One-way functions are not cryptographic hash functions, which have the three properties preimage resistance, second preimage resistance, and collision resistanc

One-way functions are not cryptographic hash functions, which have the three properties preimage resistance, second preimage resistance, and collision resistance. For example, say f(x) is a one-way function. Then define g(0x) = f(x) g(1x) = f(x) Here given some z, it's easy to find another z that maps to the same output, just flip the first bit. However, g is still a one-way function: Assume we could break g with non-negligible probability, that some program A(y) outputs x such that g(x) = y with probability p. Then say someone gives us q = f(a) for some a. We can compute A(q) that will either give us 1a or 0a by the definition of g with probability p. In either case we can discard the first bit to find the preimage for a. By contradiction, g is a one-way function.