Paper 2021/1460

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

Itai Dinur
Nathan Keller,
Ohad Klein

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.

Available format(s)
Secret-key cryptography
Publication info
Published elsewhere. Major revision. FOCS 2021\JACM
cryptanalysisfine-grained cryptographylower boundsk-SUMk-XORWagner's algorithmgeneralized birthday problemdiscrete Fourier analysis
Contact author(s)
dinuri @ bgu ac il
2024-03-10: revised
2021-11-06: received
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Creative Commons Attribution


      author = {Itai Dinur and Nathan Keller and Ohad Klein},
      title = {Fine-Grained Cryptanalysis: Tight Conditional Bounds for Dense k-SUM and k-XOR},
      howpublished = {Cryptology ePrint Archive, Paper 2021/1460},
      year = {2021},
      note = {\url{}},
      url = {}
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