Several variants of Rainbow or UOV using sparse private keys have been proposed to reduce the private key size.
As UOV can be considered as a single layer Rainbow, these two methods can also be applied to UOV.
Our Contributions: In this paper, we propose Circulant UOV with higher signature generating efficiency and shorter private key.
In this section, we will introduce UOV and its variants.
UOV is a modified version of the Oil and Vinegar scheme designed by J.
To figure out what UOV is, first of all, we'd like to introduce the concept of Oil-Vinegar polynomial with the following form:
For example, in the recent paper  the authors undertook an attempt to reduce the public key size, based on yet unbroken (under proper parameter choice) Unbalanced Oil and Vinegar (UOV) scheme .
Traditionally, basing on the four basic schemes MI , HFE , STS  and UOV , the MQ public key cryptosystems are divided in four groups.
For "generic" quadratic systems, experts believe that the Unbalanced Oil-Vinegar (UOV) scheme  is a feasible signature scheme with good efficiency and acceptable security.
In Section 2, we describe the preliminaries about hyper-spheres, signature models, multivariate cryptography and basic UOV signature scheme.
The UOV scheme is a single field construction, so we work solely in the polynomial ring K[X], where X = [x.sub.1],...,[x.sub.n].
The public key P of the UOV scheme consists of o quadratic polynomials in n variables: