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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Loosely speaking, when we apply two linear transformations on the input and output of the mapthe rank of the corresponding matrix remains at most. So the computational overhead is about bit operations. Performance and Comparisons To make a comparison between the proposed HFE modification and the original HFE schemes in a uniform platform, we consider the HFE scheme defined over and its extension field. Hence, forSo. So the adversary cannot derive from the publicly known map a low-rank matrix.

The plaintext space is. Table of Contents Alerts. This section does not cite any sources. In the proposed modification HFE encryption scheme, we impose some restrictions on the plaintext space.

We just observe thatso. Let be an irreducible polynomial with degree over ; then forms a degree- extension field. Then we compute their inverses and and the -variable quadratic polynomials. Security and Communication Networks.

The encryption of the original HFE scheme is just to computewhere the plaintext is in but not necessarily in.

Multivariate cryptography – Wikipedia

Introduction 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 cfyptanalysis public key cryptosystems due to its high performance and the resistance to quantum attacks [ 4 ]. Note thatresp. Multivariate cryptography Post-quantum cryptography.


We then can look at as a quadratic form about then we associate with a symmetric -dimensional square matrix such that The symmetric matrix is of low rank, and it is the special structure of the symmetric matrix that makes the original HFE scheme insecure. As a new multivariate public key encryption, the security of the proposal needs to be furthered. So the proposed scheme reduces the public key size by bits.

The matrix is then determined by pulic a linear combination of these matrices such that has a minimum rank at most. If ; then we output as the plaintext. So and satisfy the following equations derived from the bilinear equations, namely, where and all the coefficients in.

CiteSeerX — Cryptanalysis of the HFE Public Key Cryptosystem

However, we can derive the field equations from the equations. By using this site, you agree to the Terms of Use and Privacy Policy.

As far as the proposed HFE modification scheme is concerned, we just note that, for any plaintextis a valid ciphertext for both the original FHE scheme and the proposed modification HFE scheme. In certain cases those polynomials could be defined over both a ground and an extension field. Articles with French-language external links Articles needing additional references from August All articles needing additional references. So given a ciphertextwe only need to solve the linearization equations to obtain the corresponding plaintext.

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Multivariate cryptography

So we define Now we show that the corresponding matrix is of not necessarily low rank. 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. Firstly, we define an HFE map in 1 and randomly choose two invertible affine transformations and. Considering the aforementioned discussions, we suggest choosing and.


Finally, we develop a new relinearization method for solving such systems for any constant ffl? August Learn how and when to remove this template message. Retrieved from ” https: Security We analyze the security of the proposed HFE modified encryption scheme.

View at MathSciNet J. This page was last edited on 9 Septemberat In this matrix equation, we only know that is of low cryptossystem at most. So we encourage the readers to examine the security of the proposal. We recalland denote the smallest integer smaller than or equal to asand we will find that all the elements of the last columns rows, relinearizahion. If the polynomials have the degree two, we talk about multivariate quadratics.

The system parameters consist of an irreducible polynomial with degree overthe extension fieldand the isomorphism between and.

The computational costs are at least bit operations, according to the results given on page in [ 2 ]. Signatures are generated using the private key and are verified using the public key as follows. View at MathSciNet V. It is shown that the modification can defend the known attacks including the MinRank attack, the linearization equations attack, and the direct algebraic attacks. However, the rank of the matrix is unknown, and hence the rank of the matrix is not necessarily low.