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How to prove all NP-statements in zero-knowledge, and a methodology of cryptographic protocol design
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Protocols are presented allowing someone with a secret discrete logarithm to release it, bit by bit, such that anyone can verify each bit's correctness as they receive it. This new notion of release of secrets generalizes and extends that of the already known exchange of secrets protocols. Consequently, the protocols presented allow exchange of secret discrete logs between any number of parties.The basic protocol solves an even more general problem than that of releasing a discrete log. Given any instance of a discrete log problem in a group with public group operation, the party who knows the solution can make public some interval I and convince anyone that the solution belongs to I, while releasing no additional information, such as any hint as to where in I the solution is.This can be used directly to release a discrete log, or to transfer it securely between different groups, i.e. prove that two instances are related such that knowledge of the solution to one implies knowledge of the solution to the other.We show how this last application can be used to implement a more efficient release protocol by transferring the given discrete log instance to a group with special properties. In this scenario, each bit of the secret can be verified by a single modular squaring, and unlike the direct use of the basic protocol, no interactive proofs are needed after the basic setup has been done.Finally, it is shown how the basic protocol can be used to release the factorization of a public composite number.