Unbalanced Oil and Vinegar Signature Schemes

Abstract

Abstract. In [16], J. Patarin designed a new scheme, called "Oil and Vinegar", for computing asymmetric signatures. It is very simple, can be computed very fast (both in secret and public key) and requires very little RAM in smartcard implementations. The idea consists in hiding quadratic equations in n unknowns called "oil" and v = n unknowns called "vinegar" over a finite field K, with linear secret functions. This original scheme was broken in [10] by A. Kipnis and A. Shamir. In this paper, we study some very simple variations of the original scheme where v > n (instead of v = n). These schemes are called "Unbalanced Oil and Vinegar" (UOV), since we have more "vinegar" unknowns than "oil" unknowns. We show that, when v n, the attack of [10] can be extended, but when v ≥ 2n for example, the security of the scheme is still an open problem. Moreover, when v n 2 2 , the security of the scheme is exactly equivalent (if we accept a very natural but not proved property) to the problem of solving a random set of n quadratic equations in n 2 2 unknowns (with no trapdoor). However, we show that (in characteristic 2) when v ≥ n 2 , finding a solution is generally easy. Then we will see that it is very easy to combine the Oil and Vinegar idea and the HFE schemes of [14]. The resulting scheme, called HFEV, looks at the present also very interesting both from a practical and theoretical point of view.

A Practical Multivariate Blind Signature Scheme

“…As the other MQ signature schemes rely on the same construction, our technique applies to those cryptosystems as well. We do not use any property of Rainbow that is not shared by, e.g., HFEv − [24], pC * [7], or UOV [15]. The exceptions are the MQ signature schemes that do not have the construction of Fig.…”

“…Multivariate schemes are in general very fast and require only modest computational resources, which makes them attractive for the use on low cost devices like smart cards and RFID chips [5,6]. However, while there exist many practical multivariate standard signature schemes such as UOV [15], Rainbow [9] and Gui [24], there is a lack of multivariate signature schemes with special properties such as blind, ring, and group signatures. Blind signature schemes allow a user, who is not in charge of the private signing key, to obtain a signature for a message d by interacting with the signer.…”

Improved Algebraic Cryptanalysis of QUAD, Bivium and Trivium via Graph Partitioning on Equation Systems

“…Poisoned Equations and QUAD One could imagine the following scenario, which is inspired by Jacques Patarin's system "Oil and Vinegar" [38]. A malicious manufacturer does not generate the system at random, but rather creates a system that is sparse and has vertex connectivity of 20, for some vertex partition with β ≈ 0.6.…”

Comparison Between XL and Gröbner Basis Algorithms

“…They try to recover a secret key by solving a system of algebraic equations. Algebraic attacks were first applied to Matsumoto-Imai Public Key Scheme in [19] by Jacques Patarin and a similar attack was also applied in [15]. Algebraic attacks were also applied to block ciphers in [6], where the complexity for attacking AES and Serpent was evaluated.…”