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11. Groups & Cyclic Groups
Groups & Curves · Lesson 12 of 18 · 3:23
The lightest algebraic structure in this series — a set plus one operation — and the special property, cyclic, that makes a single generator enough to reach every element. The exact foundation elliptic curve groups are built on.

That closes out phase two, polynomials. Phase three starts here: groups, curves, and pairings — the foundation everything from elliptic curves to bilinear pairings gets built on. Today stays simple: what is a group, and what makes one cyclic?

What is a group?

A group is a much lighter structure than the finite fields built back in the finite fields lesson. It strips things down to the bare minimum that’s still useful.

Group vs. field

A field needs all four operations — add, subtract, multiply, divide. A group needs only a set and one operation, plus a short list of rules that operation has to follow.

The four group rules

Those rules have a name: the group axioms. Every one of them has to hold for a set and an operation to count as a group.

the group axioms — all four must hold Closure combine two elements, stay in the set Identity one element that changes nothing Inverses every element has an undo Associativity grouping doesn't change the result
Fig. 1 — the four group axioms. Miss any one of these and it isn't a group.

A concrete example

This should feel familiar — this series has already been working with a group, just without the name. Addition mod 7, the exact structure from the modular arithmetic lesson, satisfies these axioms directly.

checking three axioms against addition mod 7 {0, 1, 2, 3, 4, 5, 6} under addition mod 7 Closure 3 + 5 ≡ 1 (mod 7) stays in the set Identity 3 + 0 = 3 adding zero changes nothing Inverses 3 + 4 ≡ 0 (mod 7) 4 undoes 3 this is exactly the additive structure from topic 4
Fig. 2 — three of the four axioms, checked against addition mod 7.

What makes a group cyclic?

Some groups have a special property: every element can be reached by repeatedly applying the operation to just one starting element. Starting from 11 and adding 11 over and over, under addition mod 7, visits every element in the group before landing back on the start.

C₇ — generated entirely by the element 1 1 2 3 4 5 6 7 1 is the generator — start here 1 → 2 → 3 → 4 → 5 → 6 → 7 → 1 (mod 7), visiting every element
Fig. 3 — C₇, the cyclic group of order 7. Starting from 1 and following the arrows reaches every element before wrapping back around.

An element that does this — reaches the whole group on its own — is called a generator.

Generating the group from one element

Watching the running total confirms it directly: adding 11 to itself, again and again, counts straight through every element of the group.

generating C₇ from one element start at 1, keep adding 1, mod 7 1 + 1 = 2 1 + 1 + 1 = 3 … continues … 1 × 6 = 6 1 × 7 = 7 mod 7 = 0 back to start 1 visits every element before returning to start — a generator
Fig. 4 — the running total of 1 + 1 + 1 + ... mod 7, landing on every element before wrapping back to the start.

Seven additions of 11 visit every element of C7C_7 exactly once, then land back on the identity — 11 really does generate the whole group.

Why this matters for ZK

The elliptic curve groups used throughout ZK cryptography are built on this exact same cyclic structure — just with points on a curve standing in for numbers on a ring.

elliptic curve groups used in ZK are cyclic too generator point G, scalar K → compute K · G easily but going backward from K · G to K is HARD this one-way difficulty is the discrete log problem
Fig. 5 — the same one-way gap as C₇'s generator, scaled up to a curve: easy forward, hard backward.

There’s a generator point GG, and for a scalar KK, computing K⋅GK \cdot G is easy. Going the other direction — recovering KK from K⋅GK \cdot G — is hard. That one-way difficulty is the discrete log problem, and it’s what every elliptic curve scheme in this series will lean on from here forward.

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