hash - Is there an MD5 Fixed Point where md5(x) == x? -

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is there fixed point in md5 transformation, i.e. there exist x such md5(x) == x?

since md5 sum 128 bits long, fixed point have 128 bits long. assuming md5 sum of string uniformly distributed on possible sums, probability given 128-bit string fixed point 1/2128.

thus, probability no 128-bit string fixed point (1 − 1/2128)2128, probability there fixed point 1 − (1 − 1/2128)2128.

since limit n goes infinity of (1 − 1/n)n 1/e, , 2128 large number, probability 1 − 1/e ≈ 63.21%.

of course, there no randomness involved – either there fixed point or there isn't. but, can 63.21% confident there fixed point. (also, notice number not depend on size of keyspace – if md5 sums 32 bits or 1024 bits, answer same, long it's larger 4 or 5 bits).


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