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Over the past 20 years, elliptic curves have become
an integral part of many
cryptographic protocols. As a consequence, methods
for constructing elliptic
curves suitable for cryptography have been studied
intensively. This book gives
a new and very efficient method to construct an
elliptic curve of prescribed
order. It includes a basic introduction to elliptic
curves, written for
beginning graduate students in mathematics. The
presented construction
naturally leads into a discussion of the relationship
between elliptic curves
and algebraic number theory. This more advanced
topic, known as complex
multiplication theory, was first pioneered by
Kronecker in the 19th century and
it has recently regained popularity. The second part
of this book contains
various new p-adic algorithms in this area and it
gives a description of p-adic
class invariants.
an integral part of many
cryptographic protocols. As a consequence, methods
for constructing elliptic
curves suitable for cryptography have been studied
intensively. This book gives
a new and very efficient method to construct an
elliptic curve of prescribed
order. It includes a basic introduction to elliptic
curves, written for
beginning graduate students in mathematics. The
presented construction
naturally leads into a discussion of the relationship
between elliptic curves
and algebraic number theory. This more advanced
topic, known as complex
multiplication theory, was first pioneered by
Kronecker in the 19th century and
it has recently regained popularity. The second part
of this book contains
various new p-adic algorithms in this area and it
gives a description of p-adic
class invariants.