International Association
for Cryptologic Research

The Unbalanced Oil and Vinegar construction (UOV) has been the backbone of multivariate cryptography since the fall of HFE-based schemes. In fact, 7 UOV-based schemes have been submitted to the NIST additional call for signatures, and 4 of these made it to the second round. For efficiency considerations, most of these schemes are defined over a field of characteristic 2. This has as a side effect that the polar forms of the UOV public maps are not only symmetric, but also alternating.

In this work, we propose a new key-recovery attack on UOV in characteristic 2 that makes use of this property. We consider the polar forms of the UOV public maps as elements of the exterior algebra. We show that these are contained in a certain subspace of the second exterior power that is dependent on the oil space. This allows us to define relations between the polar forms and the image of the dual of the oil space under the Plücker embedding. With this, we can recover the secret oil space using sparse linear algebra.

This new attack has an improved complexity over previous methods and reduces the security by 4, 11, and 20 bits for uov-Ip, uov-III, and uov-V, respectively. Furthermore, the attack is applicable to MAYO$_2$ and improves on the best attack by 28 bits.