Higher residuosity problem
In cryptography, most public key cryptosystems are founded on problems that are believed to be intractable. The higher residuosity problem (also called the n th-residuosity problem) is one such problem. This problem is easier to solve than integer factorization, so the assumption that this problem is hard to solve is stronger than the assumption that integer factorization is hard.
From the isomorphism above, we have
as an isomorphism of groups. Since p and q were assumed to be prime, the groups and are cyclic of orders p-1 and q-1 respectively. If d is a divisor of p-1, then the set of dth powers in form a subgroup of index d. If gcd(d,q-1) = 1, then every element in is a dth power, so the set of dth powers in is also a subgroup of index d. In general, if gcd(d,q-1) = g, then there are (q-1)/(g) dth powers in , so the set of dth powers in has index dg. This is most commonly seen when d=2, and we are considering the subgroup of quadratic residues, it is well known that exactly one quarter of the elements in are quadratic residues (when n is the product of exactly two primes, as it is here).
The important point is that for any divisor d of p-1 (or q-1) the set of dth powers forms a subgroup of
Given an integer n = pq where p and q are unknown, an integer d such that d divides p-1, and an integer x < n, it is infeasible to determine whether x is a dth power (equivalently dth residue) modulo n.
Notice that if p and q are known it is easy to determine whether x is a dth residue modulo n because x will be a dth residue modulo p if and only if
When d=2, this is called the quadratic residuosity problem.
- Zhang, Yuliang; Tsutomu Matsumoto; Hideki Imai (1988). "Cryptographic Applications of th-Residuosity Problem with an Odd Integer". Transactions of the IEICE. 71 (8): 759–767. Retrieved 14 February 2014.