Title, Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. Booktitle, Advances in Cryptology – CRYPTO ’99, 19th Annual International. Download Citation on ResearchGate | Cryptanalysis of the HFE Public Key Finally, we develop a new relinearization method for solving such systems for any. Finally, we develop a new relinearization method for solving such systems for any constant ffl? Cryptanalysis of the HFE Public Key Cryptosystem ().

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Multivariate cryptography

The encryption of the original HFE relienarization is just to computewhere the plaintext is in but not necessarily in. The plaintext block also satisfies the field equation. If ; then we output as the plaintext. We impose some restrictions on the plaintext space and can use cryptanalsis restriction to merge the coefficients of the linear part and the square part.

Conclusions In this paper, we proposed a novel modified HFE encryption scheme. Considering the aforementioned discussions, we suggest choosing and. To illustrate why the proposed modification of the HFE scheme is secure against the Relinearizatioh attack [ 78 ], we just need to show that when lifted to the extension fieldthe quadratic part of the public key is not connected with a low-rank matrix.

This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We first review the basic idea of known attacks and then illustrate why the proposal is cryptosyxtem against these attacks.

We recalland denote the smallest integer smaller than or equal to asand we will find that all the elements of the last columns rows, resp. However, the original HFE crytposystem was insecure, and the follow-up modifications were shown to be still vulnerable to attacks.


However, some simple variants of HFE, such as the minus variant and the vinegar variant allow one to strengthen the basic HFE against all known attacks. In fact, the quadratic polynomial map is exactly the public key of the original HFE scheme, and the secret key of the original scheme also consists of, and.

We first note that the 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. So given a ciphertextwe only need to solve the linearization equations to obtain the corresponding plaintext.

El Din, and P. Security and Communication Networks. Multivariate Quadratics involves a public and a private key.

That is to say Or equivalently, The above equation says that we can lift the quadratic part of the public key to the extension field under some unknown linear transformations to derive and hence. Public key cryptography [ 1 ] built from the NP-hardness of solving multivariate quadratic equations over finite filed [ 23 ] 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 ].

Algebraic Cryptanalysis of GOST Encryption Algorithm

So some modifications are needed to repair the bu HFE scheme [ 10 — cryptosjstem ]. View at MathSciNet V. To receive news and publication updates for Security and Communication Networks, enter your email address in the box below.

Under the suggested parameters andthe degree of regularity of the quadratic equations is. Advanced Search Include Citations. We observe that the equation can be used to further destroy the special structure of the underlying central map of the HFE scheme.

Then two invertible affine transformations are applied to hide the special structure of the central map [ 25 ]. If the polynomials have the degree two, we talk about multivariate quadratics. In this paper, we propose a new variant of the HFE scheme by considering the special equation defined over the finite field when. This section does not cite any sources.


View at MathSciNet Y. The construction admits a standard isomorphism between the extension field and the vector space ; namely, for an elementwe have. These equations are called linearization equations and can be efficiently computed from the public polynomials. So the rank of the symmetric matrix is at most. Indexed in Science Citation Index Expanded. Correspondence should be addressed to Thee Wang ; moc. Thus we have some additional equations that associate with the plaintext ; namely, forwe have.

So both schemes have the same secret key sizes and decryption costs. Given a ciphertextwe want to recover the corresponding plaintext. Multivariate cryptography has been very productive in terms of design and cryptanalysis.

In certain cases those polynomials could be defined over both a ground and an extension field.

Overall, the situation is now more stable and the strongest schemes have withstood the test of time. By doing this, we can impose a fully nonlinear transformation on the central map of the HFE encryption scheme. However, we can derive the field equations from the equations.

We just observe thatso. Given the ciphertextwe want to solve the plaintext from the quadratic equations: Articles with French-language external links Articles needing additional references from August All articles needing additional references. If we lift to the extension field and find that the corresponding matrix is not of low rank, we can claim our proposal is secure against the MinRank attack [ 78 ].