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**2017/1262**

**Title:**

A New Index Calculus Algorithm for the Elliptic Curve Discrete Logarithm Problem and Summation Polynomial Evaluation

**Authors:**
Gary McGuire, Daniela Mueller

**Abstract:**

The introduction of summation polynomials for elliptic curves by Semaev has opened up new avenues of investigation in index calculus type algorithms for the elliptic curve discrete logarithm problem, and several recent papers have explored their use. We propose an index calculus algorithm to solve the Elliptic Curve Discrete Logarithm Problem that makes use of a technique for fast evaluation of the summation polynomials, and unlike all other algorithms using summation polynomials, does not involve a Gröbner basis computation. We further propose another algorithm that does not involve Gröbner basis computations or summation polynomials. We give a complexity estimate of our algorithms and provide extensive computational data.

**ePrint:**
https://eprint.iacr.org/2017/1262

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