**Low Weight Discrete Logarithms and Subset Sum in $2^{0.65n}$ with Polynomial Memory**

*Andre Esser and Alexander May*

**Abstract: **We propose two polynomial memory collision finding algorithms for the low Hamming weight discrete logarithm problem in any abelian group $G$. The first one is a direct adaptation of the Becker-Coron-Joux (BCJ) algorithm for subset sum to the discrete logarithm setting. The second one significantly improves on this adaptation for all possible weights using a more involved application of the representation technique together with some new Markov chain analysis. In contrast to other low weight discrete logarithm algorithms, our second algorithm's time complexity interpolates to Pollard's $|G|^{\frac 1 2}$ bound for general discrete logarithm instances.

We also introduce a new subset sum algorithm with polynomial memory that improves on BCJ's $2^{0.72n}$ time bound for random subset sum instances $a_1, \ldots, a_n, t \in \mathbb{Z}_{2^n}$. Technically, we introduce a novel nested collision finding for subset sum -- inspired by the NestedRho algorithm from Crypto '16 -- that recursively produces collisions. We first show how to instantiate our algorithm with run time $2^{0.649n}$. Using further tricks, we are then able to improve its complexity down to $2^{0.645n}$.

**Category / Keywords: **public-key cryptography / Low weight dlog, subset sum, representations, Nested Rho

**Original Publication**** (in the same form): **IACR-EUROCRYPT-2020

**Date: **received 15 Aug 2019, last revised 6 Feb 2020

**Contact author: **andre esser at rub de

**Available format(s): **PDF | BibTeX Citation

**Version: **20200206:160155 (All versions of this report)

**Short URL: **ia.cr/2019/931

[ Cryptology ePrint archive ]