Special prime numbers and discrete logs in finite prime fields

A set $A$ of primes $p$ involving numbers such as $ab^t+c$, where $\vert a\vert,\vert b\vert,\vert c\vert=O(1)$ and $t\to\infty$, is defined. An algorithm for computing discrete logs in the finite field of order $p$ with $p\in A$ is suggested. Its heuristic expected running time is $L_p[\frac13;(\frac{32}{9})^{1/3}]$ for $(\frac{32}{9})^{1/3}=1.526\dotsb$, where $L_p[\alpha;\beta]=\exp((\beta+o(1))\ln^\alpha p(\ln\ln p)^{1-\alpha})$ as $p\to\infty$, $0<\alpha<1$, and $0<\beta$. At present, the most efficient algorithm for computing discrete logs in the finite field of order $p$ for general $p$ is Schirokauer's adaptation of the Number Field Sieve. Its heuristic expected running time is $L_p[\frac13;(\frac{64}{9})^{1/3}]$ for $(\frac{64}{9})^{1/3}=1.9229\dotsb$. Using $p\in A$ rather than general $p$does not enhance the performance of Schirokauer's algorithm. The definition of the set $A$ and the algorithm suggested in this paper are based on a more general congruence than that of the Number Field Sieve. The congruence is related to the resultant of integer polynomials. We also give a number of useful identities for resultants that allow us to specify this congruence for some $p$.