Hidden field equation (HFE) multivariable cryptosystems were first suggested by Patarin. Kipnis and Shamir showed that to make the cryptosystem secure, a special parameter D of any HFE cryptosystem can not be too small. Consequently Kipnis, Patarin and Goubin proposed an enhanced variant of the HFE cryptosystem by combining the idea of Oil and Vinegar construction with the HFE construction. Essentially they “perturb” the HFE system with some external variables. In this paper, we will first present a new cryptanalysis method for the HFEv schemes. We then use the idea of internal perturbation to build a new cryptosystem, an internally perturbed HFE cryptosystem (IPHFE).

A review of the current state of multivariate public-key cryptosystems compares and contrasts the most promising multivariate schemes in digital signatures and public-key encryption as well as their security.

In 2017, Ward Beullens et al. submitted Lifted Unbalanced Oil and Vinegar (LUOV) [4], a signature scheme based on the famous multivariate public key cryptosystem (MPKC) called Unbalanced Oil and Vinegar (UOV), to NIST for the competition for post-quantum public key scheme standardization. The defining feature of LUOV is that, though the public key P works in the extension field of degree r of F_2, the coefficients of P come from F_2. This is done to significantly reduce the size of P. The LUOV scheme is now in the second round of the NIST PQC standardization process. In this paper we introduce a new attack on LUOV. It exploits the “lifted” structure of LUOV to reduce direct attacks on it to those over a subfield. We show that this reduces the complexity below the targeted security for the NIST post-quantum standardization competition.

The SimpleMatrix encryption scheme as proposed by Tao et al. \cite{TD13} is one of the very few existing approaches to create a secure and efficient encryption scheme on the basis of multivariate polynomials. However, in its basic version, decryption failures occur with non-negligible probability. Although this problem has been addressed in several papers \cite{DP14,TX15}, a general solution to it is still missing. In this paper we propose an improved version of the SimpleMatrix scheme, which eliminates decryption failures completely and therefore solves the biggest problem of the SimpleMatrix encryption scheme. Additionally, we propose a second version of the scheme, which reduces the blow-up factor between plain and ciphertext size to a value arbitrary close to 1.

We consider a two-dimensional particle of charge e interacting with a homogeneous magnetic eld perpendicular to the plane and a potential well which is transported along a closed loop in the plane. We show that a bound state corresponding to a simple isolated eigenvalue acquires at that Berry's phase equal to 2 sgne times the number of ux quanta through the oriented area encircled by the loop. We also argue that this is a purely quantum e ect since the corresponding Hannay angle is zero.