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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, ( Z / n Z ) ∗ , where n is of the form p2q and p and q are large primes.

Contents

Scheme definition

Like many public key cryptosystems, this scheme works in the group ( Z / n 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 ∈ ( Z / n Z ) ∗ such that g p − 1 ≠ 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 2 k − 1

  • Select r ∈ Z / n Z at random. Set

    • Message decryption

    If we define L ( x ) = x − 1 p , then decryption becomes

    m = L ( C p − 1 mod p 2 ) L ( g p − 1 mod p 2 ) mod p

    How the system works

    The group

    ( Z / n Z ) ∗ ≃ ( Z / p 2 Z ) ∗ × ( Z / q Z ) ∗ .

    The group ( Z / p 2 Z ) ∗ has a unique subgroup of order p, call it H. By the uniqueness of H, we must have

    H = { x : x p ≡ 1 mod p } .

    For any element x in ( Z / p 2 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 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

    ( g m h r ) p − 1 = ( g m g n r ) p − 1 = ( g p − 1 ) m g p ( p − 1 ) r p q = ( 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

    L ( ( g p − 1 ) m ) 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 ( Z / n Z ) ∗ is in the subgroup of order p. This is very similar to the quadratic residuosity problem and the higher residuosity problem.

    References

    Okamoto–Uchiyama cryptosystem Wikipedia
    (Text) CC BY-SA