Engineering

Generating a Random k-Bit Prime with Miller-Rabin

Assembling bit-length constraint, trial division filtering and Miller-Rabin into a complete prime generator.

Generating a Random Non-Increasing Sequence

Sampling a random non-increasing sequence in a bounded range, and its role as a subroutine in generating factored numbers.

Generating a Random Number from a Given Interval

Sampling uniformly from an arbitrary range using a source of random bits, and controlling the resulting bias.

Generating a Random Prime

Generating a random prime by repeated candidate testing, the expected number of trials, and the sieving optimisation.

Generating a Random Prime Between 2 and M

Generating a uniform random prime below a bound, the analysis of the retry loop, and the resulting output distribution.

Group Homomorphisms and Isomorphisms

Structure-preserving maps between groups, isomorphisms, and what it means for two groups to be the same.

Hash Function Families

Families of hash functions, keyed selection, and the properties required of them in algorithm design.

Hash Tables

Hash tables analysed with universal families: expected chain length, load factor and collision resolution.

Hensel Lifting for Polynomial Factors

Lifting a factorisation modulo a prime to a factorisation modulo a prime power, the quadratic variant, and the coprimality condition.

Engineering Mathematics13 Aug 2026 · 21 min read

Homogeneous Systems of Equations

Linear systems with every constant term zero: why they are always consistent, what the trivial solution is, and why more unknowns than equations guarantees infinitely many solut…

Ideal Multiplication and Division

Multiplying, inverting and dividing ideals as module operations, and controlling the growth these operations cause.

Ideal Norm Computation

The norm of an ideal as its index in the order, its multiplicativity, and its use as a size measure and consistency check.

Ideal Reduction in Number Fields

Finding a small ideal in a given ideal class by lattice reduction, and why reduction is the enabling step for relation collection.

Ideal Representation by Two Elements

Representing an ideal by two generators, why two always suffice, and the trade-off against the canonical matrix form.

Engineering Mathematics13 Aug 2026 · 9 min read

Ideals and Greatest Common Divisors of Integers

Greatest common divisors defined through ideals, Bezout's identity, and why the ideal-theoretic view is the one that generalises.

Ideals and Quotient Rings

Ideals, principal ideal domains, quotient rings, and the condition under which a quotient is a field.

Ideals of the Maximal Order

Ideals and fractional ideals, unique factorisation into primes, and the group structure that makes the class group possible.

Imaginary Quadratic Class Numbers by Counting Reduced Forms

Computing class numbers and group structure for imaginary quadratic fields by enumerating reduced forms, and where the method runs out.

Implementation Pitfalls and Testing Strategy

The recurring implementation errors in this subject and the testing discipline that catches them.

Implementing the Jacobi Sum Test

Practical implementation of the Jacobi sum test: precomputation, cyclotomic arithmetic, and the final divisor search.

Independent and Dependent Variables

Pivot columns make variables dependent, non-pivot columns make them free: how to partition the unknowns, count degrees of freedom and describe an infinite solution set exactly.

Infinite Discrete Probability Distributions

Countably infinite sample spaces, convergence conditions, and the geometric distribution arising from unbounded loops.

Engineering Mathematics13 Aug 2026 · 21 min read

Injective Linear Transformations

Injective (one-to-one) linear transformations: the defining condition, how to prove injectivity by reducing to a homogeneous system, and how to disprove it.

Injective Transformations, Independence and Dimension

How injective linear transformations preserve linear independence, why the images of a basis decide injectivity, and why domain dimension cannot exceed codomain.

Engineering Mathematics13 Aug 2026 · 20 min read

Inner Products of Vectors

The complex inner product of two column vectors: its definition with a conjugate on the second argument, additivity, scalar rules, conjugate symmetry and positivity.

Integer Addition and Subtraction

Multiprecision addition and subtraction: carry and borrow propagation, sign handling, and why both are linear.

Integer Division with Remainder

Multiprecision division: the normalisation step, digit estimation, correction, and why division is harder to implement than multiplication.

Integer Kernel and Image via LLL

Computing a reduced basis of the integer kernel and image of a matrix, and why this is not the same as clearing denominators from a rational kernel.

Integer Multiplication

Schoolbook multiplication of multiprecision integers, the accumulator requirement, and where the quadratic cost comes from.

Integer Square Root and Perfect Power Detection

Newton iteration for integer square roots, exact perfect power detection, and why these cheap tests belong at the front of every factoring routine.

Integral LLL: Avoiding Floating Point

Running LLL entirely in integer arithmetic using scaled Gram-Schmidt quantities, and why this matters for exact downstream computation.

Inverse Image and Supplementation of Subspaces

Solving for preimages of a subspace, extending an independent set to a basis, and the linear algebra primitives these support.

Engineering Mathematics13 Aug 2026 · 25 min read

Invertibility Criteria for Linear Transformations

A linear transformation is invertible exactly when it is injective and surjective. Criteria, kernel and range tests, composition rules and the inverse map.

Invertible Linear Transformations

Invertible linear transformations: the identity map, the two-sided composition condition defining an inverse, uniqueness, and why the inverse is itself linear.

Irreducible Polynomials

Recognising irreducible polynomials, the standard criteria, and counting them over a finite field.

Isogenies and Endomorphism Rings

Isogenies as maps of curves respecting the group law, and the two possible endomorphism rings over a field of characteristic zero.

Kernel and Image of a General Matrix

Computing kernel and image bases for rectangular matrices over a field, and why the integer analogue is a different problem.

Kernel of a Linear Transformation

The kernel of a linear transformation: definition, why it is always a subspace, its role as the preimage of zero, and why a trivial kernel means injectivity.

Kernels, Images and the Isomorphism Theorems

The kernel and image of a homomorphism, and the first isomorphism theorem relating them to a quotient.

L-Functions and the Birch-Swinnerton-Dyer Conjecture

The L-function of an elliptic curve, the Birch-Swinnerton-Dyer conjecture, and what can and cannot be computed about rank.

Language Recognition and Complexity Classes

Randomised complexity classes RP, co-RP, BPP and ZPP, and where primality testing sits among them.

Lattice Definitions and Quadratic Forms

Lattices as discrete subgroups, the Gram matrix, and the correspondence between lattices with a basis and positive definite quadratic forms.

Lattice Determinant and the Hadamard Bound

The lattice determinant as a basis-independent invariant, the Hadamard inequality, and using the orthogonality defect to measure basis quality.

Lattices, Complex Tori and the Weierstrass p-Function

The identification of elliptic curves over the complex numbers with complex tori, and the Weierstrass function that realises it.

Learning Pathways in Computational Number Theory

Suggested routes through the collection for cryptography, computer algebra, coding theory and pure mathematics readers.

Learning Pathways Through Computational Number Theory

Four routes through this 182-page collection, chosen by what you need to build rather than by chapter order.

Legendre, Jacobi and Kronecker Symbol Computation

The three quadratic symbols, their differing meanings, and the reciprocity-based algorithm that computes them without factoring.

Lehmer's Accelerated GCD Computation

Lehmer's method: running many GCD steps on single-precision leading digits before touching the full multiprecision operands.