Abstract: In many cryptographic applications it is
necessary to generate elliptic curves (ECs) whose order
possesses certain properties. The method that is usually
employed for the generation of such ECs is the
so-called ComplexMultiplication method. This method
requires the use of the roots of certain class field polynomials
defined on a specific parameter called the discriminant.
The most commonly used polynomials are
the Hilbert and Weber ones. The former can be used
to generate directly the EC, but they are characterized
by high computational demands. The latter have usually
much lower computational requirements, but they
do not directly construct the desired EC. This can be
achieved if transformations of their roots to the roots of the corresponding (generated by the same discriminant)
Hilbert polynomials are provided. In this paper we present
a variant of the ComplexMultiplicationmethod that
generates ECs 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 aWeber
polynomial to the roots of its corresponding Hilbert
polynomial for all values of the discriminant. In addition,
we prove a theoretical estimate of the precision required
for the computation ofWeber 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 the theoretical estimates.We further
investigate the time efficiency of the new ComplexMultiplication variant under different implementations
of a crucial step of the variant. Our results can serve as
useful guidelines to potential implementers of EC cryptosystems
involving generation of ECs of a desirable
order on resource limited hardware devices or in systems
operating under strict timing response constraints

Abstract: We consider the generation of prime order elliptic curves
(ECs) over a prime field Fp using the ComplexMultiplication (CM)
method. A crucial step of this method is to compute the roots of a special
type of class field polynomials with the most commonly used being the
Hilbert and Weber ones, uniquely determined by the CM discriminant
D. In attempting to construct prime order ECs using Weber polynomials
two difficulties arise (in addition to the necessary transformations of the
roots of such polynomials to those of their Hilbert counterparts). The
first one is that the requirement of prime order necessitates that D ≡ 3
(mod 8), which gives Weber polynomials with degree three times larger
than the degree of their corresponding Hilbert polynomials (a fact that
could affect efficiency). The second difficulty is that these Weber polynomials
do not have roots in Fp. In this paper we show how to overcome
the above difficulties and provide efficient methods for generating ECs of
prime order supported by a thorough experimental study. In particular,
we show that such Weber polynomials have roots in F
p3 and present a
set of transformations for mapping roots of Weber polynomials in F
p3
to roots of their corresponding Hilbert polynomials in Fp. We also show
how a new class of polynomials, with degree equal to their corresponding
Hilbert counterparts (and hence having roots in Fp), can be used
in the CM method to generate prime order ECs. Finally, we compare
experimentally the efficiency of using this new class against the use of
the aforementioned Weber polynomials.
Keywords: Elliptic Curve Cryptosystems,

Abstract: We consider a variant of the ComplexMultiplication (CM)
method for constructing elliptic curves (ECs) of prime order with additional
security properties. Our variant uses Weber polynomials whose
discriminant D is congruent to 3 (mod 8), and is based on a new transformation
for converting roots of Weber polynomials to their Hilbert
counterparts. We also present a new theoretical estimate of the bit precision
required for the construction of the Weber polynomials for these
values of D. We conduct a comparative experimental study investigating
the time and bit precision of using Weber polynomials against the (typical)
use of Hilbert polynomials. 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.

Abstract: We present a variant of the complexmultiplication method
that generates elliptic curves of cryptographically strong order. Our variant
is based on the computation ofWeber polynomials that require significantly
less time and space resources than their Hilbert counterparts. We
investigate the time efficiency and precision requirements for generating
off-line Weber polynomials and its comparison to another variant based
on the off-line generation of Hilbert polynomials. We also investigate the
efficiency of our variant when the computation of Weber polynomials
should be made on-line due to limitations in resources (e.g., hardware
devices of limited space).We present trade-offs that could be useful to potential
implementors of elliptic curve cryptosystems on resource-limited
hardware devices.

Abstract: We consider the generation of prime-order elliptic curves (ECs) over a prime field $\mathbb{F}_{p}$ using the ComplexMultiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones. These polynomials are uniquely determined by the CM discriminant D. In this paper, we consider a variant of the CM method for constructing elliptic curves (ECs) of prime order using Weber polynomials. In attempting to construct prime-order ECs using Weber polynomials, two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that D≡3mod8), which gives Weber polynomials with degree three times larger than the degree of their corresponding Hilbert polynomials (a fact that could affect efficiency). The second difficulty is that these Weber polynomials do not have roots in $\mathbb{F}_{p}$ .
In this work, we show how to overcome the above difficulties and provide efficient methods for generating ECs of prime order focusing on their support by a thorough experimental study. In particular, we show that such Weber polynomials have roots in the extension field $\mathbb{F}_{p^{3}}$ and present a set of transformations for mapping roots of Weber polynomials in $\mathbb{F}_{p^{3}}$ to roots of their corresponding Hilbert polynomials in $\mathbb{F}_{p}$ . We also show how an alternative 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 ECs. We conduct an extensive experimental study comparing the efficiency of using this alternative class against the use of the aforementioned Weber polynomials. Finally, we investigate the time efficiency of the CM variant under four different implementations of a crucial step of the variant and demonstrate the superiority of two of them.

Abstract: In many cryptographic applications it is necessary to generate
elliptic curves (ECs) with certain security properties. These curves
are commonly constructed using the ComplexMultiplication method
which typically uses the roots of Hilbert or Weber polynomials. The former
generate the EC directly, but have high computational demands,
while the latter are faster to construct but they do not lead, directly, to
the desired EC. In this paper 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 discriminant values
on which they are defined. Moreover, we prove a theoretical estimate
of the precision required for the computation of Weber polynomials. Finally,
we experimentally assess the computational efficiency of theWeber
polynomials along with their precision requirements for various discriminant
values and compare the results with the theoretical estimates. Our
experimental results may be used as a guide for the selection of the most
efficient curves in applications residing in resource limited devices such as
smart cards that support secure and efficient Public Key Infrastructure
(PKI) services.

Abstract: Elliptic Curve Cryptography (ECC) is one of the
most promising alternatives to conventional public
key cryptography, such as RSA and ElGamal, since
it employs keys of smaller sizes for the same level
of cryptographic strength. Smaller key sizes imply
smaller hardware units for performing the arithmetic
operations required by cryptographic protocols and,
thus, ECC is an ideal candidate for implementation
in embedded systems where the major computational
resources (speed and storage) are limited.
In this paper we present a port, written in ANSI C
for maximum portability, of an open source ECCbased
cryptographic library (ECC-LIB) to ATMEL˘s
AT76C520 802.11 WLAN Access Point. One of the
major features of this port, not found in similar ports,
is that it supports ComplexMultiplication (CM) for
the construction of Elliptic Curves with good security
properties. We present some experimental results that
demonstrate that the port is efficient and can lead to generic embedded systems with robust ECC-based
cryptographic protocols using cryptographically strong
ECCs generated with CM. As an application of the
ported library, an EC Diffie-Hellman key exchange
protocol is developed as an alternative of the 4-way
key handshake protocol of the 802.11 protocol.