Abstract
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.
Highlights
Public key cryptography [1] built from the NP-hardness of solving multivariate quadratic equations over finite filed [2, 3] was conceived as a plausible candidate to traditional factorization and discrete logarithm based public key cryptosystems due to its high performance and the resistance to quantum attacks [4]
We first note that the hidden field equations (HFE) scheme [5] was proposed by Patarin to thwart the linearization equations attack and no known evidence was reported on the existence of linearization equations in the HFE scheme
To illustrate why the proposed modification of the HFE scheme is secure against the MinRank attack [7, 8], we just need to show that when lifted to the extension field F3n, the quadratic part of the public key Q is not connected with a low-rank matrix
Summary
Public key cryptography [1] built from the NP-hardness of solving multivariate quadratic equations over finite filed [2, 3] was conceived as a plausible candidate to traditional factorization and discrete logarithm based public key cryptosystems due to its high performance and the resistance to quantum attacks [4]. The hidden field equations (HFE) scheme [5] may be the most famous cryptosystem amongst all multivariate public key cryptographic schemes. Two invertible affine transformations are applied to hide the special structure of the central map [2, 5]. All known modification methods only can impose partial nonlinear transformation on the special structure of the HFE central map, and they are still vulnerable to some attacks [15,16,17]. We can impose a fully nonlinear transformation on the central map of the HFE encryption scheme. It is shown that the modification can defend the known attacks including the MinRank attack, the linearization equations attack, and the direct algebraic attacks
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