Author Archives: apgoucher

One-way permutations

One-way permutations are fascinating functions that possess a paradoxical pair of properties. They are efficiently computable functions φ from [n] := {0, 1, …, n−1} to itself that are: Mathematically invertible, meaning that φ is a bijection; Cryptographically uninvertible, meaning … Continue reading

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Cyclotomic fields

The nth cyclotomic field is the field generated by a primitive nth root of unity, ζ. It is an example of a number field, consisting of algebraic numbers, and its dimension is φ(n) when regarded as a vector space over … Continue reading

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Urbit for mathematicians

The three months since the last cp4space article have largely been spent learning about an interesting project called Urbit. My attention was drawn to its existence during a talk by Riva-Melissa Tez (formerly of Intel) on the importance of continued … Continue reading

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A curious construction of the Mathieu group M11

Previously, we discussed which regular polytopes have vertex-sets that occur as proper subsets of the vertex-set of another regular polytope in the same dimension. In particular, when there is a Hadamard matrix of order 4k, then then the (4k−1)-dimensional simplex … Continue reading

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Assorted topics

This is a digest of things that have happened this month, but which are individually too small to each warrant a separate cp4space post. Firstly, there have been a couple of exciting results in the field of combinatorics: The Erdős-Faber-Lovász … Continue reading

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Meagre sets and null sets

There are two competing notions for describing a subset of the real numbers as being ‘small’: a null set is a subset of the reals with Lebesgue measure zero; a meagre set is a countable union of nowhere-dense sets. Both … Continue reading

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Keep your public keys private

Yes, the title sounds very counterintuitive. After all, don’t digital signature schemes require the general public to know your public key so that they can verify your signatures? That is correct, but importantly they don’t need to know your public … Continue reading

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The neural network of the Stockfish chess engine

Last time, we briefly mentioned the high-level differences between Stockfish and Leela Chess. To recap, Stockfish evaluates about 100 million positions per second using rudimentary heuristics, whereas Leela Chess evaluates 40 000 positions per second using a deep neural network … Continue reading

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Rigid heptagon linkage

In January 2000, Erich Friedman considered the problem of finding a rigid unit-distance graph G containing a regular heptagon as a subgraph. That is to say, the graph is immersed in the plane such that: every edge of G must … Continue reading

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Barreto-Naehrig curves and cryptographic pairings

There’s a very elegant cryptographic construction discovered by Barreto and Naehrig in a 2005 paper. It is beautiful from a pure mathematical perspective, but also has an impressive application: it was* part of the ingenious mechanism by which Zcash supports … Continue reading

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