The complexity of MinRank

In this note, we leverage some of our results from arXiv:1706.06319 to produce a concise and rigorous proof for the complexity of the generalized MinRank Problem in the under-defined and well-defined case. Our main theorem recovers and extends previous results by Faug\`ere, Safey El Din, Spaenlehauer (arXiv:1112.4411).Comment: Corrected a typo in the formula of the main theore

Solving multivariate polynomial systems and an invariant from commutative algebra

The complexity of computing the solutions of a system of multivariate polynomial equations by means of Gr\"obner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate the solving degree of a system, focusing on systems arising within public-key cryptography. In particular, we show that it is upper bounded by, and often equal to, the Castelnuovo Mumford regularity of the ideal generated by the homogenization of the equations of the system, or by the equations themselves in case they are homogeneous. We discuss the underlying commutative algebra and clarify under which assumptions the commonly used results hold. In particular, we discuss the assumption of being in generic coordinates (often required for bounds obtained following this type of approach) and prove that systems that contain the field equations or their fake Weil descent are in generic coordinates. We also compare the notion of solving degree with that of degree of regularity, which is commonly used in the literature. We complement the paper with some examples of bounds obtained following the strategy that we describe

EFLASH: A New Multivariate Encryption Scheme

Multivariate Public Key Cryptography is a leading option for security in a post quantum society. In this paper we propose a new encryption scheme, EFLASH, and analyze its efficiency and security

Cryptanalysis of Simple Matrix Scheme for Encryption

Recently, Tao et al. presented a new simple and efficient multivariate pubic key encryption scheme based on matrix multiplica- tion, which is called Simple Matrix Scheme or ABC. Using linearization method, we propose a polynomial time algorithm, which directly solves an equivalent private key from the public key of ABC. Furthermore, our attack can also be applied to the variants of ABC since these variants have the same algebraic structure as the ABC scheme. Therefore, the ABC cryptosystem and its variants are insecure

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

Combinatorial Rank Attacks Against the Rectangular Simple Matrix Encryption Scheme

In 2013, Tao et al. introduced the ABC Simple Matrix Encryption Scheme, a multivariate public key encryption scheme. The scheme boasts great efficiency in encryption and decryption, though it suffers from very large public keys. It was quickly noted that the original proposal, utilizing square matrices, suffered from a very bad decryption failure rate. As a consequence, the designers later published updated parameters, replacing the square matrices with rectangular matrices and altering other parameters to avoid the cryptanalysis of the original scheme presented in 2014 by Moody et al. In this work, we show that making the matrices rectangular, while decreasing the decryption failure rate, actually, and ironically, diminishes security. We show that the combinatorial rank methods employed in the original attack of Moody et al. can be enhanced by the same added degrees of freedom that reduce the decryption failure rate. Moreover, and quite interestingly, if the decryption failure rate is still reasonably high, as exhibited by the proposed parameters, we are able to mount a reaction attack to further enhance the combinatorial rank methods. To our knowledge this is the first instance of a reaction attack creating a significant advantage in this context

A study of big field multivariate cryptography.

As the world grapples with the possibility of widespread quantum computing, the cryptosystems of the day need to be up to date. Multivariate Public Key Cryptography is a leading option for security in a post quantum society. One goal of this work is to classify the security of multivariate schemes, especially C*variants. We begin by introducing Multivariate Public Key Cryptography and will then discuss different multivariate schemes and the main types of attacks that have been proven effective against multivariate schemes. Once we have developed an appropriate background, we analyze security of different schemes against particular attacks. Specifically, we will analyze differential security of HFEv- and PFLASH schemes. We then introduce a variant of C* that may be used as an encryption scheme, not just as a signature scheme. Finally, we will analyze the security and efficiency of a (n,d,s,a,p,t) scheme in general. This allows for individuals to generally discuss security and performance of any C* variant

Developments in multivariate post quantum cryptography.

Ever since Shor\u27s algorithm was introduced in 1994, cryptographers have been working to develop cryptosystems that can resist known quantum computer attacks. This push for quantum attack resistant schemes is known as post quantum cryptography. Specifically, my contributions to post quantum cryptography has been to the family of schemes known as Multivariate Public Key Cryptography (MPKC), which is a very attractive candidate for digital signature standardization in the post quantum collective for a wide variety of applications. In this document I will be providing all necessary background to fully understand MPKC and post quantum cryptography as a whole. Then, I will walk through the contributions I provided in my publications relating to differential security proofs for HFEv and HFEv−, key recovery attack for all parameters of HFEm, and my newly proposed multivariate encryption scheme, HFERP

Cubic multivariate cryptosystems based on big field constructions and their vulnerability to a min-rank attack

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.Resumen: En este trabajo analizamos la seguridad de construcciones criptogr´aficas c´ubicas con respecto a la debilidad del rango. Detallamos c´omo extender la idea de campo grande de cuadr´atico a c´ubico, y mostramos que la misma ca´ıda de rango ocurre. Extendemos el problema de rango m´ınimo y proponemos un algoritmo para resolverlo en este contexto. Mostramos que para rango bajo fijo, la complejidad es incluso m´as baja que en el caso cuadr´atico. Sin embargo, el rando de un polinomio c´ubico en n variables puede ser m´as grande que n, y en este caso el algoritmo es muy ineficiente. Mostramos que el rango del diferencial no es necesariamente m´as peque˜no, lo cual vuelve in´util esta l´ınea de ataque si el rango es lo suficientemente grande. Similarmente, el ataque algebr´aico es exponencial en el rango, y por lo tanto es in´util para rango alto.Maestrí