(The Infinity Machine is really interesting, hope you don't mind me picking at it...)
Possible hash algorithm:
Find a mapping from integers to irrational numbers in [0,1). For example, find
the Nth prime, take its square root, subtract the integer part. Compute an
infinite number of bits of this number and send them as your hash. The
recipient does the same hashing his copy of the message. As more bits arrive,
the probability that you both have the same message approaches 1. However, no
matter how many bits have arrived, the number of possible integers that could
have hashed to the same value is still countably infinite.
It's still useless in the face of infinite active attackers, all of
whom can send you an infinite number of copies of all possible messages in all
possible orders in a finite amount of wall time such that you can't
distinguish them from "legitimate" messages. Sooner or later I'm going to send
the correct "transfer $1,000,000" to Mallet's bank account" message and you've
lost.
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Possible hash algorithm:
Find a mapping from integers to irrational numbers in [0,1). For example, find the Nth prime, take its square root, subtract the integer part. Compute an infinite number of bits of this number and send them as your hash. The recipient does the same hashing his copy of the message. As more bits arrive, the probability that you both have the same message approaches 1. However, no matter how many bits have arrived, the number of possible integers that could have hashed to the same value is still countably infinite.
It's still useless in the face of infinite active attackers, all of whom can send you an infinite number of copies of all possible messages in all possible orders in a finite amount of wall time such that you can't distinguish them from "legitimate" messages. Sooner or later I'm going to send the correct "transfer $1,000,000" to Mallet's bank account" message and you've lost.