TRIAD • LIVE

The Triangle

Mathematics ↔ Physics ↔ Cryptography

Groups, waves, and hardness. One structure appears in three guises: elliptic curves in calculus and pendulums, Fourier interference in quantum computers, and discrete logs on circles and curves.

Vertex
MATHEMATICS
Groups • Curves • Fourier
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PHYSICS
Hamiltonians • Interference
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CRYPTOGRAPHY
Hardness • Protocols
EDGE 1 • Mathematics → Physics

Elliptic Curves: From Torus to Pendulum

Same law, three worlds: \(y^2 = x^3+ax+b\) adds points by chord-tangent. Over $\mathbb{C}$ the curve is a torus; a pendulum's $(\theta,p)$ phase flow wraps that torus.

CLICK CURVE TO ADD P, Q
Equation
$y^2 = x^3 - 1.0x + 0.0$
Group law: line through $P,Q$ meets curve at $R$, reflect to $P+Q$.
Points
Click the curve...
Torus topology
Over $\mathbb{C}$, $E \cong \mathbb{C}/\Lambda$ is a donut. Hamiltonian flows (pendulum, integrable systems) are straight lines on that torus—same addition law.
EDGE 2 • Physics → Cryptography

Quantum Fourier Transform & Shor's Algorithm

Physics turns periodicity into constructive interference. QFT sums phasors $e^{2\pi i kx/M}$; peaks land at multiples of $M/r$, revealing the period that breaks RSA.

Periodic comb
$f(x)=7^x \bmod 15$ has period $r=4$
demo r4
k 8
Phasor sum $\sum e^{2\pi i k x/M}$
|QFT| spectrum
Peaks at $k = m\cdot M/r$. Measuring $k$ reveals $r$.
Shor (N=15, a=7)
    Quantum physics finds periods exponentially faster → factors $N$, breaks RSA.
    EDGE 3 • Cryptography → Mathematics

    Diffie-Hellman on Circles and Curves

    Same protocol, different groups. Classic DH multiplies on the circle $e^{2\pi i a/p}$; ECDH adds points on an elliptic curve. Hardness comes from the group, not the wire.

    Classic DH
    $A=g^a\bmod p$, $B=g^b\bmod p$
    p=23, g=5
    A = 8 • B = 19
    Shared s = 2
    ECDH
    $E: y^2=x^3+2x+2\ (\bmod\ 17)$, $G=(5,1)$
    same a,b
    A=aG= • B=bG=
    Shared S=
    Both are groups (Mathematics). Security = discrete log hardness.
    Classical: DLP ~ $O(p)$, ECDLP ~ $O(\sqrt p)$. Quantum (Shor): both ~ $O(\log^3 p)$ via period-finding—the physics edge cuts the triangle.

    Why the Triangle Works

    Mathematics provides

    Abelian groups, elliptic curves, and Fourier analysis. The chord-tangent law, $E(\mathbb{C})\cong \mathbb{C}/\Lambda$, and characters $e^{2\pi i kx/N}$ are the same algebraic structures in different costumes.

    Physics provides

    Hamiltonian phase space, action-angle variables, and quantum interference. A pendulum winds a torus; a quantum computer sums phasors—both are physical realizations of group translations.

    Cryptography uses

    Hard problems: factoring, DLP, ECDLP. It builds protocols from math, then physics tests them. When quantum period-finding wins, math responds with lattices and isogenies—closing the loop.

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