Abstract

The authors describe the use of a mechanical theorem-prover to check the published proof of the invertibility of the public key encryption algorithm of Rivest, Shamir and Adleman: (M mod n) mod N=M, provided n is the product of two distinct primes p and q, M<n, and e and d are multiplicative inverses in the ring of integers modulo (p-1)*(q-1). Among the lemmas proved mechanically and used in the main proof are many familiar theorems of number theory, including Fermat’s theorem: M mod p=1, when p M. The axioms underlying the proofs are those of Peano arithmetic and ordered pairs. The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules.-- Godel [11] But formalized mathematics cannot in practice be written down in full, and therefore we must have confidence in what might be called the common sense of the mathematician... We shall therefore very quickly abandon formalized mathematics...-- Bourbaki [1] 1.

Citations