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Category Archives: Uncategorized
Threelds
A field F consists of two compatible Abelian groups — an additive group on F and a multiplicative group on F \ {0} — such that multiplication distributes over addition. In certain cases, though, this multiplicative group can be the … Continue reading
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29yearold Conway conjecture settled
Ilkka Törmä and Ville Salo, a pair of researchers at the University of Turku in Finland, have found a finite configuration in Conway’s Game of Life such that, if it occurs within a universe at time T, it must have … Continue reading
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Training a random Gaussian generator
I’ve spent the last couple of months tackling the problem of designing an algorithm to rapidly generate highquality normallydistributed pseudorandom numbers on a GPU. Whilst this may seem quite pedestrian, it turned out to be much more interesting than I’d … Continue reading
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Involutions on a finite set
Suppose that R is the (assumed to be finite) set of solutions to a certain problem, and you’re interested in determining the parity of R. The following proof strategy works surprisingly often, namely in at least two different scenarios, and … Continue reading
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Hamming backups: a 2of3 variant of SeedXOR
EDIT (20211014): I’ve written a reference implementation of the Hamming backup idea introduced in this article. SeedXOR is an approach for splitting a Bitcoin wallet seed, adhering to the BIP39 standard, into N ‘parts’ (each the same size as the … Continue reading
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An efficient prime for numbertheoretic transforms
My new favourite prime is 18446744069414584321. It is given by , where . This means that, in the finite field , 2^32 functions as a primitive 6th root of unity, and therefore 2 is a primitive 192nd root of unity. … Continue reading
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Hamming cube of primes
Given two nonnegative integers, m and n, we say that they are Hammingadjacent if and only if their binary expansions differ in exactly one digit. For example, the numbers 42 and 58 are Hammingadjacent because their binary expansions 101010 and … Continue reading
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Determinacy
I’d like to take this opportunity to highly recommend Oscar Cunningham’s blog. One of the posts, entitled A Better Representation of Real Numbers, describes an elegant orderpreserving* bijection between the nonnegative reals [0, ∞) and ‘Baire space‘, , the space … Continue reading
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Oneway permutations
Oneway 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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