The Okamoto–Uchiyama cryptosystem was discovered in 1998 by T. Okamoto and S. Uchiyama. The system works in the group ${\displaystyle (\mathbb{Z}/n\mathbb{Z})^*}$, 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 ${\displaystyle (\mathbb{Z}/n\mathbb{Z})^*}$. 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 two ${\displaystyle k}$ bits primes p and q and set ${\displaystyle n=p^2 q}$.
• Choose ${\displaystyle g \in (\mathbb{Z}/n\mathbb{Z})^*}$ such that ${\displaystyle g^p \neq 1 \mod p^2}$ and ${\displaystyle g^{p-1} \neq \bmod{p^2}}$.
• Let h = gn mod n.

The public key is then (ngh, k) and the private key is the factors (pq).

Message encryption

To encrypt a message m, where m is taken to be an integer and ${\displaystyle 0< m < 2^{(k-1)}}$

• Select ${\displaystyle r \in \mathbb{Z}/n\mathbb{Z}}$ at random. Set
${\displaystyle C = g^m h^r \mod n}$

Message decryption

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

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

How the system works

The group

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

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

${\displaystyle H = \{ x\in (\mathbb{Z}/q^2\mathbb{Z})^* : x \equiv 1 \mod p \}}$.

For any element x in ${\displaystyle (\mathbb{Z}/p^2\mathbb{Z})^*}$, 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 ${\displaystyle \Z/p\Z}$, 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

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

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

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

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