Improved Quantum Algorithms for the k-XOR Problem

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) a k-tuple inputs. has become ubiquitous in cryptanalytic algorithms, including variants operation is replaced by modular addition (k-SUM) or other non-commutative operations (e.g., composition permutations). case where single solution exists on average special importance. At EUROCRYPT 2020, Naya-Plasencia and Schrottenloher defined class quantum merging algorithms for problem, obtained combining search. They represented these set trees best ones through linear optimization their parameters. In this paper, we give simplified representation that makes analysis easier. We better Single-solution relaxing one previous constraints, making use walks. Our subsume improve over all k-XOR. For example, an algorithm 4-XOR (or 4-SUM) time $$\widetilde{\mathcal {O}}(2^{7n/24})$$ .