Multivariate cryptography

Multivariate cryptography is the generic term for asymmetric cryptographic primitives based on multivariate polynomials over a finite field . In certain cases those polynomials could be defined over both a ground and an extension field. If the polynomials have the degree two, we talk about multivariate quadratics. Solving systems of multivariate polynomial equations is proven to be NP-hard or NP-complete. That's why those schemes are often considered to be good candidates for post-quantum cryptography. As of today, multivariate quadratics have been used to build signature schemes, but all attempts to build a secure encryption scheme have failed.

History

In 1988 T. Matsumoto and H. Imai presented their scheme "Matsumoto-Imai-Scheme" on the Eurocrypt conference. On later work the "Hidden Monomial Cryptosystems" was developed by (French) Jacques Patarin. It is based on a ground and an extension field. On this "Hidden Field Equations" was designed and presented in 1996. In the following years J. Patarin developed other schemes. In 1997 he presented “Balanced Oil & Vinegar” and 1999 “Unbalanced Oil and Vinegar” in cooperation with Aviad Kipnis and Louis Goubin.

Construction

Multivariate Quadratics involves a public and a private key. The private key consists of two affine transformations, S and T, and an easy to invert quadratic map P’ . We denote the by matrix of the affine endomorphisms by and the shift vector by and similarly for . In other words,

The triple is the private key, also known as the trapdoor. The public key is the composition which is by assumption hard to invert without the knowledge of the trapdoor.

Signature

Signatures are generated using the private key and are verified using the public key as follows. The message is hashed to a vector in via a known hash function. The signature is .

The receiver of the signed document must have the public key P in possession. He computes the hash y and checks that the signature x fulfils .

Applications

References

Multivariate Quadratic equations; Current Version: 2005-12-15

External links

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