Engineering

Subexponential Discrete Logarithm Algorithms

Index calculus for discrete logarithms in Z_p*, its two phases, and the precomputation asymmetry it creates.

Subexponential Integer Factoring

Factoring by congruences of squares, the relation collection and linear algebra phases, and the resulting subexponential cost.

Subfield Structure and Uniqueness of Finite Fields

The subfield lattice of a finite field, its correspondence with divisors, and the uniqueness of each subfield.

Subgroups

Subgroups, the subgroup test, generated subgroups, and the subgroup lattice of a finite group.

Submodules and Quotient Modules

Submodules, quotient modules, and the correspondence between submodules of a quotient and those of the original.

Subrings

Subrings, the subring test, and the distinction between subrings and ideals.

Subspace Constructions

Null space, column space, row space, left null space and span are all subspaces. The theorems, their one-line proofs, and how to recognise a subspace on sight.

Engineering Mathematics13 Aug 2026 · 20 min read

Subspaces

What a subspace of a vector space is, why a subset inherits six axioms free of charge, and how closure, the zero vector and additive inverses are verified.

Surjective Linear Transformations

Surjective (onto) linear transformations: the defining condition, proof by carrying a symbolic right-hand side, and disproof with one unattainable vector.

Engineering Mathematics13 Aug 2026 · 22 min read

Surjective Transformations, Spanning Sets and Dimension

How images of a spanning set span the range, why the images of a basis decide surjectivity, and why domain dimension cannot fall below codomain dimension.

Engineering Mathematics13 Aug 2026 · 25 min read

Systems of Equations and Linear Transformations

Every coefficient matrix induces a linear transformation: kernels are null spaces, ranges are column spaces, and solution sets are pre-images of a vector.

Systems of Linear Equations: Fundamentals

The formal definition of a system of linear equations, double-subscript coefficient notation, what counts as a solution, and why the solution set is the real object of study.

Test Polynomials for Galois Group Software

Why Galois group implementations need a curated test set, what a good set covers, and how to build one.

Testing and Constructing Irreducible Polynomials

Testing irreducibility over a finite field and constructing irreducible polynomials of prescribed degree.

Testing Quadratic Residuosity: Prime Modulus

Deciding quadratic residuosity modulo a prime, and why the problem is easy in this case.

Testing Quadratic Residuosity: Prime Power and Composite Modulus

Quadratic residuosity modulo prime powers and composites, the reduction by Chinese remaindering, and where the hardness enters.

Testing Subspaces

The three-condition subspace test: non-empty, closed under addition, closed under scalar multiplication. Proof, worked verifications and the standard counterexamples.

The AKS Algorithm and Its Analysis

The AKS algorithm in full, its correctness argument, complexity, and why it is not used in practice.

The Algebra of Linear Transformations

The endomorphism algebra of a vector space, minimal and characteristic polynomials, and the module view of a linear operator.

The Baby Step/Giant Step Method

Shanks' baby step giant step algorithm, its square-root running time, and the time-memory trade-off it embodies.

The Baillie-PSW Compositeness Test

The Baillie-PSW test combining a strong base-two test with a strong Lucas test, and its status as the practical standard.

The Berlekamp Factorisation Algorithm

Berlekamp's linear algebra approach to factoring over a finite field, the Berlekamp subalgebra, and when it outperforms the GCD pipeline.

The Birthday Paradox

The birthday problem, the square-root threshold for collisions, and its algorithmic consequences.

The Buchmann-Lenstra Prime Decomposition Method

Decomposing any prime, including those dividing the index, by splitting the algebra of the order modulo that prime.

Engineering Mathematics13 Aug 2026 · 22 min read

The Change-of-Basis Matrix

The change-of-basis matrix converts coordinates from one basis to another, is the representation of the identity transformation, and is always non-singular.

The Characteristic Polynomial and the Hessenberg Method

Computing the characteristic polynomial via Hessenberg reduction, and its role in producing minimal polynomials of algebraic numbers.

The Chinese Remainder Theorem

The Chinese remainder theorem as a ring isomorphism, its constructive proof, and its role in decomposing modular computation.

The Conjugate Vector Representation

Representing a field element by its images under all embeddings, the analytic information this exposes, and the precision it demands.

The Continued Fraction Factorisation Method

CFRAC: generating small quadratic residues from the continued fraction expansion, and the congruence-of-squares framework it established.

Engineering Mathematics13 Aug 2026 · 24 min read

The Coordinatization Principle

The coordinatization principle: questions about linear combinations, independence and spanning in an abstract vector space can be answered in complex n-space.

The Dedekind Criterion for p-Maximality

A cheap modular test deciding whether an order is maximal at a given prime, without computing the maximal order.

Engineering Mathematics13 Aug 2026 · 22 min read

The Determinant of a Matrix

The recursive definition of the determinant via submatrices, the 2x2 formula ad - bc, notation and cost, and why a single scalar decides whether a square matrix is singular.

The Diffie-Hellman Key Establishment Protocol

Diffie-Hellman key agreement, the assumptions it rests on, and the authentication gap that makes it vulnerable alone.

The Dirichlet Unit Theorem, Computationally

The structure of the unit group, its rank from the signature, and what computing units actually requires.

The Elliptic Curve Method: Stage One

ECM stage one: multiplying a point by a highly smooth scalar to reach the identity in one component.

The Error Term in the Prime Number Theorem

How closely li(x) approximates pi(x), the connection to zeta zeros, and what the Riemann hypothesis would give.

The Euclidean Algorithm: Classical and Binary Variants

The classical Euclidean algorithm, the binary variant that replaces division with shifts, and how to choose between them.

The Existence of Finite Fields

Construction of a field of any prime power order, and the proof that one exists for every such order.

The Extended Euclidean Algorithm

Computing Bezout coefficients alongside the gcd, the recurrence for the coefficient sequences, and the size bounds that make it practical.

The Extended Euclidean Algorithm and Bezout Coefficients

Computing Bezout coefficients alongside the GCD, modular inversion as its principal application, and controlling coefficient growth.

The Fermat Test and Carmichael Numbers

The Fermat primality test, pseudoprimes, and the Carmichael numbers that defeat it for every base.

The Field of Fractions of an Integral Domain

Constructing the field of fractions of an integral domain, its universal property, and the standard examples.

Engineering Mathematics13 Aug 2026 · 9 min read

The Four Core Computational Tasks of Number Fields

The four fundamental computational problems for a number field, their dependencies, and what counts as a complete answer to each.

Engineering Mathematics13 Aug 2026 · 24 min read

The Four Fundamental Subsets of a Matrix

The null space, row space, column space and left null space of a matrix, all obtained from the two submatrices C and L of one extended echelon form reduction.

The Frobenius Map

The Frobenius endomorphism, its fixed field, its order, and the Galois structure of finite field extensions.

The Fundamental Unit of a Real Quadratic Field

Computing the fundamental unit of a real quadratic field by continued fractions, its connection to the Pell equation, and its size.

The Galois Group Computation Problem

What computing a Galois group means, why the splitting field cannot be constructed, and the transitive group classification that makes the problem finite.