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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The plaintext block also satisfies the field equation. So and satisfy the following equations derived from the bilinear equations, namely, where and all the coefficients in. If ; then we output as the plaintext. We just observe thatso. We consider the HFE scheme over finite fields with characteristic 3. So some modifications are needed to repair the basic HFE scheme [ 10 — 14 ].

The modified HFE decryption recovers the plaintext by peeling off the composition one by one from the leftmost side. Notations Let be a -order finite field with being a prime power. oof

Multivariate cryptography

However, the central map can be represented with a low-rank matrix relinearizztion 7 ], which makes it vulnerable to MinRank attacks [ 7 — crhptosystem ]. Introduction Public key cryptography [ cryptosystme ] 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 ].

The system parameters consist of an irreducible polynomial with degree overthe extension fieldand the isomorphism between and. The proposal gains some advantages over the original HFE scheme with respect to the encryption speed and public key size. During encryption, the proposed modification HFE scheme does not need to do the square ke, so the proposed encryption reduces the computational costs by bit operations.


If the polynomials have the degree two, we talk about multivariate quadratics. Conclusions In this paper, we proposed a novel modified HFE encryption scheme. Abstract The RSA public key cryptosystem is based on a single modular equation cryptosyystem one variable. That’s why those schemes are often considered to be good candidates for post-quantum cryptography. Unsourced material may be challenged and removed. 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.

The plain version of HFE is considered to be practically broken, in the sense that secure parameters lead to an impractical scheme.

These equations are called linearization equations and can be efficiently computed from the public polynomials. So tbe define Now we show that the corresponding matrix is of not necessarily low rank. This section does not cite any sources. Hence, forSo. Kipnis and Shamir noted [ 7 ] that, by lifting the quadratic part of the public key of the HFE scheme to the extension fieldthey can find a collection of matrices.

Thus we can easily verify that So we get. Thus by solving the MinRank problem we can determine cryotosystem matrix and the coefficients of the linear transformation.

Description The encryption scheme consists of three subalgorithms: Note that the Frobenius maps for defined over are -linear; namely, when expressed in the base fieldwill be -dimensional linear functions over. So we encourage the readers to examine the security of the proposal.


The HFE scheme firstly defines a univariate map over an extension field: Let be an irreducible polynomial with degree over ; then forms a degree- extension field.

Considering the aforementioned discussions, publif suggest choosing and. Conflicts of Interest The authors declare that they have no conflicts of interest.

Multivariate cryptography – Wikipedia

We define with forand It is obvious that. Suggested Parameters Considering the aforementioned discussions, we suggest choosing and.

El Din, and P. From Wikipedia, the free encyclopedia. Algebraic Attacks Basic Idea. It is commonly cryptanslysis that Multivariate cryptography turned out to be more successful as an approach to build signature schemes primarily because multivariate schemes provide the shortest signature among post-quantum algorithms.

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.

The original HFE scheme [ 5 ] works on any field and its extension. The hidden field equations HFE scheme [ 5 ] may be the most famous cryptosystem amongst all multivariate public key cryptographic schemes. View at MathSciNet J.

History of cryptography Cryptanalysis Outline of cryptography. We analyze the security of the proposed HFE modified encryption scheme. If we fail to derive a vector in form all the preimageswe output the symbol designating an invalid ciphertext.