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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 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.
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