Explicit form for the discrete logarithm over the field \(\text{GF}(p,k)\). (English) Zbl 0818.11049

Let \(a\) be a primitive element for the field \(F= \text{GF} (p^ k)\). The purpose of this note is to give for each non-zero \(b\in F\) an explicit form of the integer \(z\) for which \(1\leq z\leq p^ k-1\), \(b= a^ z\). Writing \(z= \sum_{m=0}^{k-1} d_ m p^ m\), where \(0\leq d_ m\leq p-1\), he shows that \(d_ m= \sum_{i=1}^{q-2} b^ i/ (1+ a^ i)^{2^ m}\), \(0\leq m\leq k-1\). This is a straightforward generalization of the formula for \(d_ 0\) found by A. L. Wells [IEEE Trans. Inf. Theory IT-30, 845-846 (1984; Zbl 0558.12009)].
Reviewer: S.E.Payne (Denver)


11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11T99 Finite fields and commutative rings (number-theoretic aspects)
94A60 Cryptography


Zbl 0558.12009
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