通过痕量函数计算扰动反函数的 c 微分均匀性 $$ {{\,\textrm{Tr}\,}}\big (\frac{x^2}{x+1}\big )$$

IF 0.6 3区 数学 Q3 MATHEMATICS
Kübra Kaytancı, Ferruh Özbudak
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引用次数: 0

摘要

差分均匀性是密码学中最重要的概念之一。最近,Ellingsen 等人(IEEE Trans Inf Theory 66:5781-5789, 2020)通过扩展通常的差分概念,提出了一个新概念--c-差分分布表和 c-差分均匀性。Borisov 等人(《乘法差分》,2002 年)指出,这一新概念的动机是为了抵御一些已知的差分攻击。2022 年,Hasan 等人(IEEE Trans Inf Theory 68:679-691, 2022)通过迹函数 \( {{\,\textrm{Tr}\,}}\big (\frac{x^2}{x+1}\big )\)给出了扰动逆函数 H 的 c 微分均匀性的上界。在他们的工作中,还提出了一个关于 H 的精确 c 微分均匀性的未决问题。通过使用一种基于有限域上代数曲线的新方法,我们解决了 \( {{\,\textrm{Tr}\,}}(c)=1= {{\,\textrm{Tr}\、}}(\frac{1}{c})\) for \( c \ in {\mathbb {F}}_{2^n}setminus \{0,1\} \) 完全,并且我们证明了 H 的精确 c 微分均匀性是 8。在其余情况下,我们几乎完全解决了问题,并且证明了 H 的 c 微分均匀性是 8 或 9。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

The c-differential uniformity of the perturbed inverse function via a trace function $$ {{\,\textrm{Tr}\,}}\big (\frac{x^2}{x+1}\big )$$

The c-differential uniformity of the perturbed inverse function via a trace function $$ {{\,\textrm{Tr}\,}}\big (\frac{x^2}{x+1}\big )$$

Differential uniformity is one of the most crucial concepts in cryptography. Recently Ellingsen et al. (IEEE Trans Inf Theory 66:5781–5789, 2020) introduced a new concept, the c-Difference Distribution Table and the c-differential uniformity, by extending the usual differential notion. The motivation behind this new concept is based on having the ability to resist some known differential attacks which is shown by Borisov et. al. (Multiplicative Differentials, 2002). In 2022, Hasan et al. (IEEE Trans Inf Theory 68:679–691, 2022) gave an upper bound of the c-differential uniformity of the perturbed inverse function H via a trace function \( {{\,\textrm{Tr}\,}}\big (\frac{x^2}{x+1}\big )\). In their work, they also presented an open question on the exact c-differential uniformity of H. By using a new method based on algebraic curves over finite fields, we solve the open question in the case \( {{\,\textrm{Tr}\,}}(c)=1= {{\,\textrm{Tr}\,}}(\frac{1}{c})\) for \( c \in {\mathbb {F}}_{2^n}\setminus \{0,1\} \) completely and we show that the exact c-differential uniformity of H is 8. In the remaining case, we almost completely solve the problem and we show that the c-differential uniformity of H is either 8 or 9.

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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
67
审稿时长
>12 weeks
期刊介绍: Periodica Mathematica Hungarica is devoted to publishing research articles in all areas of pure and applied mathematics as well as theoretical computer science. To be published in the Periodica, a paper must be correct, new, and significant. Very strong submissions (upon the consent of the author) will be redirected to Acta Mathematica Hungarica. Periodica Mathematica Hungarica is the journal of the Hungarian Mathematical Society (János Bolyai Mathematical Society). The main profile of the journal is in pure mathematics, being open to applied mathematical papers with significant mathematical content.
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