## Cryptology ePrint Archive: Report 2021/1460

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

Itai Dinur and Nathan Keller and Ohad Klein

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.

Category / Keywords: secret-key cryptography / cryptanalysis, fine-grained cryptography, lower bounds, k-SUM, k-XOR, Wagner's algorithm, generalized birthday problem, discrete Fourier analysis

Original Publication (with major differences): FOCS 2021

Date: received 31 Oct 2021, last revised 2 Nov 2021

Contact author: dinuri at cs bgu ac il, nathan keller27 at gmail com

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2021/1460

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