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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
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