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Mathematical Cryptography

A substitution cipher maps each plaintext letter to a fixed ciphertext letter. Caesar's shift is a special case: c ≡ p + k (mod 26) .

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A substitution cipher maps each plaintext letter to a fixed ciphertext letter. Caesar's shift is a special case: c ≡ p + k (mod 26) . Key sections include: MATHEMATICAL CRYPTOGRAPHY; Substitution & the Statistics That Betray It; The One-Time Pad — Unbreakable, Unusable; Modular Arithmetic — Integers, Wrapped; Fermat & Euler — Why RSA Works; RSA (1977) — Factoring as a Trapdoor; Diffie–Hellman — A Secret in Public; Elliptic Curves — Same Idea, Smaller Keys; One-Way: SHA-256 and the Compression Trick; Signing — Authorship Without Disclosure.

Key sections

  • 01MATHEMATICAL CRYPTOGRAPHY
  • 02Substitution & the Statistics That Betray It
  • 03The One-Time Pad — Unbreakable, Unusable
  • 04Modular Arithmetic — Integers, Wrapped
  • 05Fermat & Euler — Why RSA Works
  • 06RSA (1977) — Factoring as a Trapdoor
  • 07Diffie–Hellman — A Secret in Public
  • 08Elliptic Curves — Same Idea, Smaller Keys
  • 09One-Way: SHA-256 and the Compression Trick
  • 10Signing — Authorship Without Disclosure
  • 11Zero-Knowledge — Convince Without Revealing
  • 12Post-Quantum — Past the Reach of Shor
  • 13Where to Go Next

Topics covered

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Slide outline
  1. 01MATHEMATICAL CRYPTOGRAPHY
  2. 02Substitution & the Statistics That Betray It
  3. 03The One-Time Pad — Unbreakable, Unusable
  4. 04Modular Arithmetic — Integers, Wrapped
  5. 05Fermat & Euler — Why RSA Works
  6. 06RSA (1977) — Factoring as a Trapdoor
  7. 07Diffie–Hellman — A Secret in Public
  8. 08Elliptic Curves — Same Idea, Smaller Keys
  9. 09One-Way: SHA-256 and the Compression Trick
  10. 10Signing — Authorship Without Disclosure
  11. 11Zero-Knowledge — Convince Without Revealing
  12. 12Post-Quantum — Past the Reach of Shor
  13. 13Where to Go Next
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2026-05-17
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Presentation Transcript

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Slide 01

MATHEMATICAL CRYPTOGRAPHY

  • CLASSIFIED
  • EYES ONLY
  • DOCUMENT // 0xC1A55
  • The math behind every secret — from Caesar's shift to lattice-based post-quantum schemes.
  • VOL. I | 13 SECTIONS | RECOMMENDED CLEARANCE: CURIOSITY
Slide 02

Substitution & the Statistics That Betray It

  • § 02 — classical ciphers
  • A substitution cipher maps each plaintext letter to a fixed ciphertext letter. Caesar's shift is a special case: c ≡ p + k (mod 26).
  • The key space looks enormous — 26! ≈ 4 × 1026 permutations — yet any natural-language ciphertext leaks structure.
  • Frequency analysis (Al-Kindi, c. 850 CE). Letter frequencies in the plaintext are preserved. Match histograms → recover the key.
  • PLAIN → THE QUICK BROWN FOX
  • CIPHER → YDW UPHKB OAJZQ MJC
  • English letter frequency
  • Lesson: security cannot rest on key-space size alone. Distribution matters.
Slide 03

The One-Time Pad — Unbreakable, Unusable

  • § 03 — perfect secrecy
  • Encrypt by XOR-ing the message with a uniformly random key of equal length:
  • c = m ⊕ k
  • Decrypt the same way: m = c ⊕ k. If k is truly random and used only once, the ciphertext reveals nothing about the plaintext.
  • For every plaintext m and ciphertext c, Pr[C = c | M = m] = Pr[C = c]. The ciphertext distribution is independent of the message: perfect secrecy.
  • The catch
  • Key must be as long as the message.
  • Key must be truly random — not pseudorandom.
  • Key must never repeat (Venona broke Soviet pads precisely because they did).
  • Key distribution is the original problem in disguise.
  • m = 01001000 01001001
  • k = 10110101 11001100
  • ⊕
  • c = 11111101 10000101
  • Perfect secrecy is achievable. But every modern system is a compromise around the impracticality of OTP.
Slide 04

Modular Arithmetic — Integers, Wrapped

  • § 04 — foundations
  • Working mod n means we identify integers that differ by a multiple of n:
  • a ≡ b (mod n) ⇔ n | (a − b)
  • When n = p is prime, the set ℤ/pℤ becomes a finite field: every nonzero element has a multiplicative inverse.
  • In ℤ/pℤ, division is well-defined. We can solve ax ≡ b (mod p) uniquely — making linear algebra, polynomials, and the entire RSA/DH/ECC machinery possible.
  • Examples
  • clock
  • 17 + 9 ≡ 2 (mod 12)
  • inverse
  • 3 · 5 ≡ 1 (mod 7), so 3−1 ≡ 5 (mod 7)
  • power
  • 210 ≡ 1024 ≡ 24 (mod 1000)
  • Computation in finite fields is fast; inversion via the extended Euclidean algorithm is the workhorse of key generation.
Slide 05

Fermat & Euler — Why RSA Works

  • § 05 — the engine
  • For prime p and a not divisible by p:
  • ap−1 ≡ 1 (mod p)
  • For any a coprime to n:
  • aφ(n) ≡ 1 (mod n)
  • where φ(n) counts integers in [1, n] coprime to n.
  • What this gives us
  • If n = p·q with primes p, q, then
  • φ(n) = (p−1)(q−1)
  • Choose e coprime to φ(n); compute its inverse d ≡ e−1 (mod φ(n)). Then for any m:
  • (me)d = med ≡ m (mod n)
  • Encryption and decryption are both modular exponentiation — cheap if you know the factorization, intractable if you don't.
Slide 06

RSA (1977) — Factoring as a Trapdoor

  • § 06 — rivest, shamir, adleman
  • Pick large primes p, q (~1024 bits each).
  • Compute n = pq and φ(n) = (p−1)(q−1).
  • Choose public exponent e (often 65537).
  • Solve ed ≡ 1 (mod φ(n)) for private d.
  • Public key: (n, e). Private key: d.
  • Recovering d from (n, e) requires φ(n), which requires factoring n. No classical polynomial-time algorithm is known.
  • encrypt: c = me mod n
  • decrypt: m = cd mod n
Slide 07

Diffie–Hellman — A Secret in Public

  • § 07 — key exchange
  • Two strangers establish a shared key over an open wire. Public parameters: a prime p and generator g.
  • Alice picks secret a, sends A = ga mod p.
  • Bob picks secret b, sends B = gb mod p.
  • Both compute s = gab mod p.
  • Eve sees: g, p, ga, gb
  • Eve wants: gab
  • Given g, p, gx mod p, recovering x is believed to require sub-exponential time. No efficient classical algorithm exists.
  • Why it's beautiful
  • Modular exponentiation is a one-way function with structure: easy forward, hard inverse, but it commutes — (ga)b = (gb)a.
  • That single algebraic property — commutativity of exponents — is what allows two parties to converge on the same secret without ever transmitting it.
  • DH (1976) was the first published public-key protocol. Every TLS handshake on the modern internet is its descendant.
Slide 08

Elliptic Curves — Same Idea, Smaller Keys

  • § 08 — elliptic curve cryptography
  • An elliptic curve over a field is the set of points satisfying:
  • y2 = x3 + ax + b
  • The points form an abelian group under a chord-and-tangent addition law. "Multiplication" of a point P by integer k is repeated addition: kP.
  • Given P and Q = kP on the curve, recovering k is the elliptic-curve discrete log problem. Believed to be even harder per bit than ordinary DLP.
  • 256-bit ECC ≈ 3072-bit RSA in security — smaller keys, faster operations, less power. Why your phone uses Curve25519.
Slide 09

One-Way: SHA-256 and the Compression Trick

  • § 09 — hash functions
  • A cryptographic hash H : {0,1}* → {0,1}n compresses any input to a fixed-length digest. Three properties matter:
  • Pre-image resistance. Given h, hard to find m with H(m) = h.
  • Second pre-image. Given m, hard to find m′ ≠ m with H(m′) = H(m).
  • Collision resistance. Hard to find any pair (m, m′) colliding.
  • For n-bit output, collision attacks require ~2n/2 work. SHA-256 targets ~128-bit collision security.
  • SHA256("hello") =
  • 2cf24dba5fb0a30e26e83b2ac5b9e29e1b161e5c1fa7425e73043362938b9824
Slide 10

Signing — Authorship Without Disclosure

  • § 10 — digital signatures
  • A signature scheme is a triple (KeyGen, Sign, Verify):
  • Sign(sk, m) → σ requires the private key.
  • Verify(pk, m, σ) → {0,1} requires only the public key.
  • No one without sk can forge a valid σ on a new m.
  • Naive RSA signing: σ = H(m)d mod n; verify: σe ≡ H(m) (mod n).
  • Signing the hash, not the message, gives constant-size signatures, randomizes the input, and prevents algebraic forgery exploits.
  • Modern schemes
  • ecdsa
  • Elliptic-curve DSA — Bitcoin, TLS certificates.
  • ed25519
  • EdDSA on Curve25519 — deterministic, fast, side-channel friendly. SSH default.
  • trust model
  • Public-key infrastructure (PKI) binds pk to identities via certificate chains rooted in trusted authorities.
  • Encryption hides; signatures bind. They are dual operations on the same key pair.
Slide 11

Zero-Knowledge — Convince Without Revealing

  • § 11 — zero-knowledge
  • Goldwasser, Micali, Rackoff (1985): a zero-knowledge proof lets a prover convince a verifier of a statement without leaking anything else.
  • Three properties:
  • Completeness. Honest prover convinces honest verifier.
  • Soundness. A cheating prover can't fool the verifier (except with negligible probability).
  • Zero-knowledge. The verifier learns nothing beyond the statement's truth.
  • The Ali Baba cave: prove you know the password to a magic door without ever speaking it — by entering one tunnel and exiting the verifier's chosen tunnel, repeatedly.
  • Modern instantiations
  • zk-snarks
  • Succinct, non-interactive proofs — tiny, constant-size; powers Zcash, zkRollups.
  • zk-starks
  • Transparent setup, post-quantum hash-based, scalable to large statements.
  • bulletproofs
  • No trusted setup; logarithmic-size range proofs.
  • Applications: anonymous credentials, private blockchains, verifiable computation, identity without disclosure.
Slide 12

Post-Quantum — Past the Reach of Shor

  • § 12 — the quantum horizon
  • Shor's algorithm (1994) factors integers and computes discrete logs in polynomial time on a sufficiently large quantum computer. RSA, DH, and ECC all fall.
  • "Harvest now, decrypt later." Adversaries are recording today's encrypted traffic, betting on future quantum capability.
  • NIST's post-quantum standardization (2022–24) selected new primitives based on problems Shor cannot crack.
  • Three families
  • lattice-based — kyber, dilithium
  • Security from Learning With Errors and shortest-vector in high-dimensional lattices. NIST's primary KEM & signature.
  • hash-based — sphincs+
  • Signatures built only from hash functions. Conservative, large signatures, smallest assumptions.
  • code & multivariate
  • McEliece (error-correcting codes), Rainbow (multivariate quadratic systems). Decades of cryptanalysis behind them.
  • The migration is the largest crypto transition in history — already underway in TLS, Signal, and OS-level key stores.
Slide 13

Where to Go Next

  • END OF
  • FILE
  • 0xC1A55
  • § 13 — further reading
  • Canonical references
  • Introduction to Modern Cryptography — Katz & Lindell.
  • A Course in Number Theory and Cryptography — Neal Koblitz.
  • Cryptography Engineering — Ferguson, Schneier, Kohno.
  • The Code Book — Simon Singh (popular).
  • NIST FIPS 186-5, 197, 202, 203, 204, 205.
  • "Every secret has a structure. Cryptography is the study of which structures keep their shape under adversarial scrutiny."
  • Watch — YouTube
  • SEARCH · YouTube
  • RSA Encryption Explained
  • Tutorials, derivations, demos.
  • SEARCH · YouTube
  • Elliptic Curve Cryptography
  • Geometry, group law, ECDSA.
  • // nav: ← → arrows, space, click
  • // 13 / 13 — transmission complete
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