# CRYPTANALYSIS OF THE HFE PUBLIC KEY CRYPTOSYSTEM BY RELINEARIZATION PDF

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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It can be easily seen that both the modified and the original HFE schemes share a common secret key and decryption algorithm. The new type of attack is quite general, and in a companion paper we use it to break other multivariate algebraic schemes, such tne the Dragon encryption and signature schemes.

### Building Secure Public Key Encryption Scheme from Hidden Field Equations

Linearization equations attack [ 18 ] was found by Patarin on the Matsumoto-Imai scheme [ 19 ]. The encryption of the original HFE scheme is just to computewhere the plaintext is in but cryptanlaysis necessarily in.

It is based on a ground and an extension field. We consider the HFE scheme over finite fields with characteristic 3. To receive news and publication updates for Security and Communication Networks, enter your email address in the box below.

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 ]. Solving systems of multivariate polynomial equations is proven to be NP-hard or NP-complete. So the adversary cannot derive from the publicly known map a low-rank cryptanakysis. To illustrate why the proposed modification of the HFE scheme is secure against the MinRank 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.

From Wikipedia, the cryptanalysie encyclopedia.

Thus by solving the MinRank problem we can determine the matrix and the coefficients of the linear transformation. If we fail to derive a vector in form all the preimageswe output the symbol designating an invalid ciphertext. Therefore, we cannot hope to derive linearization equations from the modified HFE scheme. The receiver of the signed document must have the public key P in possession.

If the polynomials have the degree two, we talk about multivariate quadratics. Articles with French-language external links Articles needing additional references from August All articles needing additional references.

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. 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.

## Security and Communication Networks

In addition to HFE, J. In the Matsumoto-Imai scheme, a permutation over with characteristic 2 is defined such thatthen using two invertible affine transformations and to disguise the central map into a quadratic map overnamely, The basic idea of the attack is as follows.

However, the original HFE scheme was insecure, and the follow-up modifications were shown to be still vulnerable to attacks. So the HFE scheme is secure against linearization equations attack. So we define Now we show that the corresponding matrix is of not necessarily low rank.

If ; then we output as the plaintext. Then two invertible affine transformations are applied to hide the special structure of the central map [ 25 ]. A natural generalization of this approach is to consider systems of several modular equations in several variables. The plaintext space is but not.

The construction admits a standard isomorphism between the extension field and the vector space ; namely, for an elementwe have. We impose some restrictions on the plaintext space and can use the restriction to merge the coefficients of the linear part and the square part.

## J-GLOBAL – Japan Science and Technology Agency

Let be a -order finite field with being a prime power. It is commonly admitted 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. Without loss of generality, we assume that the two invertible affine transformations and are linear [ 21 ] and define the terms of in in 1.

During encryption, the proposed modification HFE scheme does not need to do the square computations, so the proposed encryption reduces the computational costs by bit operations. Symmetric-key algorithm Block cipher Stream cipher Public-key cryptography Cryptographic hash function Message authentication code Random numbers Steganography. This page was last edited on 9 Septemberat Given a ciphertextwe want to recover the corresponding plaintext.

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, 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. By doing this, we can impose a fully nonlinear transformation on the central map of the HFE encryption scheme.

The HFE scheme firstly defines a univariate reliinearization over an extension field: Under the suggested parameters andthe degree of regularity of the quadratic equations is.

We represent the published system of multivariate polynomials cryptoxystem a single univariate polynomial of a special form over an extension field, and use it to reduce the cryptanalytic problem to a system of fflm 2 quadratic equations in m variables over the extension field.