Paper 2021/407

Improved Quantum Algorithms for the k-XOR Problem

André Schrottenloher


The k-XOR problem can be generically formulated as the following: given many n-bit strings generated uniformly at random, find k distinct of them which XOR to zero. This generalizes collision search (two equal elements) to a k-tuple of inputs. This problem has become ubiquitous in cryptanalytic algorithms, including variants in which the XOR operation is replaced by a modular addition ($k$-SUM) or other non-commutative operations (e.g., the composition of permutations). The case where a single solution exists on average is of special importance. At EUROCRYPT 2020, Naya-Plasencia and Schrottenloher defined a class of ``quantum merging algorithms'' for the k-XOR problem, obtained by combining quantum search. They represented these algorithms by a set of ``merging trees'' and obtained the best ones through linear optimization of their parameters. In this paper, we give a simplified representation of merging trees that makes their analysis easier. We give better quantum algorithms for the Single-solution k-XOR problem by relaxing one of the previous constraints, and making use of quantum walks. Our algorithms subsume or improve over all previous quantum algorithms for Single-solution k-XOR. For example, we give an algorithm for 4-XOR (or 4-SUM) in quantum time $\widetilde{\mathcal{O}}(2^{7n/24})$.

Note: Full version of the paper.

Available format(s)
Secret-key cryptography
Publication info
Published elsewhere. MAJOR revision.SAC 2021
Quantum algorithmsmerging algorithmsk-XORk-SUMbicomposite search.
Contact author(s)
andre schrottenloher @ m4x org
2021-09-02: revised
2021-03-27: received
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      author = {André Schrottenloher},
      title = {Improved Quantum Algorithms for the k-XOR Problem},
      howpublished = {Cryptology ePrint Archive, Paper 2021/407},
      year = {2021},
      note = {\url{}},
      url = {}
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