Constructing Functions with Low Differential Uniformity
Constructing Functions with Low Differential Uniformity
复制标题
构造低微分均匀性的函数
DOI：
10.1007/s00009-022-01980-0
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发表时间：
2022
影响因子：
1.1
通讯作者：
Coulter, Robert S.
中科院分区：
文献类型：
--
作者：
Bergman, Emily;Coulter, Robert S.
如果在替换盒中使用，函数的差分均匀性越低，它对差分密码分析的弹性就越大。APN函数和平面函数分别是在奇偶特性上具有最优微分均匀性的函数。在本文中，我们提供了两种构造低但不一定是最优的微分均匀性函数的方法。我们的第一种方法涉及改变任何已知平面函数的坐标函数，并依赖于Coulter和Matthews在1997年确定的平面函数和正交系统之间的关系。由于平面函数只存在于奇阶域上，所以该方法只适用于奇阶特征。该方法还将狄龙转换技术推广到构造APN函数。我们的第二种构造方法是由Coulter和Henderson的结果激发的，他们在2008年展示了奇数阶的交换前提域如何通过对前提域的乘法与平面Dembowski-Ostrom多项式一一对应。以这种联系为出发点，我们研究了由其他结构良好的代数对象（如非交换前域和平面近域）的乘法产生的函数。特别地，我们构造了许多无限类的函数，它们具有低的，虽然不是最优的，微分均匀性。这类函数最初源于特征2的Kantor和Williams的前提域。因此，不管特征如何，在我们的两种方法之间，我们能够构建无限多个函数，这些函数在任意大阶的域上具有低（尽管不是最优）的微分均匀性。
The lower the differential uniformity of a function, the more resilient it is to differential cryptanalysis if used in a substitution box. APN functions and planar functions are specifically those functions which have optimal differential uniformity in even and odd characteristic, respectively. In this article, we provide two methods for constructing functions with low, but not necessarily optimal, differential uniformity. Our first method involves altering the coordinate functions of any known planar function and relies upon the relation between planar functions and orthogonal systems identified by Coulter and Matthews in 1997. As planar functions exist only over fields of odd order, the method works for odd characteristic only. The approach also leads us to a generalization of Dillon’s Switching Technique for constructing APN functions. Our second construction method is motivated by a result of Coulter and Henderson, who showed in 2008 how commutative presemifields of odd order were in one-to-one correspondence with planar Dembowski–Ostrom polynomials via the multiplication of the presemifield. Using this connection as a starting point, we examine the functions arising from the multiplication of other well-structured algebraic objects such as non-commutative presemifields and planar nearfields. In particular, we construct a number of infinite classes of functions which have low, though not optimal, differential uniformity. This class of functions originally stems from the presemifields of Kantor and Williams of characteristic 2. Thus, regardless of the characteristic, between our two methods we are able to construct infinitely many functions which have low, though not optimal, differential uniformity over fields of arbitrarily large order.
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DOI：
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发表时间：
--
期刊：
影响因子：
--
作者：
E. H. Moore
通讯作者：
E. H. Moore
DOI：
10.1090/s0002-9947-1905-1500706-2
发表时间：
1905
影响因子：
1.3
作者：
L. Dickson
通讯作者：
L. Dickson
局部有限近场
影响因子：
0.7
作者：
S. Groves
通讯作者：
S. Groves
DOI：
--
发表时间：
1953
期刊：
影响因子：
--
作者：
L. Carlitz
通讯作者：
L. Carlitz
DOI：
--
发表时间：
1953
期刊：
影响因子：
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作者：
Finite Field;L. Carlitz
通讯作者：
L. Carlitz