Concept

Inverse

The element that undoes another under an operation, returning the identity. Modulo a whole number it exists exactly for the residues coprime to the modulus, and the extended Euclidean algorithm finds it in a few steps.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

eˣ and its inverse, reflected in the diagonal. A curve, the line y = x, and the curve reflected in it — which is the graph of the inverse function. Tangents are drawn at matched pairs of points, and the two slopes at each pair multiply to one.

The slope of the mirror image

Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.

analysis · The derivative

Four outputs are enough to find the rule. A row of 12 outputs of a linear generator, with the first 4 marked as given and the rest as predicted. The multiplier and increment recovered from the given ones reproduce every later output exactly.

Four numbers and the rule is yours

A linear generator can be solved. Given a few of its outputs, the multiplier and the increment fall out of two congruences, and every future output is then known exactly — which is a failure of a completely different kind from the lattice defect, and is not detected by any test of how evenly the points are spread.

computation · Pseudorandomness

Named alongside it

The objects these essays reach for when they reach for this one.

Approximation Continuity Cryptography Derivative Limit Linear system Logarithm Modular arithmetic Next bit test Prediction Pseudorandomness Randomness

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