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Is SHA(-1-2-3) a one to one function for inputs the same length as the output?
To restate the question as a concrete example:
SHA-1 has a 160 byte output, so do all 160 byte inputs have unique 160 byte outputs? Is the answer the same for SHA-2 and 3 and for all available output sizes?
Nobody knows, because nobody has proven it one way or the other, or tested every possible input at that range. That’s the simple truth.
If the functions behaved truly randomly, then the answer would almost certainly be “no” due to the birthday paradox — on average, you need to test 2^80 inputs to find a collision between any pair, for a 160-bit output.