Oil and Vinegar and Multivariate Cryptography Methods Come to the Fore

Here’s my 6 am doodles from this morning on multivariate cryptography:

Our existing methods of public-key encryption — such as discrete logs, RSA and elliptic curves — are known not to be a hard problem in a world of quantum computers. But, multivariate cryptography is a known hard problem and is robust against quantum computers. Examples of methods that use multivariate cryptography are Oil-and-Vinegar, Unbalanced Oil and Vinegar, and Rainbow. In the current round of PQC standardisation for additional signatures, we see 10 multivariate signatures: [here]:

• Multivariate Signatures (10): 3WISE, DME-Sign, HPPC (Hidden Product of Polynomial Composition), MAYO, PROV (PRovable unbalanced
  Oil and Vinegar), QR-UOV, SNOVA, TUOV (Triangular Unbalanced
  Oil and Vinegar), UOV (Unbalanced Oil and Vinegar), and VOX.

Multivariate cryptography

With multivariate cryptography, we have n variables within polynomial equations. For example, if we have four variables (w, x, y, z) and an order of two, we could have [here]: