Math Notation Reference

Every article on this site uses the same small vocabulary of symbols. These five pages collect it in one place. Each entry gives the symbol, how to read it aloud, what it means on this site, a short example, and links to the articles that use it. Start with the quick lookup below, or open the page for the area you are reading about.

8.1

Numbers, sets and modular arithmetic

ℤ, ℤ_q, set-builder notation, mod and ≡, rounding brackets, ≈ and ≪, big-O, logarithms, and the Greek alphabet as used here.

8.2

Vectors and matrices

Row and column conventions, transposes and inverses, inner products, norms, determinants, unimodular and Gram matrices.

8.3

Lattice notation

Λ(B), determinant and volume, successive minima, the Gaussian heuristic, Gram–Schmidt, dual and q-ary lattices, problem names.

8.4

Rings and polynomials

ℤ_q[X]/(Xⁿ+1), coefficient vectors, negacyclic multiplication, roots of unity, the NTT and module notation.

8.5

Probability and sampling

Sampling arrows, error distributions, discrete Gaussians, the Gaussian function ρ, smoothing, and negligible functions.

Conventions to know first

Rows or columns. The introduction and cryptography chapters (1.x, 2.x) write basis vectors as the columns of a matrix, so the lattice is $\Lambda(B) = \{Bx : x \in \mathbb{Z}^n\}$ . The embedding guides (7.x) and the g6k-rs recipes write them as rows, so the lattice is $L(B) = \{xB : x \in \mathbb{Z}^n\}$ , which is the convention of fplll and most reduction software. The two descriptions differ only by a transpose. Each article states which one it uses; 8.2 shows how to convert.

Stars and tildes. A star on a lattice always means its dual, $\Lambda^*$ . Gram–Schmidt vectors are always written with a tilde, $\tilde b_i$ . Many papers use $\Lambda^\vee$ for the dual or $b_i^*$ for Gram–Schmidt vectors; they mean the same things.

Quick lookup

Symbol	Meaning	Details
$\mathbb{Z},\ \mathbb{R},\ \mathbb{Z}^n$	Integers, real numbers, integer vectors of length n	8.1
$\mathbb{Z}_q$	Integers modulo q	8.1
$a \equiv b \pmod q$	a and b leave the same remainder mod q	8.1
$\lfloor x \rceil,\ \lfloor x \rfloor,\ \lceil x \rceil$	Round to nearest, round down, round up	8.1
$B^\top,\ B^{-1},\ B^{-\top}$	Transpose, inverse, inverse transpose	8.2
$\langle x, y \rangle$	Inner (dot) product	8.2
$\\|x\\|,\ \\|x\\|_\infty$	Euclidean length, largest absolute coordinate	8.2
$\det(B)$	Determinant of a square matrix	8.2
$\Lambda,\ \Lambda(B),\ L$	A lattice; the lattice generated by basis B	8.3
$\det(\Lambda) = \mathrm{vol}(\Lambda)$	Volume of a fundamental cell; basis-independent	8.3
$\lambda_1(\Lambda)$	Length of a shortest nonzero lattice vector	8.3
$\mathrm{gh}(\Lambda)$	Gaussian heuristic: predicted λ₁ of a random lattice	8.3
$\tilde{b}_i,\ \mu_{ij}$	Gram–Schmidt vectors and coefficients	8.3
$\Lambda^*$	The dual lattice	8.3
$\Lambda_q(A),\ \Lambda_q^\perp(A)$	q-ary lattices of a matrix A	8.3
$\beta$	BKZ block size (in most articles)	8.3
$\delta_0$	Root-Hermite factor: quality of a reduced basis	8.3
$R_q = \mathbb{Z}_q[X]/(X^n+1)$	Polynomial ring used by Kyber, Dilithium and Falcon	8.4
$\zeta$	Root of unity used by the NTT	8.4
$x \leftarrow \chi$	Sample x from the distribution χ	8.5
$D_{\mathbb{Z},\sigma}$	Discrete Gaussian on the integers	8.5
$\eta_\varepsilon(\Lambda)$	Smoothing parameter of a lattice	8.5

Letters with more than one meaning

Lattice cryptography borrows from several fields, so a few Greek letters carry different meanings in different articles. The surrounding text always makes the meaning clear, and each reference page lists every meaning with the articles that use it. The most common cases are $\beta$ (BKZ block size, and a bound in Dilithium), $\gamma$ (Hermite constant, approximation factor, and Dilithium parameters), $\delta$ (the LLL parameter, and the root-Hermite factor $\delta_0$ ), $\eta$ (smoothing parameter, LLL size-reduction bound, and a secret bound in Kyber and Dilithium), and $\lambda$ (successive minima, and the security parameter).