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

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

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

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

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

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

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

Cryptanalysis of the multivariate encryption scheme EFLASH

Post-Quantum Cryptography studies cryptographic algorithms that quantum computers cannot break. Recent advances in quantum computing have made this kind of cryptography necessary, and research in the field has surged over the last years as a result. One of the main families of post-quantum cryptographic schemes is based on finding solutions of a polynomial system over finite fields. This family, known as multivariate cryptography, includes both public key encryption and signature schemes. The majority of the research contribution of this thesis is devoted to understanding the security of multivariate cryptography. We mainly focus on big field schemes, i.e., constructions that utilize the structure of a large extension field. One essential contribution is an increased understanding of how Gröbner basis algorithms can exploit this structure. The increased knowledge furthermore allows us to design new attacks in this setting. In particular, the methods are applied to two encryption schemes suggested in the literature: EFLASH and Dob. We show that the recommended parameters for these schemes will not achieve the proposed 80-bit security. Moreover, it seems unlikely that there can be secure and efficient variants based on these ideas. Another contribution is the study of the effectiveness and limitations of a recently proposed rank attack. Finally, we analyze some of the algebraic properties of MiMC, a block cipher designed to minimize its multiplicative complexity.Doktorgradsavhandlin

A Nonlinear Multivariate Cryptosystem Based on a Random Linear Code

We introduce a new technique for building multivariate encryption schemes based on random linear codes. The construction is versatile, naturally admitting multiple modifications. Among these modifications is an interesting embedding modifier--- any efficiently invertible multivariate system can be embedded and used as part of the inversion process. In particular, even small scale secure multivariate signature schemes can be embedded producing reasonably efficient encryption schemes. Thus this technique offers a bridge between multivariate signatures, many of which have remained stable and functional for many years, and multivariate encryption, a historically more troubling area

Eliminating Decryption Failures from the Simple Matrix Encryption Scheme

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