Engineering

Measures of Randomness and the Leftover Hash Lemma

Min-entropy, randomness extraction, and the leftover hash lemma that converts weak randomness into near-uniform bits.

Mertens' Theorem

Mertens' theorems on sums and products over primes, and their role in estimating smoothness probabilities.

Message Authentication with Hash Functions

Unconditionally secure message authentication from universal hash families, and how forgery probability is bounded.

Mignotte Bounds on Polynomial Factors

Bounding the coefficients of any factor of an integer polynomial, and why such a bound makes modular and lifting methods complete algorithms.

Minkowski and Bach Bounds

Bounds on the norm of ideals needed to generate the class group, and why the conditional bound is what makes computation practical.

Modern Factoring Methods Compared

Comparing factoring methods by target size, expected factor size and available hardware, with a practical sequencing recommendation.

Modular Arithmetic and Montgomery Reduction

Modular reduction strategies, Montgomery representation, and how trading division for multiplication accelerates every exponentiation.

Modular Equations and the j-Invariant

Modular functions, modular equations relating j-invariants of isogenous curves, and their use in locating isogenies.

Modular Exponentiation by Repeated Squaring

Square-and-multiply exponentiation, its cost, windowed variants, and the side-channel hazards of the naive form.

Modular Inverses and Chinese Remaindering

Combining modular inversion with Chinese remaindering to move computations between a composite modulus and its coprime factors.

Modular Inversion and Simultaneous Inversion

Computing modular inverses, and Montgomery's trick for inverting many elements at the cost of one inversion plus multiplications.

Module Homomorphisms and Isomorphisms

Module homomorphisms, kernels and images, and the isomorphism theorems in their module form.

Module Representation by Hermite Normal Form

Representing ideals and modules as Hermite normal form matrices relative to an integral basis, with a common denominator.

Modules: Definitions, Properties and Examples

Modules over a commutative ring, the generalisation of vector spaces, and what changes when scalars need not be invertible.

Modulus of a Complex Number

The modulus of a complex number: definition as the square root of c times its conjugate, multiplicativity, the triangle inequality, and its role in norms and spectral radius.

Multi-Variate Polynomials

Polynomials in several variables, total and partial degree, and the structural differences from the univariate case.

Multiprecision Addition and Subtraction

Carry and borrow propagation, sign handling, and why addition sets the baseline cost against which every other operation is measured.

Multiprecision Division and Remainder

Knuth's division algorithm, the normalisation step that makes quotient digit estimation reliable, and why division carries a larger constant than multiplication.

Multiprecision Integer Representation

How arbitrary-precision integers are stored, why the base is chosen to match the machine word, and the consequences for every algorithm above.

Mutual Independence and Secret Sharing

Shamir's threshold secret sharing, its information-theoretic security, and the independence property that underlies it.

N+1 Tests and the Lucas-Lehmer Test

Primality tests using the factorisation of one more than the candidate, and the Lucas-Lehmer test for Mersenne numbers.

Engineering Mathematics13 Aug 2026 · 24 min read

New Linear Transformations From Old

Sums, scalar multiples and compositions of linear transformations are linear. The set of all such maps is a vector space, and matrix operations mirror each construction.

New Matrix Representations From Old

Sums, scalar multiples and compositions of linear transformations become sums, multiples and products of their matrix representations: multiplication is composition.

Newton Polygon Methods for Prime Decomposition

Using Newton polygons to decompose primes locally, handling the cases where factoring modulo p is insufficient.

Engineering Mathematics13 Aug 2026 · 24 min read

Nonsingular Matrices and the Identity Matrix

Definition of a nonsingular square matrix via the trivial null space, the identity matrix, the row-reduction test and unique solutions for any right-hand side.

Engineering Mathematics13 Aug 2026 · 21 min read

Nonsingular Matrices Are Invertible

Proof that a square matrix is nonsingular exactly when it is invertible, via products with singular matrices and the sufficiency of a one-sided inverse.

Engineering Mathematics13 Aug 2026 · 15 min read

Nonsingular Matrix Equivalences — Applications and Numerical Practice

Thirteen equivalent conditions on a square matrix: nonsingular, row-reduces to the identity, trivial null space, invertible, full rank, nonzero determinant.

Engineering Mathematics13 Aug 2026 · 18 min read

Nonsingular Matrix Equivalences — Foundations and Proof Network

Thirteen equivalent conditions on a square matrix: nonsingular, row-reduces to the identity, trivial null space, invertible, full rank, nonzero determinant.

Engineering Mathematics13 Aug 2026 · 22 min read

Null Space of a Matrix

The null space N(A) is the set of all vectors annihilated by a matrix: its definition, how to compute a spanning set from free variables, and why its dimension is n minus the rank.

Null Spaces, Spans and Linear Independence

Row reduction yields n-r linearly independent vectors spanning the null space of a matrix: a minimal, non-redundant basis for every homogeneous solution.

Number Field Sieve: Polynomial Selection and Structure

How the number field sieve achieves its complexity, why polynomial selection matters so much, and the role of number field arithmetic.

Number Fields: Definition and Basic Properties

Number fields as finite extensions of the rationals, their embeddings and signature, and the presentation on which all computation depends.

Number Theory Software Packages

The computer algebra systems and libraries implementing these algorithms, what each is suited to, and why implementing from scratch is usually the wrong choice.

Operations on Subspaces and Modules

Sum, intersection and quotient of subspaces, and how the analogous operations on modules over the integers require normal forms.

Orders in Number Fields

Orders as subrings that are full-rank lattices, the equation order, the maximal order, and the index that separates them.

Engineering Mathematics13 Aug 2026 · 22 min read

Orthogonal and Unitary Matrices

Matrices whose adjoint is their inverse: the condition A*A = I, orthonormal columns, preservation of inner products and norms, adjoints and Hermitian matrices.

Orthogonal Vectors and Orthonormal Sets

Orthogonality of complex vectors, orthogonal and orthonormal sets, why an orthogonal set of non-zero vectors is linearly independent, and how to test one.

Orthonormal Bases and Coordinates

How an orthonormal basis turns coordinate extraction into a single inner product, why every subspace has one, and how orthogonal matrices supply ready-made orthonormal bases.

Engineering Mathematics13 Aug 2026 · 24 min read

Orthonormal Diagonalization

Hermitian and real symmetric matrices diagonalise with a unitary or orthogonal change of basis: real eigenvalues, orthogonal eigenvectors and the spectral decomposition.

p-adic Root Finding and Newton Polygons

Finding roots in p-adic fields by lifting, and reading ramification structure off the Newton polygon of a polynomial.

Pairwise Independence and Universal Hash Families

Constructing pairwise independent hash families over finite fields and why the weaker independence suffices.

Parameter Sizes, Records and Live References

Where to obtain current key size recommendations, factoring records and post-quantum guidance, and why they are not reproduced here.

Engineering Mathematics13 Aug 2026 · 20 min read

Particular Solutions and Homogeneous Solutions

Every solution of a consistent linear system is one particular solution plus an element of the null space, so solution sets are affine translates of a subspace.

Perfect Power Testing and Prime Power Factoring

Detecting whether an integer is a perfect power, extracting the root, and why this precedes general factoring.

Polynomial Congruences

Congruences of polynomials modulo a fixed polynomial, and the parallel with integer congruences.

Polynomial Division with Remainder

Euclidean division of polynomials, pseudo-division over a ring without inverses, and the coefficient growth pseudo-division introduces.

Polynomial Factorisation: Overall Strategy

The three-stage pipeline used to factor polynomials over finite fields and the integers, and why the stages are ordered as they are.

Polynomial GCD over a Unique Factorisation Domain

Computing polynomial GCDs over the integers, the growth problem in remainder sequences, and the modular approach that sidesteps it.