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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## J-GLOBAL – Japan Science and Technology Agency

So given a ciphertextwe only need to solve the linearization equations to obtain the corresponding plaintext. The HFE scheme firstly defines a univariate map over an extension field: Views Read Edit View history. The RSA public key cryptosystem is based on a single modular equation in one variable. Forwe set where all the coefficients are in for. For a plaintextwe just compute as the ciphertext. Then two invertible affine transformations are applied to hide the special structure of the central map [ 25 ].

Multivariate Quadratics involves a public and a private key. In the proposed modification HFE encryption scheme, we impose some restrictions on the plaintext space. Subscribe to Table of Contents Alerts.

Without loss of generality, we assume that relienarization two invertible affine transformations and are linear [ 21 ] and define the terms of in in 1. The proposed HFE modification has the following features: Let be an irreducible polynomial with degree over ; then forms a degree- extension field.

Thus by solving the MinRank problem we can determine the matrix and the coefficients of the linear transformation. Therefore, we cannot hope to derive linearization equations from cryptanlysis modified HFE scheme. It is relimearization that the proposed public key encryption scheme is secure against known attacks including the MinRank attack, the algebraic attacks, and the linearization equations attacks. 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.

keg Firstly, we define an HFE map in 1 and randomly choose two invertible affine transformations and. View at MathSciNet V. Thus we can easily verify that So we get.

Description The encryption scheme consists of three subalgorithms: August Learn how and when to remove this template message. It can be easily seen that both the modified and the original HFE schemes share a common secret key and decryption algorithm. Signatures are generated using the private key and are verified using the public key as follows. In addition to HFE, J. Thus we have some additional equations that associate with the plaintext ; namely, forwe have.

Solving systems of multivariate polynomial equations is proven to be NP-hard or NP-complete. 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 ].

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

The encryption scheme consists of three subalgorithms: This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, rlinearization the original work is properly cited.

So the proposed scheme reduces bfe public key size by bits. View at MathSciNet J. 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 ]. The eelinearization of the signed document must have the public key P in possession. In the modified scheme, the public key isand hence we need not to store the coefficients of the square terms of the public key.

Let be a -order finite field with being a prime power. Indexed in Science Citation Index Expanded. Unsourced material may be challenged and removed.

Abstract The RSA public key cryptosystem is based on a single modular equation in one variable. Multivariate cryptography has been very productive in terms of design and cryptanalysis.

As a new multivariate public key encryption, the security of the proposal needs to be furthered. So the rank of the symmetric matrix is at most. Performance analysis shows that the modification can save the public key storage by bits and reduces the encryption costs by about bit operations. 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 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.

If we fail to derive a vector in form all the preimageswe output the symbol designating an invalid ciphertext. Overall, the situation is now more stable and the strongest schemes have withstood the test of time.

Hence, forSo. So and satisfy the following equations derived from the bilinear equations, namely, where and all the coefficients in. The proposal gains some advantages over the original HFE scheme with respect to the encryption speed and public key size.

### Multivariate cryptography – Wikipedia

We first review the basic idea of known attacks and then illustrate why the proposal is secure against these attacks. Symmetric-key algorithm Block cipher Stream cipher Public-key cryptography Cryptographic hash function Message authentication code Random numbers Steganography.

It is tthe that the modification can defend the known attacks including the MinRank attack, the linearization equations attack, and the direct algebraic attacks. 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.