Cryptography Reference
In-Depth Information
this chapter, we review optimized techniques in prime and binary fields for different
arithmetic operations on different sensor platforms.
Until a decade ago, due to the resource-constraint nature of WSN, security goals
such as authentication, confidentiality, and integrity protection were accomplished
using symmetric techniques. In recent times, however, ECC has emerged as a potential
alternative to conventional public key cryptographic techniques such as RSA, Digital
Signature Algorithm (DSA), and Diffie-Hellman (DH) (Section 3.6). More specifi-
cally, ECC provides equivalent security with a smaller key size, which results in reduced
computational time and lower memory requirements. The primary operation underly-
ing RSA is modular exponentiation in integer rings and its security is based on the dif-
ficulty in factoring larger integers. Conversely, ECC operates on groups of points over
elliptic curves, and its security is based on the hardness of the Elliptic Curve Discrete
Logarithm Problem (ECDLP) (Section 3.7). Whereas subexponential algorithms solve
the integer factorization problem, only exponential algorithms are known to solve the
ECDLP. Hence, ECC achieves higher computational efficiency for the same level of
security as RSA with a smaller key size. Consequently, the motivation for increasingly
efficient ECC-based algorithms and its implementation has increased in a resource-
constraint environment such as WSN.
9.2 Point Coordinate System
Many different coordinate systems can represent points on elliptic curves. However,
the selection of the point coordinate system has a significant influence on the perfor-
mance of the elliptic curve arithmetic operations. For instance, point multiplication
operations can be expedited by using efficient representation of elliptic curve points.
For instance, in ECC, many coordinate systems such as Affine, Projective, Jacobian,
Lopez-Dahab, and mixed coordinates can be used to represent points.
Affine coordinates : These are standard ( x , y ) coordinates commonly used in mathematics.
Projective coordinates : These may be used in some cases to accelerate certain arithme-
tic operations. The point ( X, Y, Z ) on E corresponds to the affine point ( X/Z, Y/Z )
when Z ≠ 0 and to the point at infinity = (0 , 1 , 0). It has been shown that projec-
tive coordinate systems are more efficient than affine systems when calculating point
multiplication.
Jacobian coordinates : This system improves the timings for basic arithmetic opera-
tions on a curve. The point ( X, Y, Z ) on E in Jacobian coordinates corresponds to
the affine point ( X/Z 2 , Y/Z 3 ) when Z ≠ 0 and to the point at infinity = (1 , 1 , 0). In
general for Jacobian coordinates, doublings are faster and additions slower than for
the projective coordinates.
Lopez-Dahab coordinates : These are designed for curves defined over fields of charac-
teristic two. A point ( X, Y, Z ) on an elliptic curve in Lopez-Dahab coordinates is equal
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