SubramanyaRao, SrinivasaRao

### Description

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...[Show more] 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.

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