Cryptography Reference
In-Depth Information
algorithm was proposed by Ron Rivest, Adi Shamir, and Leonard Adleman (RSA
algorithm; Rivest et al. 1978). The security of such public-key cryptographic schemes
depended on the integer factorization problem or the discrete log problem. In 1984,
ElGamal proposed public-key based signature and encryption schemes based on the
discrete log problem (ElGamal 1984).
Most security products adopt the RSA algorithm to generate digital signatures and
encryption schemes. However, in recent years elliptic curve cryptography (ECC) has
gained a lot of attention, as it provides the same level of security as RSA but for a far
smaller key size, thus leading to lower processing overhead. Hence, ECC is an excellent
substitute for RSA in systems/networks that have constrained resources. This chap-
ter focuses on ECC and, in particular, on paring-based cryptography, which is useful
in understanding identity-based cryptography and its application to WSN. The basis
for these disciplines in cryptography uses mathematical concepts taken from the field
of modern algebra. For completeness, we briefly review the basic building blocks of
modern algebra, which will be used in the mathematical derivation of elliptic curve
cryptographic techniques to be presented.
3.2 Introduction to Modern Algebra
The fundamental elements of modern algebra are groups, rings, and fields. In this
branch of mathematics, we are particularly interested in performing algebraic opera-
tions on the elements of the defined set, resulting in elements that already belong to the
same set. We will look into the specific rules applied to these operations that define the
nature of the set. In particular, we are interested in modular arithmetic operations such
as addition and multiplication.
3.2.1 Groups
Let (,)
G * denote a group in which G is a set such that the binary operation
*´
: GG G
satisfies the following properties:
(i) Closure: ,
ab
G
ab G
(3.1)
(ii) Associative: ,,
abc G
( ) ( )
abc
**=**
abc
(3.2)
(iii) Identity element:
there exists an element e Î such that
aG
,
(3.3)
ae ea a
*=*=
Search WWH ::




Custom Search