Analysis of Intermediate Field Systems

We study a new generic trapdoor for public key multivariate cryptosystems, called IFS for Intermediate Field Systems, which can be seen as dual to HFE. This new trapdoor relies on the possibility to invert a system of quadratic multivariate equations with few (logarithmic with respect to the security parameter) unknowns on an intermediate field thanks to Groebner bases algorithms. We provide a comprehensive study of the security of this trapdoor and show that it is equivalent to the security provided by HFE. Therefore, while insecure in its basic form, this trapdoor may reveal quite attractive when used with, e.g., the minus modifier

Building Secure Public Key Encryption Scheme from Hidden Field Equations

Multivariate public key cryptography is a set of cryptographic schemes built from the NP-hardness of solving quadratic equations over finite fields, amongst which the hidden field equations (HFE) family of schemes remain the most famous. However, the original HFE scheme was insecure, and the follow-up modifications were shown to be still vulnerable to attacks. In this paper, we propose a new variant of the HFE scheme by considering the special equation x2=x defined over the finite field F3 when x=0,1. We observe that the equation can be used to further destroy the special structure of the underlying central map of the HFE scheme. It is shown that the proposed public key encryption scheme is secure against known attacks including the MinRank attack, the algebraic attacks, and the linearization equations attacks. The proposal gains some advantages over the original HFE scheme with respect to the encryption speed and public key size

Extension Field Cancellation: a New Central Trapdoor for Multivariate Quadratic Systems

This paper introduces a new central trapdoor for multivariate quadratic (MQ) public-key cryptosystems that allows for encryption, in contrast to time-tested MQ primitives such as Unbalanced Oil and Vinegar or Hidden Field Equations which only allow for signatures. Our construction is a mixed-field scheme that exploits the commutativity of the extension field to dramatically reduce the complexity of the extension field polynomial implicitly present in the public key. However, this reduction can only be performed by the user who knows concise descriptions of two simple polynomials, which constitute the private key. After applying this transformation, the plaintext can be recovered by solving a linear system. We use the minus and projection modifiers to inoculate our scheme against known attacks. A straightforward C++ implementation confirms the efficient operation of the public key algorithms

Small Odd Prime Field Multivariate PKCs

We show that Multivariate Public Key Cryptosystems (MPKCs) over fields of small odd prime characteristic, say 31, can be highly efficient. Indeed, at the same design security of $2^{80}$ under the best known attacks, odd-char MPKC is generally faster than prior MPKCs over \GF{2^k}, which are in turn faster than ``traditional\u27\u27 alternatives. This seemingly counter-intuitive feat is accomplished by exploiting the comparative over-abundance of small integer arithmetic resources in commodity hardware, here embodied by SSE2 or more advanced special multimedia instructions on modern x86-compatible CPUs. We explain our implementation techniques and design choices in implementing our chosen MPKC instances modulo small a odd prime. The same techniques are also applicable in modern FPGAs which often contains a large number of multipliers

Linearity Measures for MQ Cryptography

We propose a new general framework for the security of multivariate quadratic (\mathcal{MQ}) schemes with respect to attacks that exploit the existence of linear subspaces. We adopt linearity measures that have been used traditionally to estimate the security of symmetric cryptographic primitives, namely the nonlinearity measure for vectorial functions introduced by Nyberg at Eurocrypt \u2792, and the $(s, t)$--linearity measure introduced recently by Boura and Canteaut at FSE\u2713. We redefine some properties of \mathcal{MQ} cryptosystems in terms of these known symmetric cryptography notions, and show that our new framework is a compact generalization of several known attacks in \mathcal{MQ} cryptography against single field schemes. We use the framework to explain various pitfalls regarding the successfulness of these attacks. Finally, we argue that linearity can be used as a solid measure for the susceptibility of \mathcal{MQ} schemes to these attacks, and also as a necessary tool for prudent design practice in \mathcal{MQ} cryptography