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Category Archives: Uncategorized
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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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 RivaMelissa 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 vertexsets that occur as proper subsets of the vertexset 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 ErdősFaberLovász … 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 nowheredense sets. Both … Continue reading
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