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Modular Integer Arithmetic for Public Key Cryptography

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Secure Integrated Circuits and Systems

Part of the book series: Integrated Circuits and Systems ((ICIR))

Abstract

This chapter discusses building blocks for implementing popular public key cryptosystems, like RSA, Diffie-Hellman Key Exchange (DHKE) and Elliptic Curve Cryptography (ECC). Therefore, we briefly introduce field-based arithmetic on which most of recently established public key cryptosystems rely. As most popular fields, we give examples for architecture implementing efficient arithmetic operations over prime and binary extension fields for use in cryptographic applications.

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Notes

  1. 1.

    According to [14], the discovery of public-key cryptography (PKC) in the intelligence community is attributed to John H. Ellis in 1970. The discovery of the equivalent of the RSA cryptosystem [38] is attributed to Clifford Cocks in 1973 while the equivalent of the Diffie–Hellman key exchange was discovered by Malcolm J. Williamson, in 1974. However, it is believed that these British scientists did not realize the practical implications of their discoveries at the time of their publication (see, for example, [39, 11]).

  2. 2.

    It is important to understand that NP-complete problems from computer science cannot be simply converted for use with PKC since a one-way function for PKC must guarantee hardness in all cases which is usually not the case for all NP-complete problems.

  3. 3.

    Recall that the time complexity of n-bit addition or subtraction is only in \(\mathcal{O}(n)\).

  4. 4.

    Exponentiation algorithms significantly influence the performance of cryptosystems like RSA, DHKE, and ElGamal. Please find further details how to speed up exponentiation methods in [31, 16, 44].

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Authors and Affiliations

  1. Chair for Embedded Security, Ruhr University Bochum, Bochum, Germany

    Tim Güneysu & Christof Paar

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  1. Tim Güneysu
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  2. Christof Paar
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Correspondence to Tim Güneysu .

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  1. Dept. Elektrotechniek (ESAT), Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, Leuven, 3001, Belgium

    Ingrid M.R. Verbauwhede

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Güneysu, T., Paar, C. (2010). Modular Integer Arithmetic for Public Key Cryptography. In: Verbauwhede, I. (eds) Secure Integrated Circuits and Systems. Integrated Circuits and Systems. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-71829-3_1

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  • DOI: https://doi.org/10.1007/978-0-387-71829-3_1

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