Cryptography

Bridge The number field sieve (NFS) algorithm achieves sub-exponential complexity L_n[1/3, c] = exp((c+o(1)) * (ln n)^{1/3} * (ln ln n)^{2/3}) for integer factorization, establishing the precise complexity-theoretic boundary on RSA and discrete logarithm hardness that makes modern public-key cryptography quantifiably secure against classical computation while simultaneously defining the cryptanalytic target for quantum speedup

Fields: Mathematics, Computer Science, Cryptography

The NFS algorithm for factoring n applies algebraic number theory (number fields with rings of integers, ideal factorization in class groups) to the combinatorial sieve: it finds pairs (a,b) such that...

Bridge Game theory x Cryptography - Nash equilibrium as protocol security

Fields: Economics, Computer_Science, Mathematics, Cryptography

Cryptographic protocol security (no computationally bounded adversary can profitably deviate) is a Nash equilibrium condition in a game where parties are rational agents maximizing expected utility; r...

Bridge Modern cryptography is applied number theory: RSA security rests on the hardness of integer factorization, elliptic curve cryptography on the discrete logarithm problem over finite fields, and post-quantum cryptography on the shortest vector problem in integer lattices — each translating a mathematical hardness assumption into a practical security guarantee.

Fields: Mathematics, Number Theory, Computer Science, Cryptography, Algebra, Complexity Theory

RSA (Rivest, Shamir, Adleman 1978): public key e, private key d, modulus n = pq (product of two large primes). Key relationship: ed ≡ 1 (mod φ(n)) where φ(n) = (p-1)(q-1) is Euler's totient function. ...

Bridge Elliptic curves over ℂ form complex tori (compact genus-one Riemann surfaces) where the group law comes from analytic geometry — modern ECC uses curves over finite fields where points form finite Abelian groups with no literal torus topology; pedagogy often introduces the complex picture first for intuition, then warns that cryptographic security lives in discrete logarithms on 𝔽_q-rational points.

Fields: Mathematics, Computer Science, Cryptography

The chord-and-tangent group law is uniform across fields — explaining why textbooks illustrate ℂ/Λ pictorially — but security proofs and side-channel engineering operate on Galois cohomology, embeddin...

Bridge Quantum key distribution achieves information-theoretic security (unconditional security independent of adversary computing power) by exploiting quantum measurement disturbance, bridging quantum computing and cryptography through the quantum no-cloning theorem and Shannon's one-time pad.

Fields: Quantum Computing, Cryptography, Information Theory

BB84 quantum key distribution achieves information-theoretic security (proven secure against computationally unbounded adversaries) because any eavesdropping measurement on quantum states introduces d...