论文标题
在Diffie-Hellman映射的索引上
On the Index of Diffie-Hellman Mapping
论文作者
论文摘要
令$γ$是循环群$ g $ $ n $的生成器。 $ g $的自动映射$ f $的最小索引是最大的亚组$ u $ $ g $的索引，因此，$ f（x）x^{ - r} $在$ u $的每个coset上都是恒定的，对于某些正整数〜$ r $。我们确定单变量Diffie-Hellman映射$ d（γ^a）=γ^{a^2} $，$ a = 0,1，\ ldots，n-1 $，并表明小索引的任何映射仅在〜$ d $的小$ g $中与〜$ d $相一致。此外，我们证明了双变量差异 - 赫尔曼映射$ d（γ^a，γ^b）=γ^{ab} $，$ a，b = 0,1，\ ldots，n-1 $的结果相似。在特殊情况下，$ g $是有限字段的乘法组的一个子组，我们提出了改进。
Let $γ$ be a generator of a cyclic group $G$ of order $n$. The least index of a self-mapping $f$ of $G$ is the index of the largest subgroup $U$ of $G$ such that $f(x)x^{-r}$ is constant on each coset of $U$ for some positive integer~$r$. We determine the index of the univariate Diffie-Hellman mapping $d(γ^a)=γ^{a^2}$, $a=0,1,\ldots,n-1$, and show that any mapping of small index coincides with~$d$ only on a small subset of $G$. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping $D(γ^a,γ^b)=γ^{ab}$, $a,b=0,1,\ldots,n-1$. In the special case that $G$ is a subgroup of the multiplicative group of a finite field we present improvements.
