Cryptography Reference
In-Depth Information
Table 3.3. Binary Polynomials for F 2 3
Binary polynomials
for F 2 3
0
0
1
1
2
x
3
x + 1
4
x 2
5
x 2 + 1
6
x 2 + x
7
x 2 + x + 1
a public-key cryptographic system using elliptic curves (Koblitz 1987). The algebraic
structure of these elliptic curves forms the basis for elliptic curve cryptography. The
fundamental security of this scheme relies on the difficulty of the discrete log problem
in the elliptic curve setting (Section 3.7). Ever since this invention, there has been an
enormous amount of interest in elliptic curve cryptography from diverse sectors of the
research fraternity (Cohen and Frey 2006). In the early 1990s, they played a crucial
role in the proof of Fermat's Last Theorem. Subsequently, elliptic curves were used in
the definition of public-key crypto systems and opened a new paradigm in public key
cryptography. ECC has been a proven technology and has been adopted by many stan-
dardizing bodies such as the National Institute of Standards and Technology (NIST),
the International Organization for Standardization (ISO), the American National
Standards Institute (ANSI), and the Standards for Efficient Cryptography Group
(SECG). Furthermore, it has been incorporated in many commercial products such as
e-mail systems, smart cards, and many applications in mobile technology.
3.5.1 Discrete log Problem
G ´ be a multiplicative cyclic group with generator g and order n . Let x Z
(private key) be an integer randomly selected from the interval [1, n -1]. Let y be the
public key such that
Let (,)
x
yg
=
(3.12)
The discrete logarithm problem in G is defined as the problem of determining x given
y,g and n .
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