EC Cryptography Tutorials - Herong's Tutorial Examples - Version 1.00, by Dr. Herong Yang
Multiplicative Notation of Abelian Group
This section describes the multiplicative notation of an Abelian Group. The multiplication sign, *, is used as the operator. Number 1 is used as the identity element.
An Abelian Group can be expressed in Multiplicative Notation using the multiplication sign, *, as the operator and number one, 1, as the identity element:
P * Q for F(P,Q) P * Q = Q * P for F(P, Q) = F(Q, P) (P * Q) * S = P * (Q * S) for F(F(P, Q), S) = F(P, F(Q, S)) P * I = P for F(P, I) = P P * Q = 1 for F(P, Q) = I
The Multiplicative Notation matches well with certain examples of Abelian Groups.
Let's consider the following Abelian Group:
You can verify that all 5 Abelian Group conditions are satisfied. For example:
2 * 3 = 6 The "closure" condition 1/6 * 1/6 = 1/3 The "closure" condition 2 * 3 = 3 * 2 The "associativity" condition (2 * 3) * 4 = 2 * (3 * 4) The "commutativity condition 3 * 1 = 3 The "identity" condition 3 * 1/3 = 1 The "symmetry" condition
Last update: 2019.
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