Elliptic Curve Arithmetic for Cryptography

Abstract

The advantages of using public key cryptography over secret key cryptography include the convenience of better key management and increased security. However, due to the complexity of the underlying number theoretic algorithms, public key cryptography is slower than conventional secret key cryptography, thus motivating the need to speed up public key cryptosystems. A mathematical object called an elliptic curve can be used in the construction of public key cryptosystems. This thesis focuses on speeding up elliptic curve cryptography which is an attractive alternative to traditional public key cryptosystems such as RSA. Speeding up elliptic curve cryptography can be done by speeding up point arithmetic algorithms and by improving scalar multiplication algorithms. This thesis provides a speed up of some point arithmetic algorithms. The study of addition chains has been shown to be useful in improving scalar multiplication algorithms, when the scalar is fixed. A special form of an addition chain called a Lucas chain or a differential addition chain is useful to compute scalar multiplication on some elliptic curves, such as Montgomery curves for which differential addition formulae are available. While single scalar multiplication may suffice in some systems, there are others where a double or a triple scalar multiplication algorithm may be desired. This thesis provides triple scalar multiplication algorithms in the context of differential addition chains. Precomputations are useful in speeding up scalar multiplication algorithms, when the elliptic curve point is fixed. This thesis focuses on both speeding up point arithmetic and improving scalar multiplication in the context of precomputations toward double scalar multiplication. Further, this thesis revisits pairing computations which use elliptic curve groups to compute pairings such as the Tate pairing. More specifically, the thesis looks at Stange's algorithm to compute pairings and also pairings on Selmer curves. The thesis also looks at some aspects of the underlying finite field arithmetic.