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Generalized Jacobian and discrete logarithm problem on elliptic curves. (English) Zbl 1301.14016

Summary: Let \(E\) be an elliptic curve over the finite field \(\mathbb F_{q}\), \(P\) a point in \(E(\mathbb F_{q})\) of order \(n\), and \(Q\) a point in the group generated by \(P\). The discrete logarithm problem on \(E\) is to find the number \(k\) such that \(Q = kP\). In this paper we reduce the discrete logarithm problem on \(E[n]\) to the discrete logarithm on the group \(\mathbb F^*_{q}\), the multiplicative group of nonzero elements of \(\mathbb F_q\), in the case where \(n|q-1\), using generalized Jacobian of \(E\).

MSC:

14H52 Elliptic curves
94A60 Cryptography
11G05 Elliptic curves over global fields
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