John B. Baena

ZHFE, a New Multivariate Public Key Encryption Scheme

In this paper we propose a new multivariate public key encryption scheme named ZHFE. The public key is constructed using as core map two high rank HFE polynomials. The inversion of the public key is performed using a low degree polynomial of Hamming weight three. This low degree polynomial is obtained from the two high rank HFE polynomials, by means of a special reduction method that uses Hamming weight three polynomials produced from the two high rank HFE polynomials. We show that ZHFE is relatively efficient and that it is secure against the main attacks that have threatened the security of HFE. We also propose parameters for a practical implementation of ZHFE.

Square, a New Multivariate Encryption Scheme

We propose and analyze a multivariate encryption scheme that uses odd characteristic and an embedding in its construction. This system has a very simple core map F (X ) = X 2, allowing for efficient decryption. We also discuss ways to make this decryption faster with specific parameter choices. We give heuristic arguments along with experimental data to show that this scheme resists all known attacks.

Efficient ZHFE Key Generation

In this paper we present a new algorithm to construct the keys of the multivariate public key encryption scheme ZHFE. Constructing ZHFEs trapdoor involves finding a low degree polynomial of q-Hamming-weight-three, as an aid to invert a pair of q-Hamming-weight-two polynomials of high degree and high rank. This is done by solving a large sparse linear system of equations. We unveil the combinatorial structure of the system in order to reveal the hidden structure of the matrix associated with it. When the systems variables and equations are organized accordingly, an almost block diagonal shape emerges. We then exploit this shape to solve the system much faster than when ZHFE was first proposed. The paper presents the theoretical details explaining the structure of the matrix. We also present experimental data that confirms the notable improvement of the key generation complexity, which makes ZHFE more suitable for practical implementations.

Rank Analysis of Cubic Multivariate Cryptosystems.

In this work we analyze the security of cubic cryptographic constructions with respect to rank weakness. We detail how to extend the big field idea from quadratic to cubic, and show that the same rank defect occurs. We extend the min-rank problem and propose an algorithm to solve it in this setting. We show that for fixed small rank, the complexity is even lower than for the quadratic case. However, the rank of a cubic polynomial in n variables can be larger than n, and in this case the algorithm is very inefficient. We show that the rank of the differential is not necessarily smaller, rendering this line of attack useless if the rank is large enough. Similarly, the algebraic attack is exponential in the rank, thus useless for high rank.