CryptoBase

A lightweight knowledge base for mathematics and cryptography notes.

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Graph view

See the graph for a visual map of every note and how they link together, or browse the chapters index.

Authoring reference

Everything below is live documentation: copy these patterns into any note. New chapters are folders under content/ with an _index.md (sort_by = "weight" plus a weight); pages need a weight too — weightless pages are silently dropped when the section sorts by weight. Cross-link with Zola internal links like Euler’s theorem — they become backlinks and graph edges automatically.

Callouts

Admonition syntax and kb.callout render the same CSS. All five types:

Info Info Callouts use admonition syntax, or the equivalent kb.callout shortcode.

Tip Tip Prefer $\equiv$ over $=$ for congruences.

Important Crypto relevance RSA decryption correctness rests on Euler’s theorem.

Warning Hypothesis check The coprimality condition is essential: $2^2 \equiv 1 \pmod{4}$ is false.

Example Example callout $7^{-1} \equiv 13 \pmod{15}$ since $7 \cdot 13 = 91 \equiv 1$.

Custom title

Same box via shortcode. Types: info, important, tip, warning, example.

Theorem environments (numbered, linkable)

Numbering is automatic per type (CSS counters, per-page). Add label="…" to any box to anchor it, then link with <a href="#label">text</a>. (Plain markdown [text](#label) fails zola check — it only knows header ids — so box cross-refs use raw <a href>.)

Definition (Congruence). §

$a \equiv b \pmod{n}$ iff $n \mid (a - b)$.

Theorem (Euler). §

If $\gcd(a, n) = 1$ then $a^{\varphi(n)} \equiv 1 \pmod{n}$.

Theorem above is Euler’s theorem; Definition is congruence.

Lemma . §

If $\gcd(a, n) = 1$ and $ab \equiv ac \pmod{n}$ then $b \equiv c \pmod{n}$.

Example (Inverse). §

$7^{-1} \equiv 13 \pmod{15}$.

Remark . §

Remarks share the box style with their own accent color.

Notation . §

$\varphi(n)$ counts units modulo $n$.

Proof. §

The units modulo $n$ form a group of order $\varphi(n)$; apply Lagrange.

The proof is unnumbered but linkable just like the rest (the ∎ is added automatically — don’t type it yourself).

Why does RSA work?

Long derivations fold away. The modular arithmetic note covers inverses; Euler’s theorem lifts it to full RSA correctness. Link directly: Why RSA works.

Full shortcode list: kb.definition, kb.theorem, kb.lemma, kb.example, kb.proof, kb.remark, kb.notation, kb.details. Each takes name="…" (except proof) plus optional label="…"; details takes summary="…" plus optional label="…" and renders a native <details> element.

Math and numbered equations

Inline math ($e^{i\pi} + 1 = 0$) and unnumbered display math:

$$ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} $$

Numbered equations use the equation environment with \label / \eqref:

\begin{equation} a^{\varphi(n)} \equiv 1 \pmod{n} \label{eq:euler} \end{equation}

Equation \eqref{eq:euler} is Euler’s theorem again, this time referenceable. Math renders with MathML included for screen readers.

Code (line numbers, highlights, copy button)

Basic numbered block with one highlighted line:

1def egcd(a, b):
2    if b == 0:          # highlighted line
3        return (a, 1, 0)
4    g, x1, y1 = egcd(b, a % b)
5    return (g, y1, x1 - (a // b) * y1)

Start numbering at 10 with a range highlight:

10def egcd(a, b):
11    if b == 0:
12        return (a, 1, 0)
13    g, x1, y1 = egcd(b, a % b)   # highlighted
14    return (g, y1, x1 - (a // b) * y1)  # highlighted

Hide boilerplate lines from display (still copied):

2def egcd(a, b):
3    if b == 0:
4        return (a, 1, 0)

Hover any block for the copy button. Inline code uses single backticks.

Tables (scroll on mobile)

a	n	gcd	invertible?
7	15	1	yes
6	15	3	no

Wide tables scroll horizontally inside the page on small screens.

Frontmatter (byline, both optional)

author and proofread_by live in [extra]. Each is shown only when present — omit both and no byline renders (as on this homepage):

[extra]
author = "Alice"
proofread_by = "Bob"

Diagrams

Pre-rendered SVG: crisp in both themes, click to enlarge

Commit SVGs to static/diagrams/ and reference them with kb.figure (alt is required, caption optional). Strokes follow the theme automatically; click enlarges with keyboard trap and Esc to close.

For quick drafts only, set [extra] tikz = true and embed a <script type="text/tikz">…</script> block (client-side TikZJax, needs network).