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• W147079175 abstract "Elliptic curve cryptography (ECC) has gained an increasing popularity over the years, as it emerges as a fundamental and efficient technological alternative for building secure public key cryptosystems. This stems from the fact that elliptic curves (ECs) give rise to algebraic structures that offer a number of distinct advantages (smaller key sizes and highest strength per bit) over more customary algebraic structures used in various cryptographic applications (e.g., RSA). These characteristics make ECC suitable for software as well as for hardware implementations. The latter is of particular importance, since (under certain circumstances) it involves devices with limited resources such as cell phones and Smartcards. One of the fundamental issues in ECC is the generation of elliptic curves suitable for use in various cryptographic applications. The most efficient method for generating elliptic curves over prime fields is the {em Complex Multiplication} (CM) method. This method requires the use of the roots of certain polynomials, called class polynomials. The most commonly used polynomials are the {em Hilbert} and {em Weber} ones. The former can be used to generate directly the elliptic curve, but they are characterized by high computational demands. The latter have usually much lower computational requirements, but they do not construct directly the desired elliptic curve. This can be achieved if one provides transformations of their roots to the roots of the corresponding Hilbert polynomials. The goals of this PhD thesis are the following: (i) to improve the CM method by incorporating in it Weber polynomials; (ii) to provide an efficient method for the generation of prime order ECs; and (iii) to develop a flexible and portable software library that will include all the necessary primitives and protocols required for the construction of an elliptic curve cryptosystem, especially in resource limited environments. The current thesis makes a host of new contributions towards the goals set above. In particular, to address the first goal, we present a variant of the CM method that generates elliptic curves 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 upper bound 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 our theoretical bounds. Our experiments show the superiority of Weber polynomials and that the actual precision requirements for the construction of these polynomials are close to the theoretical estimate we provide. To address the second goal, we consider the use of a new variant of the CM method for the construction of {em prime order} elliptic curves. The Weber polynomials that are used for the construction of prime order elliptic curves have degree three times larger than the degree of their corresponding Hilbert polynomials. We show that, these Weber polynomials do not have roots in the field $mathbb{F}_p$, but do have roots in the extension field $mathbb{F}_{p^3}$. We present a set of transformations for mapping roots of Weber polynomials in $mathbb{F}_{p^3}$ to the roots of their corresponding Hilbert polynomials in $mathbb{F}_p$. We also show how a new class of polynomials, with degree equal to their corresponding Hilbert counterparts (and hence having roots in $mathbb{F}_p$), can be used in the CM method to generate prime order elliptic curves. We compare experimentally the efficiency of using this new class against the use of the aforementioned Weber polynomials and show that the type of polynomial that one should use depends on the particular application. We further investigate the time efficiency of the new CM variant under four different implementations of a crucial step of the variant and demonstrate the superiority of two of them. Finally, we present an implementation of an elliptic curve cryptographic library, which includes not only the aforementioned algorithms, but also several cryptographic protocols. We provide a fully-equipped library of portable source code with clearly separated modules that allows for easy development of EC cryptographic protocols, and which can be readily tailored to suit different requirements and user needs. The small size of the library makes it appropriate for use in resource limited devices." @default.
• W147079175 created "2016-06-24" @default.
• W147079175 creator A5036929593 @default.
• W147079175 date "2005-06-15" @default.
• W147079175 modified "2023-09-27" @default.
• W147079175 title "Θεωρία και εφαρμογές κρυπτογραφικών συστημάτων δημόσιου κλειδιού βασισμένων σε ελλειπτικές καμπύλες" @default.
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