Multiplicative group
A multiplicative group is a group whose operation is written as multiplication, with an identity element and an inverse for every element.
Real-World Cryptography
A multiplicative group is a group conventionally written with a multiplication operation and a multiplicative identity element.
- How it is used
- Supports understanding Diffie–Hellman key exchange through multiplication-based group operations.
- How deep it goes
- Defines four group properties and illustrates modular examples, without developing detailed algorithmic calculations.
- Examples
- Uses inverse notation, integers modulo five as a resetting clock, and subgroups of the multiplicative group modulo five.
- What you can do after
- Evaluate/understand the algebraic structure underlying Diffie–Hellman operations.
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Hacking Cryptography
A multiplicative group modulo a prime consists of the nonzero residue classes under multiplication modulo that prime.
- How it is used
- Frames discrete logarithms and public-key constructions through modular exponentiation.
- How deep it goes
- Defines generators, demonstrates exponentiation modulo 13, and stops at the discrete-logarithm formulation.
- Examples
- Uses the complete exponentiation table for Z₁₃* and identifies 2, 6, 7, and 11 as generators.
- What you can do after
- Evaluate modular exponentiation examples and understand why reversing them produces the discrete logarithm problem.
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Related concepts
- Euler's totient functionmeasured by · Euler’s totient function maps a positive integer to the number of positive integers less than it that are coprime to it.
- modular multiplicationhas part · Modular multiplication is multiplication followed by reduction of the result modulo a specified integer.
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