Fine-Grained Cryptanalysis: Tight Conditional Bounds for Dense k-SUM and k-XOR

TL;DR

This paper establishes tight bounds for the complexity of dense $k$-SUM and $k$-XOR problems, demonstrating the optimality of existing algorithms under certain conjectures and introducing a novel self-reduction and obfuscation technique.

Contribution

It proves the optimality of known algorithms for $k=3,4,5$ and the $k$-tree algorithm's optimality for larger $k$ within certain parameters, using a new self-reduction and obfuscation method.

Findings

Algorithms are essentially optimal for $k=3,4,5$.

The $k$-tree algorithm is optimal for larger $k$ in some parameter ranges.

Obfuscation with noise removes correlations in oracle outputs, ensuring robustness.

Abstract

An average-case variant of the $k$-SUM conjecture asserts that finding $k$ numbers that sum to 0 in a list of $r$ random numbers, each of the order $r^k$, cannot be done in much less than $r^{\lceil k/2 \rceil}$ time. On the other hand, in the dense regime of parameters, where the list contains more numbers and many solutions exist, the complexity of finding one of them can be significantly improved by Wagner's $k$-tree algorithm. Such algorithms for $k$-SUM in the dense regime have many applications, notably in cryptanalysis. In this paper, assuming the average-case $k$-SUM conjecture, we prove that known algorithms are essentially optimal for $k= 3,4,5$. For $k>5$, we prove the optimality of the $k$-tree algorithm for a limited range of parameters. We also prove similar results for $k$-XOR, where the sum is replaced with exclusive or. Our results are obtained by a self-reduction that, given an instance of $k$-SUM which has a few solutions, produces from it many instances in the dense regime. We solve each of these instances using the dense $k$-SUM oracle, and hope that a solution to a dense instance also solves the original problem. We deal with potentially malicious oracles (that repeatedly output correlated useless solutions) by an obfuscation process that adds noise to the dense instances. Using discrete Fourier analysis, we show that the obfuscation eliminates correlations among the oracle's solutions, even though its inputs are highly correlated.