It is well known that quantum computers are a threat for the current cryp- tosystem. In this direction, the National Institute of Standards and Technology (alias NIST) decided to start a competition for select algorithms that are “quan- tum resistant”. This has been translated into finding mathematical problems on which base such cryptosystems. Nowadays, the most promising algorithms are based on

• Lattices;
• Multivariate Polynomial Systems; • Coding Theory;
• Hash Functions.

In particular, we are focused on Multivariate Quadratic (MQ) problem that had good results in the NIST competitition, although in the last call “Rainbow” (a multivariate digital signature scheme) has been discarded. Analyzing the timeline of the MQ problem, we believed that is possible build a MQ-system with trapdoor, obtained by mixing of OV-type equations and fully quadratic ones, such that, up to the current knowledge, there is no known attack running in a time bounded by polynomial function of the decryption time. This system, called OliVier, could be used for a public encryption scheme, as we proposed in our paper, but maybe it could be also used for digital signature scheme.