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

The Shortest Signatures Ever

Multivariate Cryptography is one of the main candidates for creating post quantum public key cryptosystems. Especially in the area of digital signatures, there exist many practical and secure multivariate schemes. In this paper we present a general technique to reduce the signature size of multivariate schemes. Our technique enables us to reduce the signature size of nearly all multivariate signature schemes by 10 to 15 % without slowing down the scheme significantly. We can prove that the security of the underlying scheme is not weakened by this modification. Furthermore, the technique enables a further reduction of the signature size when accepting a slightly more costly verification process. This trade off between signature size and complexity of the verification process can not be observed for any other class of digital signature schemes. By applying our technique to the Gui signature scheme, we obtain the shortest signatures of all existing digital signature schemes

On the Complexity of the Hybrid Approach on HFEv-

The HFEv- signature scheme is one of the most promising candidates for post-quantum digital signatures. Most notably here is the short signature size of the scheme. It has long been known that direct attacks against HFEv- systems work more efficiently than against random systems. The reason for this was found by Jintai Ding et al., who proved an upper bound on the degree of regularity of these systems. However, not much is known about the efficiency of the hybrid approach against the HFEv- scheme. In order to find suitable parameter sets for HFEv- for higher levels of security, this topic has to be studied in more detail. In this article we consider this question by performing a large number of computer experiments. As our experiments show, guessing variables does not help to speed up direct attacks against HFEv- systems. Therefore, in the parameter selection of these schemes, we do not have to consider the hybrid approach. Furthermore, we develop in this article a simple formula to estimate the degree of regularity of a determined HFEv- system. Together with our results on the behavior of the hybrid approach, this formula gives us an easy way to estimate the complexity of direct attacks against HFEv- systems

Improved Cryptanalysis of HFEv- via Projection

The HFEv- signature scheme is one of the most studied multivariate schemes and one of the major candidates for the upcoming standardization of post-quantum digital signature schemes. In this paper, we propose three new attack strategies against HFEv-, each of them using the idea of projection. Especially our third attack is very effective and is, for some parameter sets, the most efficient known attack against HFEv-. Furthermore, our attack requires much less memory than direct and rank attacks. By our work, we therefore give new insights in the security of the HFEv- signature scheme and restrictions for the parameter choice of a possible future standardized HFEv- instance

Implementing 128-bit Secure MPKC Signatures

Multivariate Public Key Cryptosystems (MPKCs) are often touted as future-proofing against Quantum Computers. In 2009, it was shown that hardware advances do not favor just ``traditional\u27\u27 alternatives such as ECC and RSA, but also makes MPKCs faster and keeps them competitive at 80-bit security when properly implemented. These techniques became outdated due to emergence of new instruction sets and higher requirements on security. In this paper, we review how MPKC signatures changes from 2009 including new parameters (from a newer security level at 128-bit), crypto-safe implementations, and the impact of new AVX2and AESNI instructions. We also present new techniques on evaluating multivariate polynomials, multiplications of large finite fields by additive Fast Fourier Transforms, and constant time linear solvers

Improved Key Recovery of the HFEv- Signature Scheme

The HFEv- signature scheme is a twenty year old multivariate public key signature scheme. It uses the Minus and the Vinegar modifier on the original HFE scheme. An instance of the HFEv- signature scheme called GeMSS is one of the alternative candidates for signature schemes in the third round of the NIST Post Quantum Crypto (PQC) Standardization Project. In this paper, we propose a new key recovery attack on the HFEv- signature scheme. We show that the Minus modification does not enhance the security of cryptosystems of the HFE family, while the Vinegar modification increases the complexity of our attack only by a polynomial factor. By doing so, we show that the proposed parameters of the GeMSS scheme are not as secure as claimed. Our attack shows that it is very difficult to build a secure and efficient signature scheme on the basis of HFEv-

On Roots Factorization for PQC Algorithms

In this paper we consider several methods for an efficient extraction of roots of a polynomial over large finite fields. The problem of computing such roots is often the performance bottleneck for some multivariate quantum-immune cryptosystems, such as HFEv-based Quartz, Gui, etc. We also discuss a number of techniques for fast computation of traces as part of the factorization process. These optimization methods could significantly improve the performance of cryptosystems where roots factorization is a part thereof

On the Security and Key Generation of the ZHFE Encryption Scheme

At PQCrypto\u2714 Porras, Baena and Ding proposed a new interesting construction to overcome the security weakness of the HFE encryption scheme, and called their new encryption scheme ZHFE. They provided experimental evidence for the security of ZHFE, and proposed the parameter set $(q,n,D)= (7,55,105)$ with claimed security level $2^{80}$ estimated by experiment. However there is an important gap in the state-of-the-art cryptanalysis of ZHFE, i.e., a sound theoretical estimation for the security level of ZHFE is missing. In this paper we fill in this gap by computing upper bounds for the Q-Rank and for the degree of regularity of ZHFE in terms of $\log_q D$, and thus providing such a theoretical estimation. For instance the security level of ZHFE(7,55,105) can now be estimated theoretically as at least $2^{96}$. Moreover for the inefficient key generation of ZHFE, we also provide a solution to improve it significantly, making almost no computation needed