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Efficient generation of secure elliptic curves.

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Title: Efficient generation of secure elliptic curves.
Authors: Konstantinou, Elisavet ekonstantinou@aegean.gr, Stamatiou, Yannis istamat@cc.uoi.gr, Zaroliagis, Christos zaro@ceid.upatras.gr
Source: International Journal of Information Security. Jan2007, Vol. 6 Issue 1, p47-63. 17p. 9 Charts, 5 Graphs.
Subjects: Cryptography, Complex multiplication, Elliptic curves, Numerical roots, Polynomials, Algebraic curves
Abstract: In many cryptographic applications it is necessary to generate elliptic curves (ECs) whose order possesses certain properties. The method that is usually employed for the generation of such ECs is the so-called Complex Multiplication method. This method requires the use of the roots of certain class field polynomials defined on a specific parameter called the discriminant. The most commonly used polynomials are the Hilbert and Weber ones. The former can be used to generate directly the EC, but they are characterized by high computational demands. The latter have usually much lower computational requirements, but they do not directly construct the desired EC. This can be achieved if transformations of their roots to the roots of the corresponding (generated by the same discriminant) Hilbert polynomials are provided. In this paper we present a variant of the Complex Multiplication method that generates ECs of cryptographically strong order. Our variant is based on the computation of Weber polynomials. We present in a simple and unifying manner a complete set of transformations of the roots of a Weber polynomial to the roots of its corresponding Hilbert polynomial for all values of the discriminant. In addition, we prove a theoretical estimate of the precision required for the computation of Weber polynomials for all values of the discriminant. We present an extensive experimental assessment of the computational efficiency of the Hilbert and Weber polynomials along with their precision requirements for various discriminant values and we compare them with the theoretical estimates. We further investigate the time efficiency of the new Complex Multiplication variant under different implementations of a crucial step of the variant. Our results can serve as useful guidelines to potential implementers of EC cryptosystems involving generation of ECs of a desirable order on resource limited hardware devices or in systems operating under strict timing response constraints. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Information Security is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="AR" term="%22Konstantinou%2C+Elisavet%22">Konstantinou, Elisavet</searchLink><i> ekonstantinou@aegean.gr</i><br /><searchLink fieldCode="AR" term="%22Stamatiou%2C+Yannis%22">Stamatiou, Yannis</searchLink><i> istamat@cc.uoi.gr</i><br /><searchLink fieldCode="AR" term="%22Zaroliagis%2C+Christos%22">Zaroliagis, Christos</searchLink><i> zaro@ceid.upatras.gr</i>
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  Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Information+Security%22">International Journal of Information Security</searchLink>. Jan2007, Vol. 6 Issue 1, p47-63. 17p. 9 Charts, 5 Graphs.
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  Data: <searchLink fieldCode="DE" term="%22Cryptography%22">Cryptography</searchLink><br /><searchLink fieldCode="DE" term="%22Complex+multiplication%22">Complex multiplication</searchLink><br /><searchLink fieldCode="DE" term="%22Elliptic+curves%22">Elliptic curves</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+roots%22">Numerical roots</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Algebraic+curves%22">Algebraic curves</searchLink>
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  Data: In many cryptographic applications it is necessary to generate elliptic curves (ECs) whose order possesses certain properties. The method that is usually employed for the generation of such ECs is the so-called Complex Multiplication method. This method requires the use of the roots of certain class field polynomials defined on a specific parameter called the discriminant. The most commonly used polynomials are the Hilbert and Weber ones. The former can be used to generate directly the EC, but they are characterized by high computational demands. The latter have usually much lower computational requirements, but they do not directly construct the desired EC. This can be achieved if transformations of their roots to the roots of the corresponding (generated by the same discriminant) Hilbert polynomials are provided. In this paper we present a variant of the Complex Multiplication method that generates ECs of cryptographically strong order. Our variant is based on the computation of Weber polynomials. We present in a simple and unifying manner a complete set of transformations of the roots of a Weber polynomial to the roots of its corresponding Hilbert polynomial for all values of the discriminant. In addition, we prove a theoretical estimate of the precision required for the computation of Weber polynomials for all values of the discriminant. We present an extensive experimental assessment of the computational efficiency of the Hilbert and Weber polynomials along with their precision requirements for various discriminant values and we compare them with the theoretical estimates. We further investigate the time efficiency of the new Complex Multiplication variant under different implementations of a crucial step of the variant. Our results can serve as useful guidelines to potential implementers of EC cryptosystems involving generation of ECs of a desirable order on resource limited hardware devices or in systems operating under strict timing response constraints. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of International Journal of Information Security is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10207-006-0009-3
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        Text: English
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        Type: general
      – SubjectFull: Complex multiplication
        Type: general
      – SubjectFull: Elliptic curves
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              Text: Jan2007
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