## A note on discrete logarithms in finite fields.(English)Zbl 0749.11055

Let $$F_ q$$ denote the finite field of order $$q$$, $$q=p^ n\geq 3$$, $$p$$ prime and $$F^*_ q$$ the cyclic multiplicative group of nonzero elements. If $$\alpha\in F^*_ q$$ is primiitve, the discrete logarithm of $$\beta\in F^*_ q$$ to base $$\alpha$$ is the unique integer $$x$$ such that $$\beta=\alpha^ x$$. It was previously shown that for $$q=p$$ a prime $\log_ \alpha\beta=\sum^{p-2}_{i=1}(1-\alpha^ i)^{-1}\beta^ i.$ This note generalizes this formula to cases where the base is not necessarily a primitive element and to prime power fields.

### MSC:

 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 94A60 Cryptography

### Keywords:

cryptography; finite field; discrete logarithm
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### References:

 [1] Lidl, R., Niederreiter, H.: Finite Fields, Encyclo. Math. Appls., Vol. 20. Addison-Wesley, Reading, MA 1983. (Now distributed by Camb. Univ. Press) · Zbl 0554.12010 [2] Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge: Camb. Univ. Press 1986 · Zbl 0629.12016 [3] Mullen, G. L., White, D.: A polynomial representation for logarithms in GF(q). Acta Arith.47, 255-261 (1986) · Zbl 0562.12018 [4] Niederreiter, H.: A short proof for explicit formulas for discrete logarithms in finite fields. App. Alg. Eng. Comm. Comp.1, 55-57 (1990) · Zbl 0726.11079 [5] Odlyzko, A. M.: Discrete logarithms in finite fields and their cryptographic significance. Proc. EUROCRYPT, Vol. 84, pp. 224-314. Lect. Notes in Comp. Sci., Vol. 209. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0594.94016 [6] Pollard, J. M.: The fast Fourier transform in a finite field. Math. Computation25, 365-374 (1971) · Zbl 0221.12015 [7] Wells, A. L., Jr.: A polynomial form for logarithms modulo a prime. IEEE Trans. Inform. TheoryIT-30, 845-846 (1984) · Zbl 0558.12009
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