The Okamoto–Uchiyama cryptosystem was discovered in 1998 by Tatsuaki Okamoto and Shigenori Uchiyama. The system works in the multiplicative group of integers modulo n, , where n is of the form p2q and p and q are large primes.
Like many public key cryptosystems, this scheme works in the group . A fundamental difference of this cryptosystem is that here n is an integer of the form p2q, where p and q are large primes. This scheme is homomorphic and hence malleable.
A public/private key pair is generated as follows:
- Generate large primes p and q and set .
- Choose such that .
- Let h = gn mod n.
The public key is then (n, g, h) and the private key is the factors (p, q).
To encrypt a message m, where m is taken to be an element in
- Select at random. Set
If we define , then decryption becomes
How the system works
The group has a unique subgroup of order p, call it H. By the uniqueness of H, we must have
For any element x in , we have xp−1 mod p2 is in H, since p divides xp−1 − 1.
The map L should be thought of as a logarithm from the cyclic group H to the additive group , and it is easy to check that L(ab) = L(a) + L(b), and that the L is an isomorphism between these two groups. As is the case with the usual logarithm, L(x)/L(g) is, in a sense, the logarithm of x with base g.
So to recover m we just need to take the logarithm with base gp−1, which is accomplished by
The security of the entire message can be shown to be equivalent to factoring n. The semantic security rests on the p-subgroup assumption, which assumes that it is difficult to determine whether an element x in is in the subgroup of order p. This is very similar to the quadratic residuosity problem and the higher residuosity problem.