On the index of the Diffie–Hellman mapping
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CitationIşık, L., & Winterhof, A. (2020). On the index of the Diffie–Hellman mapping. Applicable Algebra in Engineering, Communication and Computing, 1-9.
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(γa)=γa2, a= 0 , 1 , … , 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(γa, γb) = γab, a, b= 0 , 1 , … , n- 1. In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements