‎this article presents a systematic study for structure of finite wavelet frames‎ ‎over prime fields‎. ‎let $p$ be a positive prime integer and $mathbb{w}_p$‎ ‎be the finite wavelet group over the prime field $mathbb{z}_p$‎. ‎we study theoretical frame aspects of finite wavelet systems generated by‎ ‎subgroups of the finite wavelet group $mathbb{w}_p$.

‎This article presents a systematic study for structure of finite wavelet frames‎ ‎over prime fields‎. ‎Let $p$ be a positive prime integer and $mathbb{W}_p$‎ ‎be the finite wavelet group over the prime field $mathbb{Z}_p$‎. ‎We study theoretical frame aspects of finite wavelet systems generated by‎ ‎subgroups of the finite wavelet group $mathbb{W}_p$.

A set A of primes p involving numbers such as abt + c, where |a|, |b|, |c| = O(1) and t → ∞, is defined. An algorithm for computing discrete logs in the finite field of order p with p ∈ A is suggested. Its heuristic expected running time is Lp[ 1 3 ; ( 32 9 )1/3] for ( 32 9 )1/3 = 1.526 · · · , where Lp[α;β] = exp((β + o(1)) ln α p(ln ln p)1−α) as p → ∞, 0 < α < 1, and 0 < β. At present, the mo...