Group

As explained in Real-World Cryptography

A group is an algebraic structure consisting of a set and an associative operation with an identity element and inverses for all elements.

A group combines a set of elements with a binary operation such as addition or multiplication. Together, they support the algebraic behavior that makes key-exchange algorithms work. section 5.2.1

The set and operation form a group only when they satisfy four properties:

• Closure: operating on two elements produces another element in the same set. section 5.2.1
• Associativity: regrouping several operations does not change the result, so a(bc) equals (ab)c. section 5.2.1
• Identity: one element leaves every other element unchanged; multiplication commonly uses 1. section 5.2.1
• Inverses: every element has a partner that combines with it to produce the identity. section 5.2.1

Diffie–Hellman uses a multiplicative group, so multiplication is the chosen operation and the notation may omit the multiplication symbol. section 5.2.1

Figure 5.4 The four properties of a group: closure, associativity, identity element, and inverse element.

Use the figure to compare closure, associativity, identity, and inverse as four separate checks. section 5.2.1

How Real-World Cryptography covers it

How deep it goes

Defines four group properties and connects them to multiplicative notation, but stops short of detailed algorithm mechanics.

Examples

Uses multiplication, identity 1, inverses, modular arithmetic modulo 5, and figures illustrating group properties and subgroups.

What you can do after

Evaluate whether a set and operation satisfy the basic requirements for a group used in key exchange.

How it connects

Connects groups with group theory, Diffie–Hellman key exchange, multiplicative groups, and public-key cryptography.

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Related concepts

• algebraic structurekind of · An algebraic structure is a set of elements equipped with operations and laws governing those operations.
• Diffie–Hellman key exchangeused by · Diffie–Hellman key exchange is a public-key key-agreement protocol that enables parties to derive a shared secret over an insecure channel without transmitting it.
• zk-SNARKused by · A zk-SNARK is a zero-knowledge, succinct, non-interactive argument of knowledge that enables efficient verification of a prover’s knowledge of a computation’s secret witness without revealing it.
• matrix-matrix multiplicationproduced by · Matrix multiplication combines compatible matrices to form a result whose entries are sums of products between rows of the first matrix and columns of the second.
• elliptic curveincludes · An elliptic curve is a nonsingular algebraic curve defined by a cubic equation whose points, including a point at infinity, support a group operation.

This explainer was generated by Manning's knowledge graph from the text of Real-World Cryptography. The section links point to the parts of the book it draws on.