Okamoto–Uchiyama cryptosystem

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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.

Scheme definition[edit]

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.

Key generation[edit]

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 (ngh) and the private key is the factors (pq).

Message encryption[edit]

To encrypt a message m, where m is taken to be an element in

  • Select at random. Set

Message decryption[edit]

If we define , then decryption becomes

How the system works[edit]

The group


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.

We have

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.