1} $在$ \ mathbb F_ {p ^ n} $上的微分频谱，其中$ p $是一个奇质数$ n，k，e $是整数$ \ gcd（n，k）= e $并且$ \ frac {n} {e} $为奇数。特别是，当$ n $为奇数且$ e = 1 $时，作为特殊情况，我们的结果包括最近的结果（IEEE Trans。Inform。Theory 65（10）：6819-6826）。