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**2004/031**

**Title:**

Summation polynomials and the discrete logarithm problem on elliptic curves

**Authors:**
Igor Semaev

**Abstract:**

The aim of the paper is the construction of the index calculus

algorithm for the discrete logarithm problem on elliptic curves.

The

construction presented here is based on the problem of finding

bounded solutions to some explicit modular multivariate

polynomial equations. These equations arise from the elliptic

curve summation polynomials introduced here and may be computed

easily. Roughly speaking, we show that given the algorithm for

solving such equations, which works in polynomial or low

exponential time in the size of the input, one finds discrete

logarithms faster than by means of Pollardâ€™s methods.

**ePrint:**
https://eprint.iacr.org/2004/031

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