For a hash function that generates an n-bit message digest, the ideal preimage and 2nd preimage resistance on a single hash function invocation implies on the order of 2"n" operations to find a collision under a classical computing model. According to Grover's algorithm, finding a preimage collision on a single invocation of an ideal hash function is upper bound on O(2"n"/2) operations under a quantum computing model. In Lamport signatures, each bit of the public key and signature is based on short messages requiring only a single invocation to a hash function.