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Category Archives: Uncategorized
Hamming backups: a 2-of-3 variant of SeedXOR
EDIT (2021-10-14): 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 number-theoretic 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 Hamming-adjacent if and only if their binary expansions differ in exactly one digit. For example, the numbers 42 and 58 are Hamming-adjacent 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 order-preserving* bijection between the nonnegative reals [0, ∞) and ‘Baire space‘, , the space … Continue reading
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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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