“Okamoto–Uchiyama cryptosystem”的意思、由来-开放百科全书

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Okamoto–Uchiyama cryptosystem

释义

Scheme definition
Key generation Message encryption Message decryption
How the system works
Security
References

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

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

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 (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

Select at random. Set

Message decryption

If we define , then decryption becomes

How the system works

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 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 , 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 g^p−1, which is accomplished by

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 is in the subgroup of order p. This is very similar to the quadratic residuosity problem and the higher residuosity problem.

References

{{cite book |first=Tatsuaki |last=Okamoto |first2=Shigenori |last2=Uchiyama |chapter=A new public-key cryptosystem as secure as factoring |title=Advances in Cryptology — EUROCRYPT'98 |series=Lecture Notes in Computer Science |volume=1403 |location= |publisher=Springer |year=1998 |pages=308–318 |doi=10.1007/BFb0054135 }}

1 : Public-key encryption schemes

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