Space--Time Tradeoffs for Subset Sum: An Improved Worst Case Algorithm

TL;DR

This paper extends the improved space-time tradeoff algorithms for the SUBSET SUM problem from random instances to worst-case scenarios, using a novel approach involving random composite moduli and parallelization techniques.

Contribution

It removes the randomness assumptions from the dissection method, achieving similar tradeoffs in worst-case instances and introduces a bailout mechanism for explicit control.

Findings

Achieves worst-case space-time tradeoffs comparable to random-instance algorithms.

Demonstrates near-full parallelization of the dissection algorithm for small space.

Introduces a bailout mechanism for explicit space-time management.

Abstract

The technique of Schroeppel and Shamir (SICOMP, 1981) has long been the most efficient way to trade space against time for the SUBSET SUM problem. In the random-instance setting, however, improved tradeoffs exist. In particular, the recently discovered dissection method of Dinur et al. (CRYPTO 2012) yields a significantly improved space--time tradeoff curve for instances with strong randomness properties. Our main result is that these strong randomness assumptions can be removed, obtaining the same space--time tradeoffs in the worst case. We also show that for small space usage the dissection algorithm can be almost fully parallelized. Our strategy for dealing with arbitrary instances is to instead inject the randomness into the dissection process itself by working over a carefully selected but random composite modulus, and to introduce explicit space--time controls into the algorithm by means of a "bailout mechanism".