Okamoto–Uchiyama cryptosystem

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, $(\mathbb{Z}/n\mathbb{Z})^*$, where n is of the form p²q and p and q are large primes.

Scheme definition

Like many public key cryptosystems, this scheme works in the group $(\mathbb{Z}/n\mathbb{Z})^*$. A fundamental difference of this cryptosystem is that here n is a of the form p²q, where p and q are large primes. This scheme is homomorphic and hence malleable.

Key generation

A public/private key pair is generated as follows:

• Generate large primes p and q and set $n=p^2 q$.
• Choose $g \in (\mathbb{Z}/n\mathbb{Z})^*$ such that $g^p \neq 1 \mod p^2$.
• Let h = gⁿ mod n.

The public key is then (n, g, h) and the private key is the factors (p, q).

Message encryption

To encrypt a message m, where m is taken to be an element in $\mathbb{Z}/p\mathbb{Z}$

• Select $r \in \mathbb{Z}/n\mathbb{Z}$ at random. Set

$C = g^m h^r \mod n$

Message decryption

If we define $L(x) = \frac{x-1}{p}$, then decryption becomes

$m = \frac{L\left(C^{p-1} \mod p^2\right)}{L\left(g^{p-1} \mod p^2 \right)} \mod p$

How the system works

The group

$(\Z/n\Z)^* \simeq (\mathbb{Z}/p^2\mathbb{Z})^* \times (\mathbb{Z}/q\mathbb{Z})^*$.

The group $(\mathbb{Z}/p^2\mathbb{Z})^*$ has a unique subgroup of order p, call it H. By the uniqueness of H, we must have

$H = \{ x : x^p \equiv 1 \mod p \}$.

For any element x in $(\mathbb{Z}/p^2\mathbb{Z})^*$, we have x^p−1 mod p² is in H, since p divides x^p−1 − 1.

The map L should be thought of as a logarithm from the cyclic group H to the additive group $\mathbb{Z}/p\mathbb{Z}$, 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

$(g^mh^r)^{p-1} = (g^m g^{nr})^{p-1} = (g^{p-1})^m g^{p(p-1)rpq} = (g^{p-1})^m \mod p^2.$

So to recover m we just need to take the logarithm with base g^p−1, which is accomplished by

$\frac{L \left( (g^{p-1})^m \right) }{L(g^{p-1})} = m \mod p.$

Security

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 $(\mathbb{Z}/n\mathbb{Z})^*$ is in the subgroup of order p. This is very similar to the quadratic residuosity problem and the higher residuosity problem.