# On the first fall degree of summation polynomials

## TL;DR

This paper improves the bound on the first fall degree of polynomial systems derived from Weil descent and Semaev's summation polynomials, which are used in solving the Elliptic Curve Discrete Logarithm Problem with Gr"obner basis methods.

## Contribution

It provides a tighter bound on the first fall degree for these polynomial systems, enhancing understanding of their complexity in cryptographic applications.

## Key findings

- Improved the first fall degree bound for specific polynomial systems.
- Enhanced analysis of the complexity of solving elliptic curve discrete logs.
- Contributed to the theoretical understanding of polynomial system behavior in cryptography.

## Abstract

We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev's summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gr\"obner basis algorithms.