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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Advanced Search Include Citations Disambiguate. The hidden field equations HFE scheme [ 5 ] may be the most famous cryptosystem amongst all multivariate public key cryptographic schemes. However, all known modification methods only can impose partial nonlinear transformation on the special structure of the HFE central map, and hence they are still vulnerable to some attacks [ 15 — 17 ].

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 based public key cryptosystems due to its high performance and the resistance to quantum attacks [ 4 ].

Views Read Edit View history. However, the central map can be represented with a low-rank matrix [ 7 ], which makes it vulnerable to MinRank attacks [ 7 — 9 ].

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

Symmetric-key algorithm Block cipher Stream cipher Public-key cryptography Cryptographic hash function Message authentication code Random numbers Steganography.

Considering the aforementioned discussions, we suggest choosing and. 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 encryption of the original HFE scheme is just to computewhere the plaintext is in but not necessarily in. Security We analyze the security of the proposed HFE modified encryption scheme.

In addition to HFE, J. Then two invertible affine transformations are applied to hide the special structure of the central map [ 25 ]. Table of Contents Alerts. So given a ciphertextwe only need to solve the linearization equations to obtain the corresponding plaintext.

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

This section does not cite any sources. Retrieved from ” https: Linearization Equations Attack Basic Idea. The RSA public key cryptosystem is based on a single modular equation in one variable. We represent the published system of multivariate polynomials by a single univariate polynomial of a special form over an eky field, and use it hve reduce the cryptanalytic problem to a system of fflm 2 quadratic equations in m variables over the extension field.

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.

Notations Let be a -order finite field with being a prime power. In this paper we consider Patarin’s Hidden Field Equations HFE scheme, which is believed to be one of the strongest schemes of this type. Multivariate Quadratics relinearizaation a public and a private key. We can see from the security analysis that the proposed HFE modification encryption scheme can obtain a security level of 80 bits under the suggested parameters.

Forwe set where all the coefficients are in for. However, the original HFE scheme was insecure, and the follow-up modifications were shown to be still vulnerable to attacks. Thus we can easily verify that So we get. However, the rank of the matrix is unknown, and hence the rank of the matrix is not necessarily low.

Finally, we develop a new relinearization method for solving such systems for any constant ffl? The plaintext space is but not. So the computational overhead is about bit operations. However, some cryptanaoysis variants of HFE, such as the minus variant and the vinegar variant allow one to strengthen the basic HFE against all known attacks.

The HFE scheme firstly defines a univariate map over an extension field: In this paper, we proposed a novel modified HFE encryption scheme.

Therefore, we cannot hope to derive linearization equations from the modified HFE scheme. In certain cases those polynomials could be defined over both a ground and an extension field.

It is based on a ground and an extension field. We set the quadratic part of the public key as with for.

## Multivariate cryptography

We first relinearizatioj 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 both schemes have the same secret key sizes and decryption costs. We define with forand It is obvious that. Though the MinRank problem is proven to be NP-complete [ 2223 ], the reduction to the MinRank problem does impose a serious security threat on the security of the HFE scheme [ 78 ].

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 cryptosysfem follows. Schmidt, Multivariate Public Key Cryptosystemsvol.

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. Signatures are generated using the private key and are verified using the public key as follows. 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.

The encryption scheme consists of three subalgorithms: The matrix is hve determined by finding a linear combination of these matrices such that has a minimum rank at most.

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.