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ZK Scholars Assembly Revision 1 - Group Theory Group is a set of elements that satisfied interesting properties that are useful in cryptography such as - Closure - Associativity - Identity Element - Inverse Element Closure means for any elements a, b ∈ G (group), the result of the operation between them will be result in an element that is of the same group, example a • b = c ∈ G. Associativity means sequence of running operation doesn’t matter for any elements a, b, c ∈ G, for example the result of (a • b) • c will be equal to a • (b • c). Identity Element means there exists an element e ∈ G such that for any element a ∈ G, the operation with e will always result in itself, example e • a = a • e = a. One can imagine e as 0 in simple additive group or 1 in simple multiplicative group as 3 + 0 = 3 and 3 * 1 = 3.
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Inverse Element means for every element a ∈ G, there exists an inverse element a^-1 such that a • a^-1 = a^-1 • a = e, where e is the identity element. Example 3 + (-3) = 0 and 3 * 1/3 = 1. An Abelian Group is also called Field has two binary operations which are + and *. It has all the properties from Group + 1 extra property which is called commutativity. Commutativity means the position of elements in operation doesn’t matter, example for any elements a, b ∈ G, a • b = b • a. For the case of multiplicative group, it has another property called Distributive Law, meaning for a, b, c ∈ G, a • (b + c) is also = (a • b) + (a • c).
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