Okamoto–Uchiyama cryptosystem


The Okamoto–Uchiyama cryptosystem is a public key cryptosystem proposed 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.

Background

Let be an odd prime. Define.
is a subgroup of with .
Define by
is a homomorphism between and the additive group : that is,. Since is bijective, it is an isomorphism.
One can now show the following as a corollary:
Let such that and for. Then
The corollary is a direct consequence of.

Operation

Like many public key cryptosystems, this scheme works in the group. This scheme is homomorphic and hence malleable.

Key generation

A public/private key pair is generated as follows:
  1. Generate two large primes and.
  2. Compute.
  3. Choose a random integer such that.
  4. Compute.
The public key is then and the private key is.

Encryption

A message can be encrypted with the public key as follows.
  1. Choose a random integer.
  2. Compute.
The value is the encryption of.

Decryption

An encrypted message can be decrypted with the private key as follows.
  1. Compute.
  2. Compute. and will be integers.
  3. Using the Extended Euclidean Algorithm, compute the inverse of modulo :
  4. :.
  5. Compute.
The value is the decryption of.

Example

Let and. Then. Select. Then.
Now to encrypt a message, we pick a random and compute.
To decrypt the message 43, we compute
And finally.

Proof of correctness

We wish to prove that the value computed in the last decryption step,, is equal to the original message. We have
So to recover we need to take the discrete logarithm with base. This can be done by applying, as follows.
By Fermat's little theorem,. Since one can write with. Then and the corollary from earlier applies:.

Security

Inverting the encryption function can be shown to be as hard as factoring n, meaning that if an adversary could recover the entire message from the encryption of the message they would be able to factor 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.