Rings

Rings

1. What a Ring Is (Group + Second Operation)

A ring $(R, +, \times)$ is an abelian group under $+$ plus a second operation $\times$ with closure, associativity, distributivity:

• $(R, +)$: abelian group (identity $0$, inverses $-a$).
• $(R, \times)$: closed, associative; distributes: $a(b+c) = ab + ac$.
• Commutative ring: $ab = ba$ (all crypto examples: $\mathbb{Z}$, $\mathbb{Z}_n$, polynomials).
• With unity: multiplicative identity $1
  e 0$ exists.
• Integral domain: commutative + unity + no zero divisors ($ab = 0 \Rightarrow a = 0$ or $b = 0$) — e.g. $\mathbb{Z}$, but NOT $\mathbb{Z}_6$ ($2 \times 3 = 0$).

Hierarchy: group (1 op) ⊂ ring (2 ops) ⊂ field (2 ops + ×-inverses) — each topic adds one property.

2. Worked Examples

Ring	Zero / unity	Inverses?	Notes
$(\mathbb{Z}, +, \times)$	$0$ / $1$	Only $\pm 1$ have ×-inverses	Integral domain, not a field
$(\mathbb{Z}_6, +, \times)$	$0$ / $1$	$1^{-1} = 1$, $5^{-1} = 5$; 2,3,4 none	Zero divisors ($2 \times 3 = 0$) — needs care in RSA-style mod $n$
$(\mathbb{Z}_7, +, \times)$	$0$ / $1$	All nonzero (it's a field too)	$\mathbb{Z}_p$ prime $\Rightarrow$ field
Polynomials $\mathbb{Z}_2[x]$	$0$ / $1$	Only $1$	AES field built as quotient of this ring (Module 3 Topic 3–4)

Zero-divisor check: mod 6: $2 \times 3 = 6 \equiv 0$ with neither factor $0$ — so cancellation fails ($2a = 2b
ot\Rightarrow a = b$); mod 7:$2a = 2b \Rightarrow a = b$always (multiply by$2^{-1} = 4$). Exams contrast these two cases.

3. Why Rings Matter for AES and RSA

• AES S-box arithmetic happens in $GF(2^8) = \mathbb{Z}_2[x]/(m(x))$ — a quotient of the polynomial ring $\mathbb{Z}_2[x]$ by irreducible $m(x) = x^8+x^4+x^3+x+1$. Ring theory (ideals, quotients) is what makes "$\bmod m(x)$" rigorous.
• RSA works in ring $\mathbb{Z}_n$ ($n = pq$ composite): elements coprime to $n$ form group $\mathbb{Z}_n^*$ ($\phi(n)$ of them — Euler), non-coprimes are zero divisors. RSA correctness uses ring + group facts together.
• Homomorphic encryption (Paillier, BFV) computes directly on ring structure: $E(a) \oplus E(b) = E(a+b)$ uses the additive group of a ring.

4. Subrings, Ideals, Quotients (One-Line Each for Exams)

• Subring: subset closed under both ops (e.g. even integers $2\mathbb{Z} \subset \mathbb{Z}$).
• Ideal: subring absorbing multiplication ($r \in R, a \in I \Rightarrow ra \in I$) — multiples $(m(x))$ in $\mathbb{Z}_2[x]$: the modulus of AES arithmetic.
• Quotient ring $R/I$: cosets with mod-$I$ arithmetic — $\mathbb{Z}_n = \mathbb{Z}/(n)$ and $GF(2^8) = \mathbb{Z}_2[x]/(m(x))$ are both quotients. This single idea builds every crypto field.

Exam Points

• Ring axioms + commutative/unity/integral-domain refinements — 2-mark opener.
• $\mathbb{Z}_6$ zero-divisor ($2 \times 3 = 0$) vs $\mathbb{Z}_7$ cancellation — standard contrast numerical.
• Hierarchy group ⊂ ring ⊂ field with one added property each.
• Quotient preview: $\mathbb{Z}_n$, $GF(2^8)$ as quotients (bridge to Modules 3–4).