Orchestra and cryptography
The Fragile Security of Classical Cryptography
Today’s public‑key cryptography – RSA, Diffie–Hellman, elliptic‑curve schemes – rests on a single assumption:
Some mathematical problems are so hard that even the fastest classical computers cannot solve them in any reasonable time.
For RSA, that “hard problem” is integer factorisation. For Diffie–Hellman, it is the discrete logarithm.
Both reduce to finding a hidden periodicity – an integer r – buried inside modular exponentiation. Classically, finding this r is like searching for a single grain of sand in a desert. The search space grows exponentially; the computational cost becomes astronomical.
This is why your bank account, your medical records, your private messages remain safe.
For now, that is.
Q‑Day: When the Assumption Breaks
If a universal, fault‑tolerant quantum computer is ever built, Shor’s algorithm will run on it. And when it does, the exponential hardness of factorisation collapses to polynomial time.
That day – often called Q‑Day – is the moment when all classical public‑key cryptography becomes decryptable. Not theoretically vulnerable. Not gradually weakened. Broken in a bang.
This is why post‑quantum cryptography exists. This is why governments are already migrating. This is why Shor’s algorithm is not just an academic curiosity. It’s also my assignment for this week for my course in Quantum Algorithm.
The Heart of Shor’s Algorithm: A Quantum Fourier Transform
What fascinates me most is that Shor’s algorithm does not directly factor numbers. It does something more subtle, more elegant:
It transforms the problem of factorisation into a problem of period‑finding, ie like finding a common geometry in all the sand structures in the world that the oceans make.
And the engine that extracts that hidden period is the Quantum Fourier Transform (QFT).
This is where my past and present converge.
From Classical Fourier Analysis to Quantum Fourier Symmetry
During my first PhD, working with the scanning proton microprobe, Fourier transforms were familiar tools. They decomposed spatial patterns, noise spectra and beam instabilities into their constituent frequencies.
The classical Fourier transform takes a messy signal and reveals the pure tones inside it. It is the mathematical equivalent of taking a chord and identifying the individual notes.
The quantum Fourier transform does something analogous, but on a superposition. Like on magic mushrooms.
Classical FT
A function → a spectrum of frequencies.
Quantum FT
A quantum state → a spectrum of amplitudes encoding hidden periodicity.
Where the classical transform analyses data, the quantum transform analyses possibility.
A Music Analogy: Why the QFT Works
Imagine striking a piano chord. You hear a rich, complex sound. But a Fourier transform reveals the underlying notes – the pure frequencies that give the chord its identity.
Shor’s algorithm does the same thing with modular arithmetic.
The modular exponentiation function
is like a complex chord. It looks irregular, jagged, unpredictable.
But hidden inside it is a perfect periodicity – the order r.
The QFT is the instrument that “listens” to the quantum state and extracts those pure frequency components. The peaks in the QFT output correspond to rational multiples of 1/r. From those peaks, a classical post‑processing step recovers r.
And once you know r, the factors of N fall out almost effortlessly.
Why This Matters to Me
I am autistic – neurodiverse – and the world often arrives as an overwhelming superposition of signals in ways my mind cannot always disentangle. Hyperfocus is my refuge, but even in that narrowed beam, the background cacophony never fully disappears. When we lived next to the Royal Albert Hall and my older daughter played in an orchestra, practising her cello for hours, the sound of the orchestra warming up – all those instruments tuning, colliding, searching for their pitch – felt exactly like my internal landscape: a hundred frequencies competing, unresolved, vibrating against the edges of my perception. I wrote about that in my book Live Patanjali – Yoga Wisdom for Everyday Living, trying to make sense and find peace in my world.
But then I met my partner – my mathematician – and something shifted. In the midst of the noise, he became the single, steady tone that allowed the rest of the spectrum to fall into place. When I look into his blue eyes, the world that usually feels like an unfiltered Fourier series suddenly collapses into coherence. The elusive period – the one Shor’s algorithm searches for in the chaos of modular exponentiation – reveals itself effortlessly. In the vast, noisy superposition of life, he is the underlying rhythm, the fundamental frequency that makes the whole signal intelligible. The one word that resolves the entire equation. And that word is his name.
There is something profoundly beautiful about this. A problem that is intractable in the classical world becomes solvable when we have the right system or person – when we look at it through the lens of quantum interference and Fourier symmetry.
