An archive of lecture notes and topics for Fall 2026 will accumulate here as the semester progresses.

Daily Post Solutions

• Daily post due Monday, August 24th — classical ciphers (problems 9–12)
• Daily post due Wednesday, August 26th — modular arithmetic practice sheet, XOR, Sage worksheet, cancellation, and exponents (problems 3–8)
• Daily post due Friday, August 28th — affine known-plaintext attack, modular inverses, and the $3^{519}$ competition (problems 3–5)
• Daily post due Monday, August 31st (AI generated) — unit groups and $\varphi(n)$, a badly chosen affine key, powers of $a$ repeating, Hill cipher key recovery, and double-and-add (problems 2–7)
• Daily post due Wednesday, September 2nd (AI generated) — the $\varphi$ formula, successive squaring, Euler’s theorem and tower exponents (problems 1, 3–7)
• Daily post due Friday, September 4th (AI generated) — a complete worked example of the Diffie-Hellman key exchange exercise, message to ciphertext and back (problems 3–4)
• Daily post due Wednesday, September 9th (AI generated) — big-oh comparisons, and baby-step giant-step for $34 = 3^x \pmod{113}$ (problems 3–4)
• Daily post due Friday, September 11th (AI generated) — the parity of a discrete log, baby-step giant-step for $51 = 2^x \pmod{83}$, the tower $2^{(3^{2026})} \pmod{33}$, and linear versus binary search (problems 3–6)
• Daily post due Monday, September 14th (AI generated) — the Index Calculus for $2^x \equiv 17 \pmod{131}$, why you can’t divide by $2$ modulo $p-1$, and the runtime of the Index Calculus versus baby-step giant-step (problems 4–6)
• Daily post due Wednesday, September 16th (AI generated) — the optional El Gamal encryption chain worked end to end, breaking a key with Pohlig-Hellman and baby-step giant-step, and the repeated ephemeral key (problem 2)
• Daily post due Friday, September 18th (AI generated) — how $x^2 \equiv y^2 \pmod n$ factors $n$, and what the dynamical portraits say about the modulus (problems 2–3)
• Daily post due Monday, September 21st (AI generated) — El Gamal malleability and how to fight it, and the RSA ciphertext chain worked end to end (problems 1–2)
• Daily post due Wednesday, September 23rd (AI generated) — square roots $x^2 \equiv y^2$ and which $n$ admit them, the Ceres birthday problem, and Fermat’s test on 2001 and 2003 (problems 1, 5, 6; problem 7 is in the textbook, Section 3.7)
• Daily post due Friday, September 25th (AI generated) — Miller-Rabin on $90751$ and $3057601$, Fermat factoring, factoring from relations and from $\varphi(n)$, the quadratic sieve for $31861$, the small-message attack on RSA, and Carmichael numbers (problems 1–10)
• Daily post due Monday, September 28th (AI generated) — one equation gives both inverses, Alice’s RSA signature on $65$ and its verification, existential forgery, multiplying signatures, SHA-256 avalanche counts, breaking the parity and XOR toy hashes, and the birthday bound $2^{k/2}$ (problems 2–10)
• Daily post due Wednesday, September 30th (AI generated) — the Euclidean algorithm on $(135, 170)$ and $(183, 105)$, inverses from $7x + 15y = 1$ by inspection, recipe cards for $183x + 105y = 3$, all solutions of $3x + 5y = 0$ and $= 1$, the halving argument for the runtime, Fibonacci inputs as the slow case, $12x + 42y$ and $1071x + 462y = 21$, and $\gcd(2^a - 1, 2^b - 1)$ (problems 2–10)
• Daily post due Friday, October 2nd (AI generated) — the inverse of $35$ modulo $61$ by extended Euclid, the CRT system modulo $244$ and $495$, Sunzi’s problem, non-coprime moduli, inverting by Fermat, RSA decryption sped up by CRT, and Håstad’s broadcast attack (problems 1–8)
• Daily post due Monday, October 5th (AI generated) — square roots modulo $p \equiv 3 \pmod 4$ via $a^{(p+1)/4}$, square roots modulo $pq$ from the factorization, the four square roots of $71$ modulo $77$, square roots give factoring, and $\sqrt{3}$ modulo $11$ on the dynamical portrait (problems 1–4, 6)
• Daily post due Wednesday, October 7th (AI generated) — Pollard’s $\rho$ on $8051$ (the tortoise-and-rabbit table, why $97$ appears before $83$), the runs for $10403$ and $455459$, and the birthday-paradox $\sqrt{p}$ step count (problems 1–2)
• Daily post due Monday, October 12th (AI generated) — elliptic curve addition on $y^2 = x^3 + x + 1 \pmod 7$ ($P+Q$, $P+\infty$, $-P$, doubling by the tangent line, why $y = 0$ gives $2P = \infty$), a sum modulo $5$, the vertical-line failure modulo $7$, and the non-invertible denominator modulo $35$ that factors it (problems 1–8)
• Daily post due Wednesday, October 14th (AI generated) — four vectors with the same image in $\mathbb{P}^1_{\mathbb{R}}$, a colour class of $\mathbb{P}^2_{\mathbb{F}_3}$ and the count of $13$, the six points of $\mathbb{P}^1_{\mathbb{F}_5}$ with all their vectors, the sizes $p - 1$ and $(p^{k+1} - 1)/(p - 1)$ in general, the two points of $y^2 = xy - 1$ at infinity, and why vertical lines pass through $[0,1,0]$ (problems 2, 3, 5–8)
• Daily post due Friday, October 16th (AI generated) — reading the multiples of $P = (1,5)$ on $y^2 = x^3 + x + 1 \pmod{11}$: $\#E = 14$ and Hasse’s bound, the range $7$–$21$ modulo $13$, addition and negation by table lookup, points of order $2$, $7$ and $14$, the ECDLP $kP = (6,6)$ and why it doesn’t scale, and Elliptic Curve Diffie-Hellman (problems 1–7)
• Daily post due Monday, October 19th (AI generated) — a worked pass through the Elliptic Curve El Gamal chain (keys, padding the message onto the curve, encrypting, decrypting), breaking it by Pohlig–Hellman since the order of $P$ is $3 \cdot 40197392597 \cdot 511877268341$, why Pollard $p-1$ works, and the Sage loop factoring $16637 = 127 \cdot 131$ at $B = 7$ (problems 1–3)
• Daily post due Friday, October 23rd (AI generated) — the length of $\frac35 + \frac45 i$, normalizing and measuring $i|0\rangle + \sqrt{3}|1\rangle$, the $V$ basis states measured in the $L$ basis, and $\frac{1}{\sqrt 5}|\uparrow\rangle + \frac{2}{\sqrt 5}|\rightarrow\rangle$ measured in both bases (problems 2–5)
• Daily post due Monday, October 26th (AI generated) — BB84 on paper (the photons, the key $11000$, why photon 2 isn’t an error, what Eve learns from the bases), expected key length and the binomial, Eve intercepting in the wrong basis, the $\frac14$ disagreement rate and $(3/4)^{20}$ (problems 1, 3–5)
• Daily post due Friday, October 30th (AI generated) — two qubits measured separately and as one product state, the Bell state’s probabilities and why it isn’t a product, and normalizing $|00\rangle + 2i|01\rangle - 2|11\rangle$ (problems 1–4)
• Daily post due Monday, November 2nd (AI generated) — probabilities of one bit of $(|01\rangle + |10\rangle)/\sqrt 2$, entangled or not for four states, measuring the first qubit (outcome states and probabilities), conjugate transpose and a non-unitary matrix, Pauli $Z$ on a state, Hadamard distinguishing $|+\rangle$ from $|-\rangle$, and a gate sending $|0\rangle$ to $(|0\rangle + i|1\rangle)/\sqrt 2$ (problems 1–9)

Lectures

Friday, August 21st, 2026:

• First day slides: A Whirlwind Tour of Cryptography

Monday, August 24th, 2026:

• Introduction to modular arithmetic, including addition and multiplication and reducing as you go, NOT reducing exponents, and writing caesar cipher in terms of modular arithmetic.

Wednesday, August 26th, 2026:

• Multiplicative inverses and Hill cipher.

Friday, August 28th, 2026:

• Formal mathematics of inverses and gcd.

Friday, September 25th, 2026 (pre-recorded):

• Lecture video, notes part 1 and part 2.
• The quadratic sieve worked visually ($n = 2201 = 31 \cdot 71$), digital signatures and RSA signatures, existential forgery, and cryptographic hash functions (preimage resistance, collision resistance, the avalanche effect; MD5, SHA-1, SHA-2, SHA-3).

Friday, October 9th, 2026:

• Lecture notes: elliptic curves $y^2 = x^3 + ax + b$, the point at infinity, the chord-and-tangent group law and its formulas, inverses, and adding points modulo $p$.

Monday, October 12th, 2026:

• Lecture notes: projective space. Lines through the origin and slopes give the projective line $\mathbb{P}^1_{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$; vectors modulo scaling and the notation $[a,b]$; $\mathbb{P}^1_{\mathbb{F}_p}$ with the example $\mathbb{P}^1_{\mathbb{F}_3}$ (4 elements); the projective plane $\mathbb{P}^2_{\mathbb{R}}$ as lines through the origin in space and as a sphere with antipodal points identified; $\mathbb{P}^2_{\mathbb{F}_3}$ (26 nonzero vectors, 13 points); and elliptic curves in projective space, $Y^2 Z = X^3 + aXZ^2 + bZ^3$, with the affine part $Z \neq 0$ and the single point $[0,1,0] = \infty$ on the line at infinity $Z = 0$.

Wednesday, October 21st, 2026 (Zoom; recording link on discord):

• Lecture notes: complex numbers (addition, multiplication, length, conjugates and inverses, polar form $re^{i\theta}$ where lengths multiply and angles add); qubits $a|0\rangle + b|1\rangle$ with $|a|^2 + |b|^2 = 1$, superposition and global phase; measurement and collapse; the $L$ basis $|\uparrow\rangle, |\rightarrow\rangle$ and $V$ basis $|\nearrow\rangle, |\nwarrow\rangle$ for polarized light, and the change of basis between them; polarizing filters and the three filter experiment ($1/8$ of the light gets through).

Friday, October 23rd, 2026 (Zoom; recording link on discord):

• Lecture notes: normalizing and measuring a qubit (the daily problem $i|0\rangle + \sqrt{3}|1\rangle$); photons in the $V$ and $L$ bases and changing between them; Quantum Key Distribution (BB84, Bennett and Brassard) with the encoding $|\uparrow\rangle = 0$, $|\rightarrow\rangle = 1$ in $L$ and $|\nwarrow\rangle = 0$, $|\nearrow\rangle = 1$ in $V$: Phase I (Alice sends random bits in random bases, Bob measures in random bases), Phase II (compare bases publicly, keep the rows where they agree), an eavesdropper corrupts each shared bit with probability $1/4$, and Phase III (reveal and compare some bits; Eve escapes $k$ checks with probability $(3/4)^k$).

Monday, October 26th, 2026:

• Slides: Here comes the Hackpocalypse: cybersecurity in the AI era. Password storage (salted slow hashes), password strength and guessing times, reuse and password managers, second factors (TOTP, passkeys and YubiKeys, and why passkeys can’t be phished), memory safety, the padlock in the URL bar and certificate authorities, bad randomness, reading a URL, AI capabilities, authentication for humans, identity theft, and quantum computers.