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Efficient generation of quadratic cyclotomic classes for shortest quadratic decompositions of polynomials
Cryptography and Communications ( IF 1.4 ) Pub Date : 2021-07-14 , DOI: 10.1007/s12095-021-00512-z
Kamil Otal 1 , Eda Tekin 2
Affiliation  

Nikova et al. investigated the decomposition problem of power permutations over finite fields \(\mathbb {F}_{2^{n}}\) in (Cryptogr. Commun. 11:379–384, 2019). In particular, they provided an algorithm to give a decomposition of a power permutation into quadratic power permutations. Their algorithm has a precomputation step that finds all cyclotomic classes of \(\mathbb {F}_{2^{n}}\) and then use the quadratic ones. In this paper, we provide an efficient and systematic method to generate the representatives of quadratic cyclotomic classes and hence reduce the complexity of the precomputation step drastically. We then apply our method to extend their results on shortest quadratic decompositions of \(x^{2^{n}-2}\) from 3 ≤ n ≤ 16 to 3 ≤ n ≤ 24 and correct a typo (for n = 11). We also give two explicit formulas for the time complexity of the adaptive search to understand its efficiency with respect to the parameters.



中文翻译:

多项式最短二次分解的二次分圆类的有效生成

尼科娃等人。研究了(Cryptogr. Commun. 11:379–384, 2019) 中有限域\(\mathbb {F}_{2^{n}}\)上幂排列的分解问题。特别是,他们提供了一种算法来将幂置换分解为二次幂置换。他们的算法有一个预计算步骤,可以找到\(\mathbb {F}_{2^{n}}\) 的所有分圆类,然后使用二次类。在本文中,我们提供了一种有效且系统的方法来生成二次分圆类的代表,从而大大降低预计算步骤的复杂性。然后我们应用我们的方法来扩展他们对\(x^{2^{n}-2}\)从 3 ≤ n 的最短二次分解的结果≤ 16 到 3 ≤ n ≤ 24 并更正一个错字(对于n = 11)。我们还给出了自适应搜索的时间复杂度的两个明确公式,以了解其在参数方面的效率。

更新日期:2021-07-14
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