J. Patarin. Hidden field equations (HFE) and isomorphism of polynomials (IP): Two new families of asymmetric algorithms. In U. Maurer, editor, Eurocrypt'96, LNCS, volume 1070, pages 33--48. Springer, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....dimensional) is expressed with respect to a eld basis to achieve an expression of several variables (i.e. higher dimensional) Their attempt is noble, however, unsuccessful, since it has been cracked by Patarin. Patarin proposed another public key system using Hidden Field Equations (HFE) in [8]. Its decryption involves solving equations, and hence the process is slow. The TTM cryptosystem (cf [6] 7] is a truly higher dimensional method. It is given by the composition of tame mappings ( i i ) from K where K is a nite eld and n m. The public key is the composition ....

Patarin, J. Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP): two new Families of Asymmetric Algorithms. Eurocrypt'96, 1996.

....interpolation where n denotes the number of variables. Moreover, the running time of polynomial interpolation is approximately 30 higher than for base transformation. 1 Introduction During the past years, several asymmetric cryptographic systems were proposed based on multivariate cryptography [MI88, Pat96a, Pat96b, CGP02a, CGP02b]. There are systems both for signing and encryption. The general idea of these systems is very similar: using polynomials over nite elds of di erent size, they exploit the intractability of the MQ problem , i.e. multivariate quadratic equations over nite elds are dicult to solve [GJ79, p. ....

....[Pat95, Fau02, CDF03] the area is still vital and promises interesting algorithms with rather low computational costs. Moreover, they also allow rather short signature sizes (e.g. only 128 bits for Quartz [CGP02a] For an encryption algorithm based on HFE, we can expect similar short messages [Pat96b]. This would be of interest for the transmission of session keys without overhead (e.g. for equivalent security, we would have more than 800 bits overhead for an RSA system) Although several systems were proposed, the question of ecient public key generation did not receive much attention yet. ....

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Jacques Patarin. Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of asymmetric algorithms. In Advances in Cryptology { EUROCRYPT 1996, volume 1070 of Lecture Notes in Computer Science, pages 33{ 48. Ueli Maurer, editor, Springer, 1996. Extended Version: http://www.minrank. org/hfe.pdf.

....an adversary can derive a new valid signature for a previously signed message. Their respective complexities are 26 and 2 computations of the public function. The second one also requires 2 signatures, for any messages. 1 Introduction PREVIOUS WORKS. The HFE schemes have been introduced in [4] at Eu rocrypt 96. They are based on the difficulty of solving large systems of quadratic multivariate equations over a finite field. Quartz is a deterministic HFE scheme, with two perturbations respectively denoted V and , and a Fcistcl Patarin construction. In [1] some conjectures on the ....

....a HFEV algorithm composed with four rounds chained as in a CBC mode of operations, called a Fcistcl Patarin construction. We will not give the details of the rounds since, only the mode of operation will be needed in the following. The reader can see the submission [5] the NESSIE report [3] and [4] for further details on the HFE algorithm and on all the variations that can be introduced in the scheme. The private key of the signer consists in two aftinc bijcctions s : F2oo F2oo and t: F2o7 F2o7, a family of functions (Fv)v o,1 4 from F2oo to F2oo with a special form, and a secret ....

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Jacques Patarin. Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IF): two new Families of Asymmetric Algorithms. In Advances in Cryptology - Eurocrypt'96, volume 1070 of LNCS, pages 33 - 48. Springer Verlag, Berlin, 1996.

....that recovering an AES key is equivalent to solving particular systems of extremely sparse multivariate quadratic equations by expressing a BES (and hence an AES) encryption as such a system. The problem of solving such systems of equations lies at the heart of several public key cryptosystems [3, 22], and there has been some progress in providing solutions to such problems [4, 5, 14] Recently, Courtois and Pieprzyk [6] have suggested the use of a system of multivariate quadratic equations over GF (2) to analyse the AES. However, such a GF (2) system derived directly from the AES is far more ....

J. Patarin. Hidden field equations (HFE) and isomorphisms of polynomials (IP): Two new families of asymmetric algorithms. In U. Maurer, editor, Proceedings of Eurocrypt '96, LNCS 1070, pages 33--48, Springer-Verlag, 1996.

....these polynomials, contains a special hidden structure (the secret key) which can be used to decrypt, i.e. solve the system of equations. Several schemes were proposed and some were broken. For instance, Shamir s scheme [78] was broken in [23] and Patarin s hidden fields equations (HFE) scheme [70] was cryptanalysed in [48] We refer to [49] for a discussion of some such algebraic cryptosystems. Lattices 13 are playing an important role in cryptanalysis: many cryptosystems have been broken using lattice reduction techniques. Recently, a number of public key cryptosystems were proposed ....

J. Patarin, Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms, Advances in Cryptology --- EUROCRYPT '96, Lecture Notes in Computer Science, vol. 1070, pp. 33--48, Springer-Verlag, 1996.

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Jacques Patarin. Hidden fields equations (HFE) and isomorphisms of polynomials (IP): Two new families of asymmetric algorithms. EUROCRYPT, pages 33--48, 1996.

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J. Patarin. Hidden Field Equations (HFE) and Isomorphism of Polynomials (IP): Two New Families of Asymmetric Algorithms. In U. Maurer, editor, Advances in Cryptology--Eurocrypt 1996, LNCS, volume 1070, pages 33--48, Springer, 1996. Extended Version: http://www.minrank.org/hfe.pdf.

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J. Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of asymmetric algorithms, in Advances in Cryptology, Proceedings of EUROCRYPT'96, LNCS n # 1070, Springer, 1996, pp. 33-48.

.... cryptosystems is as old as the idea of public key cryptosystems itself [1] Many of the most efficient proposed schemes were based on multivariate polynomials [3,9,2] but they were usually broken later [8,10,4] In a series of papers Patarin proposes new secure and efficient public key systems [5,6] based on hiding the structure of polynomials in a difficult to analyze encryption code, and analyzes other similar schemes [4] One of his more promising scheme is the very efficient 2 Round Public Key System with S Boxes (shortly called 2R) 7] The design of this scheme is unique as it uses ....

Jacques Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms, Lecture Notes in Computer Science, Advances in Cryptology, proceedings of EUROCRYPT'96, pp. 33- 48, 1996.

....Universit at Karlsruhe, Am Fasanengarten 5, 76 131 Karlsruhe, Germany Fachhochschule Aargau, Klosterzelgstrasse, 5210 Windisch, Switzerland Abstract. At EUROCRYPT 96 J. Patarin introduced the Isomorphisms of Polynomials (IP) problem as a basis of authentication and signature schemes [4, 5]. We describe an attack on the secret key of IP with one secret and demonstrate its eciency through examples with realistic parameter sizes. To prevent our attack, additional restrictions on the suggested parameters should be imposed. Keywords: cryptanalysis, multivariate polynomials 1 ....

....(IP) and demonstrated how to use this problem for constructing an authentication scheme and an asymmetric signature scheme. Besides being of interest in their own, IP based schemes are also of interest with regard to other cryptographic schemes using multivariate polynomials: as explained in [4, 5], an ecient algorithm for IP would yield a new, more powerful, attack on the Matsumoto Imai algorithm [3] and it would have strong implications on the design and security of HFE [4, 5] There are di erent variants of the IP problem, and in [6, 7] several techniques for solving the so called IP ....

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J. Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms, in Advances in Cryptology | EUROCRYPT '96, U. Maurer, ed., vol. 1070 of Lecture Notes in Computer Science, Springer, 1996, pp. 33-48.

....polynomials over a finite field. One of these schemes was later presented in [7] under the name C # , and was based on the idea of hiding a monomial field equation. This scheme was broken in [8] by Jacques Patarin, due to unexpected algebraic properties. J. Patarin and L. Goubin then suggested ([9], 10] 11] 12] some schemes to repair C # , but this was done at the cost of slightly more complex public key or secret key computations. In part I of this paper, we will study some very simple variations of the C # scheme, where the attack of [8] is avoided, and where the very simple secret ....

....variation of the Matsumoto Imai C # algorithm that we will present in the part I of this paper. The C # algorithm was presented in [4] and [7] and was broken in [8] due to unexpected algebraic properties. However, it is possible to imagine many ways of avoiding the cryptanalysis of [8] In [9], J. Patarin suggested to use a hidden polynomial instead of a hidden monomial . These HFE algorithms are still unbroken. However, the secret key computations in HFE schemes are sensibly more complex than in the original C # scheme. In [10] 11] and [12] J. Patarin and L. Goubin also ....

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J. Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP) : Two New Families of Asymmetric Algorithms, Advances in Cryptology, Proceedings of EUROCRYPT'96, Springer, pp. 33-48.

....improved algorithms to solve Isomorphisms of Polynomials (IP) problems. These problems were first explicitly related to the problem of finding the secret key of some asymmetric cryptographic algorithms (such as Matsumoto and Imai s C # scheme of [13] or some variations of Patarin s HFE scheme of [15]) Moreover, in [15] it was shown that IP can be used in order to design an asymmetric authentication or signature scheme in a straightforward way. We also introduce the more general Morphisms of Polynomials problem (MP) As we see in this paper, these problems IP and MP have deep links with ....

....to solve Isomorphisms of Polynomials (IP) problems. These problems were first explicitly related to the problem of finding the secret key of some asymmetric cryptographic algorithms (such as Matsumoto and Imai s C # scheme of [13] or some variations of Patarin s HFE scheme of [15] Moreover, in [15], it was shown that IP can be used in order to design an asymmetric authentication or signature scheme in a straightforward way. We also introduce the more general Morphisms of Polynomials problem (MP) As we see in this paper, these problems IP and MP have deep links with famous problems such as ....

[Article contains additional citation context not shown here]

Jacques Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): two new Families of asymmetric Algorithms, EUROCRYPT'96, Springer-Verlag, pp. 33-48.

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J. Patarin. Hidden field equations (HFE) and isomorphism of polynomials (IP): Two new families of asymmetric algorithms. In U. Maurer, editor, Eurocrypt'96, LNCS, volume 1070, pages 33--48. Springer, 1996.

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J. Patarin. Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. Proc. EUROCRYPT '96, Lecture Notes in Computer Science, Vol.1070, pp.33--48, Springer, 1996. 10

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J. Patarin. Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms. In Eurocrypt 1996, LNCS 1070:33--

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J. Patarin. Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In Eurocrypt'96, LNCS, pages 33--48, 1996.

No context found.

Jacques Patarin. Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of asymmetric algorithms. In Advances in Cryptology | EUROCRYPT 1996, volume 1070 of Lecture Notes in Computer Science, pages 33-48. Ueli Maurer, editor, Springer, 1996. Extended Version: http://www.minrank.org/hfe.pdf.

No context found.

Jacques Patarin. Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of asymmetric algorithms. In Advances in Cryptology | EUROCRYPT 1996, volume 1070 of Lecture Notes in Computer Science, pages 33-48. Ueli Maurer, editor, Springer, 1996. Extended Version: http://www.minrank.org/hfe.pdf.

No context found.

J. Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms, EUROCRYPT'96, LNCS V. 1070, pp. 33--48.

No context found.

Jacques Patarin. Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms. In Ueli M. Maurer, editor, Proceedings of EUROCRYPT'96, volume 1070 of Lecture Notes in Computer Science, pages 33--48. Springer-Verlag, 1996.

No context found.

J. Patarin, Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms, Advances in Cryptology --- EUROCRYPT '96, Lecture Notes in Computer Science, 1070 (1996), Springer-Verlag, pp. 33-48.

No context found.

J. Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms, EUROCRYPT'96, LNCS V. 1070, pp. 33--48.

No context found.

J. Patarin. Hidden field equations (HFE) and isomorphisms of polynomials (IP): Two new families of asymmetric algorithms. In U. Maurer, editor, Proceedings of Eurocrypt '96, LNCS 1070, pages 33--48, Springer-Verlag, 1996.

No context found.

J. Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms, EUROCRYPT'96, LNCS V. 1070, pp. 33--48.

No context found.

Patarin, J Hidden elds equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. Advances in Cryptology- Euricrypt '96, SpringerVerlag, 33-48, 1996.

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