Nostr notes by MagicInternetMath

📐 x² + x + 1 has no root in GF(32) The polynomial $x^2 + x + ...

2026-10-08T09:28:55Z

📐 x² + x + 1 has no root in GF(32)

The polynomial $x^2 + x + 1$ has no root in GF(32). Since it has degree 2, this also makes it irreducible over GF(32).

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 Fresh-secret generation Fix a threshold $k$ with $2 \\\\le k ...

2026-10-08T02:17:13Z

📖 Fresh-secret generation

Fix a threshold $k$ with $2 \\\\le k \\\\le 9$, an identifier of four bech32 symbols, and a supported master seed length. Generate $k$ payloads independently and uniformly at random, and assign them to the first $k$ of the required initial indices. The master seed is the value the resulting polynomials take at index s.

From: codex32
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📖 Generator polynomial and the code it defines Fix a monic ...

2026-10-07T22:17:29Z

📖 Generator polynomial and the code it defines

Fix a monic polynomial $g(X)$ of degree $c$ with coefficients in GF(32), called the generator polynomial. For a length $n > c$, the code it defines is $C = \\\\{\\\\, u(X) : \\\\deg u < n \\\\text{ and } g(X) \\\\mid u(X) \\\\,\\\\}$ that is, the strings of length $n$ whose polynomial is a multiple of $g$.

From: codex32
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📖 Computational versus information-theoretic security A scheme ...

2026-10-07T09:17:36Z

📖 Computational versus information-theoretic security

A scheme is computationally secure if recovering the secret from what the attacker observes is possible in principle but believed to require infeasible work. It is information-theoretically secure if the observation does not determine the secret at all: more than one secret is exactly as consistent with it, so no amount of computation, and no future machine, can decide between them.

From: codex32
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📐 ℤn is not a field when n is composite For composite $n$, ...

2026-10-07T01:49:31Z

📐 ℤn is not a field when n is composite

For composite $n$, the set $\\\\mathbb{Z}_n$ contains zero divisors and therefore fails the field axioms.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 The register as a polynomial Write the register contents as ...

2026-10-06T21:54:27Z

📖 The register as a polynomial

Write the register contents as $R(x) = r_{12}x^{12} + \\\\cdots + r_1 x + r_0$, where each $r_j$ is the five-bit symbol in slot $j$. Shifting the whole register left by five bits moves every coefficient up one slot, so in polynomial terms a five-bit left shift is multiplication by $x$.

From: codex32
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📐 Any k shares recover the secret If $k$ shares of the same ...

2026-10-06T17:25:32Z

📐 Any k shares recover the secret

If $k$ shares of the same secret are held, the secret symbol is determined and is computed by evaluating the Lagrange sum at $\\\\sigma$.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📐 β = 5z has order exactly 93 $\\\\beta^{93} = 1$, while ...

2026-10-06T10:47:57Z

📐 β = 5z has order exactly 93

$\\\\beta^{93} = 1$, while $\\\\beta^{3}$ and $\\\\beta^{31}$ are not 1. Therefore $\\\\beta$ has multiplicative order 93.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📐 Symbol count and surplus A seed of $n$ bytes occupies $m = ...

2026-10-06T03:20:41Z

📐 Symbol count and surplus

A seed of $n$ bytes occupies $m = \\\\lceil 8n/5 \\\\rceil$ symbols, with a surplus of $r = 5m - 8n$ bits, and $0 \\\\le r \\\\le 4$.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 Expanded codeword The expanded codeword of a codex32 string ...

2026-10-05T22:55:44Z

📖 Expanded codeword

The expanded codeword of a codex32 string is the five expansion symbols 3, 3, 0, 13, 19 followed by every data symbol after the separator — header, payload and checksum alike. Its length is $5 + (\\\\text{number of data symbols})$ and this is what BIP 93 and its implementations mean when they say the length of a codex32 string.

From: codex32
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📐 The five constants are one generator scaled by a basis Let ...

2026-10-05T16:24:23Z

📐 The five constants are one generator scaled by a basis

Let $g$ be the generator, packed as $G_0 = \\\\texttt{0x19dc500ce73fde210}$. Then $G_i$ is $g$ with every coefficient multiplied by $2^i$ in GF(32), and for a departing symbol $t$ the XOR of the selected constants equals $t \\\\cdot g$.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 Master seed format Generic codex32, plus the requirement ...

2026-10-05T07:42:05Z

📖 Master seed format

Generic codex32, plus the requirement that the payload be one of the six application lengths: 26, 32, 39, 45, 52 or 103 symbols — corresponding to master seeds of 16, 20, 24, 28, 32 or 64 bytes.

From: codex32
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📖 Polynomial over GF(32) A polynomial over GF(32) is a finite ...

2026-10-04T22:32:57Z

📖 Polynomial over GF(32)

A polynomial over GF(32) is a finite expression $f(X) = a_0 + a_1 X + a_2 X^2 + \\\\cdots + a_d X^d$ in which every coefficient $a_i$ is an element of GF(32). If $a_d \\\\ne 0$, the degree of $f$ is $d$. The zero polynomial has no degree.

From: codex32
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📐 Eight-symbol detection Any two equal-length valid codex32 ...

2026-10-04T18:48:30Z

📐 Eight-symbol detection

Any two equal-length valid codex32 strings within a checksum period that differ in at most eight symbols are in fact the same string.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📐 ℤp is a field when p is prime For $p$ prime, ...

2026-10-04T14:59:14Z

📐 ℤp is a field when p is prime

For $p$ prime, $\\\\mathbb{Z}_p$ is a field with $p$ elements.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📐 A field has no zero divisors If $ab = 0$ in a field and $a ...

2026-10-04T08:49:11Z

📐 A field has no zero divisors

If $ab = 0$ in a field and $a \\\\ne 0$, then $b = 0$.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 Derivation Computing the share at an index not currently ...

2026-10-04T02:00:01Z

📖 Derivation

Computing the share at an index not currently held, by interpolating the held shares at that index — column by column, exactly as in Part 3.

From: codex32
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📐 A one-symbol one-time pad over GF(32) has perfect secrecy ...

2026-10-03T21:47:21Z

📐 A one-symbol one-time pad over GF(32) has perfect secrecy

With a uniformly random key used once, every ciphertext is equally likely under every message, so observing the ciphertext leaves the attacker\

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📐 Every order divides 1023 The order of any nonzero element of ...

2026-10-03T18:17:55Z

📐 Every order divides 1023

The order of any nonzero element of GF(1024) divides $1023 = 3 \\\\cdot 11 \\\\cdot 31$, so it is one of 1, 3, 11, 31, 33, 93, 341, 1023.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📐 The conjugation map The map $t \\\\mapsto t^{32}$ is a field ...

2026-10-03T13:56:27Z

📐 The conjugation map

The map $t \\\\mapsto t^{32}$ is a field automorphism of GF(1024). In coordinates it is $(a + bz) \\\\mapsto (a+b) + bz$. It fixes exactly the elements of GF(32), it swaps the two roots of $x^2 + x + 1$, and applying it twice returns the original element.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📐 Any single transcription error is detected If a valid ...

2026-10-03T08:34:20Z

📐 Any single transcription error is detected

If a valid codex32 string is transcribed with one wrong symbol, the result fails verification. The wallet that would have been silently wrong is loudly wrong instead.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 The bech32 alphabet The 32 characters, in field-value order: ...

2026-10-03T02:20:02Z

📖 The bech32 alphabet

The 32 characters, in field-value order: qpzry9x8gf2tvdw0s3jn54khce6mua7l The character at position $i$ in that string represents the field element $i$, counting from zero.

From: codex32
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📐 Creation always produces a string that verifies Any string ...

2026-10-02T23:16:37Z

📐 Creation always produces a string that verifies

Any string assembled by appending the checksum that the creation procedure produces passes verification. In the formalization this is Checksum.verify_create; the companion result encoded_seed_checksum_valid carries it through the encoder, proving that every serialized master seed has a valid checksum.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 Weight The weight of a polynomial is the number of its ...

2026-10-02T19:32:35Z

📖 Weight

The weight of a polynomial is the number of its nonzero coefficients. A polynomial of weight at most 8 is one that differs from zero in at most eight places, however spread out.

From: codex32
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📐 The long-only verifier is strictly more permissive There are ...

2026-10-02T14:09:14Z

📐 The long-only verifier is strictly more permissive

There are strings that Checksum.verifyLong accepts and that Checksum.verify rejects. The BIP gives one explicitly.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 Parity check A parity check on strings of length $n$ is a ...

2026-10-02T07:32:15Z

📖 Parity check

A parity check on strings of length $n$ is a linear equation $h_1 u_1 + h_2 u_2 + \\\\cdots + h_n u_n = 0$ with fixed coefficients $h_i$ in GF(32). A code defined as the set of strings satisfying a fixed collection of parity checks is automatically linear, since the solution set of a system of homogeneous linear equations is a subspace.

From: codex32
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📖 Detection, correction, and erasure filling Detection is ...

2026-10-02T01:05:29Z

📖 Detection, correction, and erasure filling

Detection is answering "this string is not valid". It requires no knowledge of what the right string was. Correction is answering "the right string was this one", when neither the number of damaged positions nor their locations is known in advance. Erasure filling is answering "the missing symbols were these", when the damaged positions are known and only their contents are unknown.

From: codex32
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📖 A validated share set A collection of shares carrying ...

2026-10-01T21:24:14Z

📖 A validated share set

A collection of shares carrying evidence that: the threshold is nonzero and identical across all of them; the identifiers match; the payload lengths match; the share indices are pairwise distinct; and none of them is the secret index.

From: codex32
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🏆 Magic Internet Math - Daily Top 10 🥇 Ape Mithrandir - ...

2026-10-01T18:07:41Z

📐 Nonzero elements of GF(32) are invertible For every nonzero ...

2026-10-01T16:42:27Z

📐 Nonzero elements of GF(32) are invertible

For every nonzero $a \\\\in \\\\mathrm{GF}(32)$ there is exactly one $b$ with $a b = 1$.

Proof: See the collapsible proof in the section itself, where it is presented in full.

From: codex32
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📖 Generic codex32 A prefix, a six-symbol header, a payload of ...

2026-10-01T09:26:31Z

📖 Generic codex32

A prefix, a six-symbol header, a payload of anywhere from 0 to 997 symbols, and a checksum of the size the length rules demand. Nothing constrains what the payload means.

From: codex32
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📖 Zero divisor A nonzero element $a$ is a zero divisor if ...

2026-10-01T02:27:10Z

📖 Zero divisor

A nonzero element $a$ is a zero divisor if there is some nonzero $b$ with $a \\\\times b = 0$.

From: codex32
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📖 Irreducible Polynomial A polynomial $f \\in F[X]$ of degree ...

2026-09-10T05:19:34Z

📖 Irreducible Polynomial

A polynomial $f \\in F[X]$ of degree $\\geq 1$ is irreducible over $F$ if it cannot be written as a product $f = gh$ with $\\deg(g), \\deg(h) \\geq 1$.

From: gal-weintraub
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📖 Solvable Group (Preview) A group $G$ is solvable if there ...

2026-09-10T00:37:11Z

📖 Solvable Group (Preview)

A group $G$ is solvable if there exists a chain of subgroups $\\{e\\} = G_0 \\triangleleft G_1 \\triangleleft G_2 \\triangleleft \\cdots \\triangleleft G_k = G$ where each $G_i$ is normal in $G_{i+1}$ and each quotient $G_{i+1}/G_i$ is cyclic of prime order. $S_3$ and $S_4$ are solvable, but $S_5$ is not.

From: gal-edwards
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📐 Hilbert Theorem 90 (Additive Form) Let $K/F$ be a cyclic ...

2026-09-09T21:58:03Z

📐 Hilbert Theorem 90 (Additive Form)

Let $K/F$ be a cyclic Galois extension with generator $\\sigma$. An element $\\beta \\in K$ has $T_{K/F}(\\beta) = 0$ if and only if $\\beta = \\alpha - \\sigma(\\alpha)$ for some $\\alpha \\in K$.

From: gal-morandi
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📐 Constructibility Criterion A length $\\alpha$ is ...

2026-09-09T18:51:37Z

📐 Constructibility Criterion

A length $\\alpha$ is constructible by straightedge and compass if and only if $\\alpha$ lies in a field extension of $\\mathbb{Q}$ of degree $2^n$ for some $n \\geq 0$.

From: gal-weintraub
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📐 Cramer If the determinant $D$ of the coefficient matrix is ...

2026-09-09T14:48:50Z

📐 Cramer

If the determinant $D$ of the coefficient matrix is nonzero, the system $\\sum_j a_{ij}x_j = b_i$ has the unique solution $D \\cdot x_k = \\sum_i A_{ik} b_i$, where $A_{ik}$ are the cofactors.

Proof: Multiply the $i$-th equation by $A_{ik}$ and sum over $i$. By the orthogonality relations of cofactors, $\\sum_i a_{ij} A_{ik} = D$ if $j=k$ and $0$ otherwise. This gives $D \\cdot x_k = \\sum_i A_{ik} b_i$.

From: gal-artin
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📐 Dimension of a Sum If $U_1$ and $U_2$ are subspaces of a ...

2026-09-09T10:08:14Z

📐 Dimension of a Sum

If $U_1$ and $U_2$ are subspaces of a finite-dimensional vector space, then $\\dim(U_1 + U_2) = \\dim U_1 + \\dim U_2 - \\dim(U_1 \\cap U_2)$.

From: linalg-axler
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