Far Eastern Mathematical Journal

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Asymmetric cryptography and hyperelliptic sequences


Illarionov A.A.

2019, issue 2, Ñ. 185–196


Abstract
We study sequences $\{A_n\}_{n=-\infty}^{+\infty}$ of elements of a field $\FF$ that satisfy decompositions of the form $$A_{m+n}A_{m-n}=a_1(m)b_1(n)+a_2 (m)b_2 (n),$$ where $a_1, a_2, b_1, b_2:\ZZ \to\FF$. The results are used to build analogues of the Diffie-Hellman and El-Gamal algorithms. The discrete logarithm problem is posed in the group $(S, +)$, where the set $S$ consists of fours $S(n)=(A_{n-1}, A_n, A_{n+1}, A_{n+2})$, $n\in \ZZ$, and $S(n)+S(m)=S(n+m)$.

Keywords:
hyperelliptic sequences, nonlinear recurrence sequences, asymmetric cryptography

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