A Generalized Birthday Problem

We study a k-dimensional generalization of the birthday problem: given k lists of n-bit values, find some way to choose one element from each list so that the resulting k values xor to zero. For k = 2, this is just the extremely well-known birthday problem, which has a square-root time algorithm with many applications in cryptography. In this paper, we show new algorithms for the case k > 2: we show a cube-root time algorithm for the case of k = 4 lists, and we give an algorithm with subexponential running time when k is unrestricted.We also give several applications to cryptanalysis, describing new subexponential algorithms for constructing one-more forgeries for certain blind signature schemes, for breaking certain incremental hash functions, and for finding low-weight parity check equations for fast correlation attacks on stream ciphers. In these applications, our algorithm runs in O(22√n) time for an n-bit modulus, demonstrating that moduli may need to be at least 1600 bits long for security against these new attacks. As an example, we describe the first-known attack with subexponential complexity on Schnorr and Okamoto-Schnorr blind signatures over elliptic curve groups.