International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 800
ISSN 2229-5518
IJSER © 2013
http://www.ijser.org
Performance Comparism Of Finite Fields
Arithmetic In Elliptic Curve Based Cryptographic
Schemes.
Aliyu Danladi Hina
Abstract
Finite fields are well studied discrete structures with a vast array of useful properties and are indispensable in the theory and application of
cryptography. Arithmetic in finite field is an integral part of many public key algorithms. The performance of elliptic curve based schemes
depends on the efficient arithmetic in the underlying field. ; Cryptography is one of the most prominent application areas of finite field
arithmetic. Most of public-key cryptographic algorithms including the recent algorithms such as elliptic curve and pairing-based cryptography
rely heavily on finite field arithmetic, which needs to be performed efficiently to meet the execution speed and design space constraints. These
objectives constitute massive challenges that necessitate research efforts that will render the best algorithms, architectures, implementations, and
design practices. This paper aims to provide a concise perspective for efficient finite field arithmetic in the most widely used finite field for
usage in cryptography, The Optimal Extension Field.
Key words: cryptography, discrete structures, elliptic curve, Finite field, finite field arithmetic.
1.0 INTRODUCTION
To implement an ECC, one must select an underlying
finite field in which to perform arithmetic calculations. A
finite field is identified with the notation GF(p
m
) for p a
prime and m a positive integer. It is well known that
there exists a finite field for all primes p and positive
integers m. Any such field is isomorphic to
GF(p)[x]/(P(x)), where P(x) =
+ ∑
,
−1
=0
(), is a monic-irreducible
polynomial of degree m over GF(p). In the following, each
residue class will be identified with the unique
polynomial of least degree in this class.
Various finite fields admit the use of different algorithms
for arithmetic. Unsurprisingly, the choices of p, m, and
P(x) can have a dramatic impact on the performance of
the ECC. In particular, there are generic algorithms for
arithmetic in an arbitrary finite field and there are
specialized algorithms which provide better performance
in finite fields of a particular form. In the following, we
briefly describe field types proposed for ECC. The basic
requirement for a fast and thus energy efficient
implementation of ECC is a very fast multiplication in the
prime field. The fastest known implementation was
implemented by SUN Microsystems. [5]
2.0 FINITE FIELDS
Various finite fields admit the use of different algorithms
for arithmetic. The choice of p, m and p(x) can have a
dramatic impact on the performance of the elliptic curve
cryptography (ECC). There are generic algorithms in an
arbitrary field and there are specialized algorithms which
provide better performance in a finite field of a particular
form.
2.1 Binary Fields GF(2
m
): The finite field GF(2
m
) called a
binary finite field of 2
m
elements implying that there exist
a set of m elements {
0
,
1
,
2
,…
−1
} in GF(2
m
) such that
each (2
) can be written in the form =
∑
−1
=0
where
{0,1}.
Implementing the binary field in designing elliptic curve
based schemes, one often choose p = 2 and P(x) to be a
trinomial or pentanomial. Such choices of irreducible
polynomial lead to efficient methods for extension field
modular reduction. We will refer to this type of field as a
binary field, The elements of the subfield GF(2) can be
represented by the logical signals 0 and 1. In this way, it
is possible to construct fast and area efficient finite field
arithmetic. Binary fields are also popular for software
implementations of ECC. Many authors have suggested
the use of p = 2 and m a composite number, In this case,
the field GF(2
m
) is isomorphic to ((2
)
), for m = sr and
we call this a composite field.
2.2 Binary Composite Fields: An extension defined over
a subfield of GF(2
k
) is known as a composite field denoted
by GF((2
n
)
m
). Considering the fact that both binary and
composite fields ((2
)
) refer to same field, efficient
implementation can be obtained for composite fields,
since this field provides efficient implementations for
specific operations such as multiplication, inversion and
exponentiation.
The composite field has the advantage that its operations
are computed using arithmetic in the subfield GF(2
n
) and
the operations in the subfield can be efficiently performed
by index table look-up if n is too large [3]. Thus instead of
performing the computation in the binary field, it is more
efficient to implement the composite field to perform the
computations. This approach can provide superior
performance when compared to the case of binary fields.
However, a recent attack against ECCs over composite
fields makes their use in practice questionable.
2.3 Prime Fields: Prime fields, GF(p
m
) where m = 1 are
perhaps the most obvious finite fields to use. For ECC, a
typical prime is chosen to be larger than 2
160
, and must be
stored in multiple computer words. The problem with
this representation is that during computation, the carries
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