Math Notation: Numbers, Sets and Modular Arithmetic

Each table lists a symbol as it is rendered and as you would type it in LaTeX, how to read it aloud, what it means in the articles on this site, a small example, and the chapters that use it most. Vectors, norms and matrices are in 8.2, lattice quantities in 8.3, polynomial rings in 8.4, and sampling in 8.5.

Number systems

Symbol	Read as	Meaning on this site	Example	Used in
$\mathbb{Z}$ `\mathbb{Z}`	“zed” or “the integers”	All whole numbers …, −2, −1, 0, 1, 2, …. Lattice points have integer coordinates in a basis, so this is the most common symbol on the site.	$-3 \in \mathbb{Z},\ 0.5 \notin \mathbb{Z}$	1.6, 2.3, 2.4, 2.5, 7.5, 1.2
$\mathbb{Z}^n$ `\mathbb{Z}^n`	“zed to the n”	Vectors of length $n$ with integer entries. $\mathbb{Z}^n$ is also the simplest lattice, the integer grid.	$(2, -1, 0) \in \mathbb{Z}^3$	1.6, 7.5, 6.2, 2.5, 1.2, 2.4
$\mathbb{R},\ \mathbb{R}^n$ `\mathbb{R}^n`	“the reals”, “R to the n”	Real numbers, and real vectors of length $n$ . A lattice sits inside $\mathbb{R}^n$ ; the target point in CVP is usually an arbitrary point of $\mathbb{R}^n$ .	$(0.5, \sqrt{2}) \in \mathbb{R}^2$	1.3, 2.2, 1.6, 1.2, , 3.2
$\mathbb{Q},\ \mathbb{C}$ `\mathbb{Q},\ \mathbb{C}`	“the rationals”, “the complex numbers”	Fractions, and complex numbers. They appear rarely: $\mathbb{Q}(\zeta_{2n})$ for cyclotomic fields, and $\mathbb{C}$ for the complex embedding used by CKKS and Falcon.	$\tfrac{355}{113} \in \mathbb{Q}$	1.6, 3.2, 3.6
$\mathbb{Z}_q$ `\mathbb{Z}_q`	“zed mod q” or “zed q”	The integers modulo $q$ : the residues $0, 1, \dots, q-1$ with addition and multiplication wrapping around at $q$ . Other books write $\mathbb{Z}/q\mathbb{Z}$ ; it is the same object. $\mathbb{Z}_q^n$ means vectors of length $n$ with entries in $\mathbb{Z}_q$ .	In $\mathbb{Z}_{3329}$ (ML-KEM): $3000 + 400 = 71$	2.3, 2.4, 3.5, 3.3, 3.1, 3.4

Sets and membership

Symbol	Read as	Meaning on this site	Example	Used in
$\in,\ \notin$ `\in,\ \notin`	“is in”, “is an element of”; “is not in”	Membership. $v \in \Lambda$ says $v$ is a lattice point.	$s \in \mathbb{Z}_q^n$	3.3, 2.3, 1.6, 2.4, 3.1, 2.6
$\{\, x : P(x) \,\}$ `\{ x : P(x) \}`	“the set of x such that P(x)”	Set-builder notation. The colon and the vertical bar mean the same thing; both appear on the site, e.g. $\{Bx \mid x \in \mathbb{Z}^n\}$ in 1.1 and $\{xB : x \in \mathbb{Z}^n\}$ in 7.1.	$\{x \in \mathbb{Z} : x^2 < 5\} = \{-2,\dots,2\}$	1.3, 1.1, 1.2, 1.4, 7.1, 7.5
$\subset$ `\subset`	“is a subset of”	Every element of the left set is in the right set. Used for a lattice inside $\mathbb{R}^n$ and for sublattices such as $D_n \subset \mathbb{Z}^n$ .	$\Lambda \subset \mathbb{R}^n$	1.3, 1.6
$\setminus$ `\setminus`	“minus”, “without”	Set difference. $\Lambda \setminus \{0\}$ is every lattice point except the origin, which is how SVP excludes the trivial answer.	$\{1,2,3\} \setminus \{2\} = \{1,3\}$	2.1, 1.4
$\cap$ `\cap`	“intersect”	Elements in both sets. In 1.3, $\Lambda \cap B(0, r)$ is the lattice points inside a ball of radius $r$ .	$\{1,2,3\} \cap \{2,3,4\} = \{2,3\}$	1.3
$[a, b],\ (a, b]$ `(a, b]`	“closed interval”, “half-open interval”	A square bracket includes the endpoint, a round one excludes it. Also used for integer ranges of coefficients, e.g. $[-\eta, \eta]$ in ML-DSA.	$\delta \in (1/4, 1]$	3.3, 2.5, 1.5, 3.6

Modular arithmetic

Lattice cryptography does almost all of its arithmetic modulo an integer $q$ : 3329 in ML-KEM, 8380417 in ML-DSA, 12289 in Falcon. Two things trip readers up. First, $\pmod q$ after an equation and $\bmod q$ inside an expression look alike but play different roles: one says an equation holds modulo $q$ , the other computes a remainder. Second, whether a residue is “small” depends on the representative you choose.

Symbol	Read as	Meaning on this site	Example	Used in
$a \equiv b \pmod{q}$ `a \equiv b \pmod{q}`	“a is congruent to b mod q”	A statement: $q$ divides $a - b$ . The parenthesised `\pmod` form qualifies the whole equation.	$17 \equiv 2 \pmod{5}$	2.6, 2.4, 3.3, 1.6, 3.5, 3.1
$a \bmod q$ `a \bmod q`	“a mod q”	An operation: the remainder of $a$ after division by $q$ , a number in $\{0, \dots, q-1\}$ unless the page says it is centred. Also used for polynomials: $f \bmod (X^2 - \gamma)$ in 3.5 is the remainder of polynomial division.	$-3 \bmod 7 = 4$	3.5, 6.4, 3.6, 2.6, 3.3
$x \bmod^{\pm} q$ `\text{centred } x \bmod q`	“x mod q, centred”	The representative in $(-q/2, q/2]$ instead of $[0, q)$ . Errors and secrets are small only in this form: 3000 is huge in $[0, 3329)$ but its centred value is −329. Kyber, Dilithium and NTRU all decode by centring first.	For $q = 3329$ : $3000 \mapsto -329$ , $1665 \mapsto -1664$	2.6, 3.1, 3.3, 2.3
$x^{-1} \bmod q$ `x^{-1} \bmod q`	“the inverse of x mod q”	The residue $y$ with $xy \equiv 1 \pmod q$ . It exists exactly when $\gcd(x, q) = 1$ .	$3^{-1} \bmod 7 = 5$	6.4, 2.6, 3.5
$\gcd(a, b)$ `\gcd(a, b)`	“g-c-d of a and b”	Greatest common divisor. In 7.1 it checks that a linear equation has integer solutions for every right-hand side.	$\gcd(271, 828, 1828) = 1$	7.1

Rounding and absolute value

Symbol	Read as	Meaning on this site	Example	Used in
$\lfloor x \rceil$ `\lfloor x \rceil`	“round x”, “nearest integer to x”	Round to the nearest integer, halves rounded up: $\lfloor x \rceil = \lfloor x + 1/2 \rfloor$ . Applied to a vector or polynomial it rounds every coordinate. Size reduction in LLL subtracts $\lfloor \mu_{k,j} \rceil b_j$ .	$\lfloor 2.5 \rceil = 3,\ \lfloor -2.5 \rceil = -2,\ \lfloor 3329/2 \rceil = 1665$	7.2, 3.2, 3.1, 2.3, 2.2, 4.1
$\lfloor x \rfloor$ `\lfloor x \rfloor`	“floor of x”	The largest integer not above $x$ .	$\lfloor 3329/2 \rfloor = 1664,\ \lfloor -2.5 \rfloor = -3$	7.3, 3.5, 3.4, 2.3, 3.6
$\lceil x \rceil$ `\lceil x \rceil`	“ceiling of x”	The smallest integer not below $x$ .	$\lceil 3329/2 \rceil = 1665,\ \lceil -2.5 \rceil = -2$	7.4
$\|x\|,\ \|S\|$ `\|x\|`	“absolute value of x”; “size of S”	For a number, its distance from zero. For a set, the number of elements. For a matrix, $\|\det(B)\|$ is the absolute value of the determinant. Vector lengths use double bars, $\\|v\\|$ ; see 8.2.	$\|-7\| = 7,\ \|\{0, 1\}^n\| = 2^n$	1.2, 7.3, 4.1, 4.6, 2.2

Estimates and growth rates

Symbol	Read as	Meaning on this site	Example	Used in
$\approx$ `\approx`	“approximately equal to”	Numerically close. On this site it usually marks an estimate from a model (Gaussian heuristic, BKZ cost), not a bound.	$\delta_0 \approx 1.022$	1.5, 7.2, 7.4, 3.4, 7.3, 2.1
$\sim$ `\sim`	“on the order of”; “is distributed as”	Two uses. Before a number it means “roughly”, as in $\sim 2^{29}$ . Between a variable and a distribution it means “sampled from”, as in $s \sim D_{\Lambda,\sigma}$ ; see 8.5.	$\sim 4 \times 10^{19}$	2.1, 1.5, 4.5
$\ll,\ \gg$ `\ll,\ \gg`	“much less than”, “much greater than”	Informal: smaller or larger by a wide margin. In LWE, $\sigma \ll q$ says the noise is tiny compared with the modulus.	$\sigma \ll q$	2.2, 2.3, 2.6, 7.3, 2.4, 3.5
$\gtrsim,\ \lesssim$ `\gtrsim,\ \lesssim`	“at least about”, “at most about”	An inequality that holds up to constants or rounding. Typical use: a threshold found by estimate or experiment, such as $N \gtrsim 2.2 \times 10^4$ in 7.2.	$\beta \gtrsim 50$	7.2, 1.3, 2.4, 1.5, 1.4, 6.4
$O(f),\ \Theta(f)$ `O(f),\ \Theta(f)`	“big-O of f”, “theta of f”	$O(f)$ : grows at most like $f$ up to a constant factor. $\Theta(f)$ : grows exactly like $f$ up to constants. Inside an exponent, $2^{O(n)}$ means “exponential in $n$ ” with the constant hidden.	$O(n \log q),\ 2^{\Theta(n \log n)}$	2.3, 1.5, 4.3, 3.4, 3.1, 2.1
$\tilde{O}(f)$ `\tilde{O}(f)`	“soft-O of f”	Big-O that also ignores logarithmic factors: $\tilde{O}(\sqrt{n})$ may hide a $\log n$ . Common in approximation factors for worst-case reductions.	$\gamma = \tilde{O}(\sqrt{n})$	2.1, 2.4

Logs, sums, products and logic

Symbol	Read as	Meaning on this site	Example	Used in
$\log,\ \log_2,\ \ln$ `\log,\ \log_2,\ \ln`	“log”, “log base 2”, “natural log”	$\log_2$ and $\ln$ are explicit. A bare $\log$ appears inside big-O and complexity expressions such as $O(n \log q)$ , where the base only changes a constant. Bit-security figures (“ $2^{182}$ operations”) are base 2.	$\log_2 3329 \approx 11.70,\ \ln 3329 \approx 8.11$	4.2, 2.3, 1.5, 6.4, 3.4, 1.4
$\textstyle\sum_{i=1}^{n},\ \prod_{i=1}^{n}$ `\sum_{i=1}^{n},\ \prod_{i=1}^{n}`	“sum over i from 1 to n”, “product over …”	Repeated addition and multiplication. The Gram–Schmidt identity $\det(\Lambda) = \prod_i \\|\tilde{b}_i\\|$ is the most frequent product on the site.	$\textstyle\sum_{i=1}^{4} i = 10,\ \prod_{i=1}^{4} i = 24$	7.2, 1.4, 1.3, 3.5, 1.2, 4.1
$n!$ `n!`	“n factorial”	$1 \cdot 2 \cdots n$ . Appears in Minkowski’s second theorem; $\Gamma(n/2 + 1)$ in the ball-volume formula generalises it.	$5! = 120$	1.3
$\implies,\ \iff$ `\implies,\ \iff`	“implies”, “if and only if”	Logical consequence and equivalence. $\Lambda(B) = \Lambda(B') \iff B' = BU$ in 1.2 says the two statements are true together or false together.	$x = 2 \implies x^2 = 4$	3.2, 1.2, 7.2, 1.3, 4.5, 4.3
$\exists,\ \forall$ `\exists,\ \forall`	“there exists”, “for all”	Quantifiers. The site mostly writes “for all” in words; $\exists$ appears in the definition of the q-ary lattice $\Lambda_q(A)$ .	$\exists\, s \in \mathbb{Z}^n : y \equiv A^\top s$	1.6
$\pm$ `\pm`	“plus or minus”	Either sign. $\{0, \pm 1\}$ is the set $\{-1, 0, 1\}$ , the usual alphabet of ternary secrets, and $\det(U) = \pm 1$ defines unimodular matrices.	$\det(U) = \pm 1$	2.6, 2.4, 3.3, 2.5, 1.6, 7.4

Greek letters

Only letters that appear on the site are listed. Several carry more than one meaning, usually because a standard or a paper fixed the name first (ML-DSA’s $\beta, \gamma_1, \eta$ clash with the BKZ block size, the Hermite constant and the smoothing parameter). The surrounding text always says which one is meant; the column below lists every use we found.

Letter	Name and pronunciation	Meanings on this site
$\alpha$	alpha (AL-fuh)	The real number being approximated or tested for relations (7.2, 7.3).
$\beta$	beta (BAY-tuh)	BKZ block size (8.3). In ML-DSA, the bound $\beta = \tau\eta$ on $\\|cs_1\\|_\infty$ (3.3).
$\gamma$	gamma (GAM-uh)	Approximation factor in $\mathrm{SVP}_\gamma$ , $\mathrm{CVP}_\gamma$ ; Hermite constant $\gamma_n$ (8.3); ML-DSA parameters $\gamma_1, \gamma_2$ ; an NTT constant in 3.5.
$\delta$	delta (DEL-tuh)	LLL parameter ( $\delta = 3/4$ or 0.99); root-Hermite factor $\delta_0$ (8.3).
$\varepsilon,\ \epsilon$	epsilon (EP-si-lon)	A small positive number: the tolerance in the smoothing parameter $\eta_\varepsilon$ (8.5), an advantage in 3.4, signs $\epsilon_i \in \{\pm 1\}$ in 4.3.
$\zeta$	zeta (ZAY-tuh)	A root of unity: $\zeta = 17$ in ML-KEM’s NTT, $\zeta_{2n}$ for cyclotomic fields (8.4).
$\eta$	eta (AY-tuh)	Smoothing parameter $\eta_\varepsilon(\Lambda)$ (8.5); the coefficient bound of secrets in ML-KEM and ML-DSA; the LLL size-reduction constant 0.501 in g6k-rs.
$\theta$	theta (THAY-tuh)	An angle between vectors (1.2, 4.5).
$\kappa$	kappa (KAP-uh)	The start index of a BKZ block in g6k’s pump (4.4).
$\lambda$	lambda (LAM-duh)	Successive minima $\lambda_1, \dots, \lambda_n$ ; $\lambda_1$ is the length of a shortest nonzero vector (8.3).
$\mu$	mu (MYOO)	Gram–Schmidt coefficients $\mu_{ij}$ (8.2); covering radius $\mu(\Lambda)$ in 1.4; a message bit or message representative in 2.3 and 3.3.
$\pi$	pi (PIE)	The constant 3.14159…, usually inside Gaussian formulas such as $\sqrt{n/(2\pi e)}$ .
$\rho$	rho (ROH)	Gaussian function $\rho_s$ (8.5); packing radius $\rho = \lambda_1/2$ in 1.4; the public 32-byte seed in ML-KEM and ML-DSA.
$\sigma$	sigma (SIG-muh)	Standard deviation of a Gaussian or of the noise (8.5); a seed in 3.1; the canonical embedding $\sigma: R \to \mathbb{C}^n$ in 3.2.
$\tau$	tau (TOW)	Number of ±1 entries in the ML-DSA challenge (3.3).
$\chi$	chi (KY)	The error distribution in LWE (8.5).
$\psi$	psi (SIGH)	A primitive $2n$ -th root of unity in the negacyclic NTT (3.5).
$\omega$	omega (oh-MAY-guh)	A root of unity in 3.5; the maximum number of hint bits in ML-DSA (3.3).
$\Gamma$	capital gamma	The gamma function in the volume of the unit ball, $V_n = \pi^{n/2}/\Gamma(n/2+1)$ (1.3, 1.4).
$\Delta$	capital delta	Packing density $\Delta(\Lambda)$ (1.4, 1.6); the CKKS scaling factor $\Delta = 2^{40}$ (3.2).
$\Lambda$	capital lambda	A lattice (8.3). The embedding guides call the same object $L$ .
$\Phi$	capital phi (FYE)	The potential function that proves LLL terminates (1.5).

Conventions we follow

Residues mod $q$ are taken in $\{0, \dots, q-1\}$ unless the text says “centred”, in which case they lie in $(-q/2, q/2]$ .
$\lfloor x \rceil$ rounds halves up, matching FIPS 203. Applied to a vector or polynomial it acts on each coordinate.
$\approx$ marks a heuristic estimate; inequalities ( $\leq$ , $<$ ) mark bounds that are proved or stated by a standard.
Bit costs such as $2^{128}$ are counts of operations in base 2.
Single bars $|x|$ are for numbers and set sizes; double bars $\|v\|$ are for vector lengths.

8.1 Numbers, sets and modular arithmetic