3.1 Fingerprinting

Observe that if a = b then Bob will always be correct. However, if b 6= a then there may be an error: this happens iff the fingerprints of a and b happen to coincide. We now show that, even for a modest value of T (exponentially smaller than a and b), if a 6= b then Pr[Fp(a) = Fp(b)] is small. First observe that, if Fp(a) = Fp(b), then a = b mod p, so p must divide |a − b|. But |a − b| is an n-bit number, so the number of primes p that divide it is (crudely) at most n (each prime is at least 2). Thus the probability of error is at most n π(T ) , where π(x) is defined as the number of primes less than or equal to x. We now appeal to a standard result in Number Theory: