How to read the math in this curriculum¶
Prereqs: Orientation recommended (Motivation → Challenges) · or none if jumping straight to symbols
Next: Why cryptography?
Learning goals. Recognize common symbols (\(=\), \(\equiv\), \(\oplus\), \(\Pr\), vectors), know that you can skip a formula on first read and come back, and practice translating several lines of math into English — slowly, on purpose.
Symbols on this page (starter kit)¶
| Symbol | English |
|---|---|
| \(=\) | Ordinary equals |
| \(\equiv\pmod{n}\) | Same leftover mod \(n\) |
| \(\oplus\) | XOR |
| \(\Pr[\ldots]\) | Probability that … |
| \(\mathbf{e}\) | A whole list (vector), not one number |
| \(2^{10}\) | Two×itself ten times (\(1024\)), not “twenty” |
Full curriculum dictionary: Glossary.
Why this page exists¶
PQC uses symbols. That does not mean you must already love advanced math. It means we need a shared shorthand. This page is a symbol survival guide. Read it once; return whenever a later page feels dense.
You are allowed to:
- Read the English first.
- Ignore a displayed formula on the first pass.
- Come back and translate the formula word-by-word.
- Write your own English under every line until it feels boring.
Feeling slow is normal. Professionals unpack formulas the same way.
Story first — one noisy message¶
Imagine Alice sends a protected bit-string \(c\). Noise flips some bits; call that noise \(e\). Bob sees
Say that out loud: “received word equals codeword XOR error.”
If you can say that, the symbols are already doing their job. Everything below is vocabulary for the same habit.
Same story, slower¶
| Symbol | Role in the story |
|---|---|
| \(c\) | What Alice meant to send (clean) |
| \(e\) | What the channel flipped (noise) |
| \(y\) | What Bob actually got |
| \(\oplus\) | “Combine by flipping where \(e\) has ones” |
You do not need to solve for \(e\) yet. Just name the pieces.
The only rules we really need early¶
1. Equals vs congruence¶
- \(a = b\) means “same number” in ordinary arithmetic.
- \(a \equiv b \pmod{n}\) means “same remainder when divided by \(n\)” (clock math).
Example: \(15 \equiv 3 \pmod{12}\) because both leave remainder \(3\) on a 12-hour clock.
Say it: “fifteen is congruent to three mod twelve.”
Another: \(10 + 3 = 13\), but \(10 + 3 \equiv 1 \pmod{12}\).
Ordinary sum and clock-sum can disagree — that is intentional.
2. Bits and XOR¶
A bit is \(0\) or \(1\).
\(a \oplus b\) means XOR: same bits → \(0\), different bits → \(1\).
In \(\mathbb{F}_2\) (our bit world), XOR is the same as addition: \(1+1=0\).
| \(a\) | \(b\) | \(a\oplus b\) |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
3. Lists of bits (vectors)¶
\((1,0,1)\) is just three bits in a row.
Bold letters like \(\mathbf{e}\) usually mean “a whole list,” not one number.
Length of \((1,0,1)\) is \(3\).
Weight (number of ones) of \((1,0,1)\) is \(2\).
4. Probability¶
\(\Pr[\text{something}]\) means “probability that something happens,” a number between \(0\) and \(1\).
Example: fair coin \(\Pr[\text{Heads}]=\tfrac12\).
Read \(\tfrac12\) as “one half,” not “twelve.”
5. Subscripts and superscripts¶
- \(a_1, a_2, a_3\) — first, second, third piece.
- \(2^{10}\) — two multiplied by itself ten times (\(1024\)), not “twenty.”
- \(X^2\) in a polynomial — the placeholder \(X\) squared (a label for position), not “mystery variable you must solve for” in our early examples.
6. Fractions and “at most”¶
\(\Pr[W \ge t] \le 2^{-128}\) uses:
- \(\ge\) “at least / greater or equal”
- \(\le\) “at most / less or equal”
- \(2^{-128}\) “one over two-to-the-one-hundred-twenty-eight” (tiny)
You do not compute that number by hand. Hear: “astronomically small.”
7. Set braces (light)¶
\(\{0,1\}\) means “the set containing only \(0\) and \(1\).”
\(\mathbf{e}\in\{0,1\}^n\) means “\(e\) is a length-\(n\) bit list.”
Skip set notation on first read if it stresses you; return later.
How to translate a formula (method)¶
Recipe: (1) name each symbol in English, (2) name the operation, (3) restate the whole line as one sentence.
Worked example — aloud #1¶
| Piece | English |
|---|---|
| \(y\) | received word |
| \(c\) | codeword (clean protected message) |
| \(e\) | error / noise bit-string |
| \(\oplus\) | XOR, position by position |
Full sentence: “The received word \(y\) is the codeword \(c\) XOR the error \(e\).”
Worked aloud #2¶
| Piece | English |
|---|---|
| \(H\) | a table of parity checks (a matrix) |
| \(\mathbf{e}\) | the error list |
| \(\mathbf{s}\) | the syndrome (fingerprint of the error) |
| \(\top\) | “stand the list up as a column” (some books omit it) |
Full sentence: “Matrix \(H\) times the error list \(\mathbf{e}\) produces the syndrome list \(\mathbf{s}\).”
Worked aloud #3 (probability)¶
Full sentence: “The probability that the weight \(W\) is at least \(t\) is at most about one in \(2^{128}\).”
You do not need to compute \(2^{128}\) yet — just hear “astronomically small.”
Worked aloud #4 (modular)¶
Full sentence: “\(a\) plus \(b\) leaves the same remainder as \(c\) when you wrap around modulus \(q\).”
Worked aloud #5 (KEM slogan)¶
Full sentence: “The session key \(K\) is what Decaps returns from the secret key and the ciphertext.”
Sans-serif names like \(\mathsf{Decaps}\) are just function names — “a recipe with that label.”
Displayed math vs inline math¶
- Inline: \(1+1=0\) sits inside a sentence.
- Displayed:
is the same idea, just centered for breathing room.
On GitHub or MkDocs with KaTeX, both should render as pretty math. If you see raw $...$ text, use mkdocs serve for a better view. This curriculum uses $...$ / $$...$$ on purpose (GitHub-friendly).
Slow-motion lab (do on paper)¶
- Translate aloud: \(c = m \oplus p\).
- What is \(11 \bmod 5\)?
- Compute \(110 \oplus 101\).
- Write one English sentence for \(\Pr[b=1]=\tfrac14\).
- Is \(2^3\) equal to \(6\) or \(8\)?
Answers
1. “Ciphertext equals message XOR pad” (or similar). 2. $1$. 3. $011$. 4. “The probability that bit $b$ is one equals one fourth.” 5. $8$.Check yourself¶
- Write in English: \(m \oplus p = c\).
- What is \(7 \bmod 5\)?
- What is \(101 \oplus 001\)?
- Say out loud what \(\Pr[W\ge 3]\) is asking.
- Translate aloud: \(K = \mathsf{Decaps}(\mathsf{sk},\mathsf{ct})\).
- True or false: you must understand every symbol on first sight before continuing.
Answers
1. Message XOR pad equals ciphertext (or similar wording). 2. $2$. 3. $100$. 4. “Probability that the weight $W$ is at least 3.” 5. “The session key $K$ is what Decaps returns from the secret key and ciphertext.” 6. False — English first, symbols on the second pass.Emotional contract¶
Feeling slow is normal. Cryptography professionals still unpack formulas line by line. Speed is not the goal — accurate English meaning is.
If a later page piles up symbols, open this page in another tab and translate one line at a time.
Next steps¶
- Why cryptography?
- Then Modular arithmetic when you are ready for clock math
- Return here anytime a formula feels opaque