Humans have been trying to hide messages from enemies for as long as we've been writing them. The ancient Greeks used a device called a scytale — a wooden rod you'd wrap a strip of leather around, write your message across it, and then unwind. The strip looked like gibberish until you wrapped it around a rod of exactly the same diameter. Julius Caesar used an even simpler trick: shift every letter of the alphabet three places. A. becomes D, B becomes E, and so on. You could break it in minutes today, but it baffled most people in 50 BC.
These were the earliest forms of what cryptographers call, substitution ciphers — replacing one symbol with another according to a rule. For centuries, the arms race between code-makers and code-breakers was essentially about making that rule more and more complicated.
By World War II, the Germans had built a machine that took this idea to a breathtaking level of complexity. The Enigma machine looked like a typewriter, but it was really a fiendish mechanical scrambler. When you pressed a key, an electrical signal passed through a series of rotating rotors — each one scrambling the signal differently — then bounced off a reflector and came back through the rotors in reverse, lighting up a different letter on a display board. The genius, and the terror, of Enigma was that the rotors rotated with every keystroke. So pressing A once might give you X, but pressing A again immediately after might give you Q. The starting position of the rotors changed every single day, based on a secret codebook. The number of possible settings ran into the hundreds of quintillions. German military commanders were absolutely convinced it was unbreakable.
Every morning, the British listening stations at Bletchley Park would intercept fresh German radio traffic, and they would essentially be staring at a completely new lock.
This is where our next character enters the story. Here is a potted history.
1930s, German Enigma machine, Rotor-based polyalphabetic cipher.
1940s, Alan Turing & the Bombe, Logical attack on Enigma, Bletchley Park.
1950s, Turing test & prosecution, Computing Machinery and Intelligence, Father of computer science
1970s, Public key cryptography, RSA, Diffie-Hellman, Math, not mechanical rotors.
2008, Bitcoin & the blockchain, Satoshi Nakamoto white paper, Cryptography goes decentralised.
Turing was a Cambridge mathematician with a mind that seemed to operate on a different plane from everyone else. Before the war, he'd already published a paper laying out the theoretical foundations of what we now call computing — the idea of a universal machine that could simulate any other machine by following rules. That paper, written in 1936, essentially invented the concept of software. When the war came and he landed at Bletchley Park, he approached Enigma not as a communications problem but as a logical problem. He figured that Jerman operators, being human, would make predictable choices — common phrases, standard openings, messages that had to contain certain words. He designed an electromechanical device called the Bombe that could rapidly test enormous numbers of Enigma settings, hunting for a configuration that would produce a known phrase. It didn't break Enigma in one shot — it narrowed the field from the impossible to the merely difficult. Analysts could then finish the job.
The intelligence this produced, was code named, Ultra, and historians estimate it shortened the war in Europe by somewhere between two and four years. Potentially millions of lives were saved by one man's genius.
After the war, Turing continued building what would become the modern computer, working on early machines at Manchester. And then in 1950 he asked one of the most provocative questions in the history of science: Can machines think? His paper proposed a test — now called the Turing Test — where a human interrogator chats with both a human and a machine, and if the interrogator can't reliably tell which is which, the machine has demonstrated something worthy of the word intelligence. We're still arguing about that question today.
Then comes the trajic part of the story that is almost unbearable to read. In 1952, Turing was prosecuted by the British government for gross indecency — meaning he was gay, and homosexuality was a criminal offence in the UK. He was convicted and given a choice: prison, or chemical castration via injections of synthetic oestrogen. He chose the injections. In 1954, he was found dead in his bedroom, a half-eaten apple beside him laced with cyanide. The coroner ruled acidental suicide.
He was 41 years old. The man who had done more than almost anyone to save his country was destroyed by it.The Apple logo — a bitten apple — was designed in 1977 by Rob Janoff. When people ask Steve Jobs whether the bite in the apple was a tribute to Alan Turing and his cyanide apple, Jobs reportedly said something along the lines of: "God, we wish it were true." Meaning: it wasn't intentional, but they loved the connection. The actual reason for the bite was reportedly just so the apple wouldn't look like a cherry at small sizes — the bite gives it a sense of scale. But the Turing interpretation has taken on a life of its own, because it feels so right — a computer company's logo as a quiet bittersweet tribute to the father of computing, who died by a poisoned apple. Rob Janoff himself has said it wasn't a conscious tribute, though he's acknowledged the poetry of the coincidence.
Britain eventually gave Turing a formal royal pardon in 2013, nearly sixty years after his death. A bit late.
The war's end didn't stop cryptographers — it lit a fire under them. If mechanical machines could be broken, what was truly unbreakable? The answer, worked out through the 1970s by mathematicians like Whitfield Diffy, Martin Hellman, and the RSA trio (Ron Rivist, Adi Shamir, and Len Adleman), was mathematical hardness. Instead of scrambling letters with rotors, you'd use mathematical problems that are trivially easy in one direction, but essentially impossible to reverse. Multiplying two huge prime numbers together takes a fraction of a second. Figuring out which two primes were multiplied to produce a 2048 digit number would take longer than the age of the universe, even with every computer on Earth working on it. That asymmetry is modern encryption. It's what protects your banking app, your messages, every HTTPS connection you make.
The modern public-key cryptography system normally taught in universities begins in the 1970s with work by Whitfield Diffie and Martin Hellman at Stanford University, followed soon after by the RSA system developed by Ron Rivest, Adi Shamir, and Leonard Adleman. Their work transformed secure communications and underpins much of today’s internet security.
But decades later, it emerged that key elements of the same ideas had already been discovered in secret inside the British signals intelligence world.
The story centres on GCHQ, the postwar successor to the wartime codebreaking operation at Bletchley Park. During the late 1960s and early 1970s, several brilliant British mathematicians working in extreme secrecy independently developed concepts astonishingly close to modern public-key cryptography.
The first breakthrough came from James Ellis. In 1969 he proposed what he called “non-secret encryption. "At the time, all practical encryption required both parties to share a secret key beforehand - a huge logistical problem. Ellis realised, theoretically, that it might be possible to encrypt messages without first exchanging a secret key. The trouble was, he could not figure out how to implement it mathematically.
Then came Clifford Cocks, a young mathematician at GCHQ. In 1973, just days after hearing Ellis's idea, Cocks devised a practical mathematical system based on the difficulty of factoring large prime numbers. It was, in essence, RSA - three years before the American RSA team published their work publicly. Cocks reportedly wrote the solution in a notebook almost casually, not fully realising how historically important it would become.
Soon after that, another GCHQ mathematician, Malcolm Williamson, developed a method closely resembling the later Diffie–Hellman key exchange system. Again, this was discovered independently and in secret before the public academic version appeared in the United States.
The tragedy, in a way, is that none of this work could be published. GCHQ classified it all. The British government regarded these techniques as too strategically valuable to reveal. As a result, the open scientific world had no knowledge of the discoveries. The Americans therefore reinvented the same core ideas independently — and because they published openly, they received the recognition.
Years later, after some of the British work was declassified in the 1990s, members of the American cryptographic community visited Britain and examined the old documents. Whitfield Diffie himself became deeply interested in the story and publicly acknowledged the significance of the British achievements. There was understandable surprise - even disbelief at first - because the GCHQ papers predated the published American work so clearly.
The connection to Bletchley Park is partly institutional and partly cultural. The actual discoveries were made after the war at GCHQ facilities rather than inside wartime Bletchley itself, but GCHQ inherited much of the mathematical and cryptanalytic tradition established there by figures like Alan Turing and his colleagues. In that sense, the intellectual lineage runs directly from wartime codebreaking to modern internet cryptography.
Today, historians generally give credit to both groups. The British invented the ideas first in secret; the Americans invented them independently and brought them into the public world, where they transformed computing, commerce and communications forever.
Clifford Cocks studied mathematics at King's College, Cambridge. He then began postgraduate work at Oxford before leaving academia to join GCHQ in 1973.
His mathematical speciality was number theory - exactly the branch of mathematics that later became central to modern cryptography. Number theory deals with the properties of integers, especially prime numbers, modular arithmetic, factorisation, congruences, and related structures.
That background was crucial. The brilliance of RSA-style public-key cryptography comes from exploiting a mathematical asymmetry: multiplying two very large prime numbers together is easy, but factoring the resulting huge composite number back into its original primes is extremely hard. Cocks immediately recognised that this "one-way function" might solve James Ellis's "non-secret encryption" problem because he was already immersed in number-theoretic thinking.
There's a wonderful detail in the story. According to later interviews, when Ellis's problem was explained to him at GCHQ, Cocks essentially solved the core idea overnight — some accounts even say within half an hour. He later remarked that the problem had been presented to him "in a very mathematical way," and because he had been working in number theory, prime factorisation naturally suggested itself.
He had also been an exceptionally gifted young mathematician. While at Manchester Grammar School he won a silver medal at the International Mathematical Olympiad in 1968. Interestingly, another future GCHQ cryptographer, Malcolm Williamson, was at the same school and also went on to Cambridge.
One of the striking things about Cocks is how understated he remained about the whole affair. By temperament he seems much more like a pure mathematician than a Silicon Valley entrepreneur. Had the work been public, he could have become enormously wealthy from one of the foundational inventions of the internet age. Instead, the discovery disappeared into classified files for nearly a quarter of a century.
"Clock arithmetic" is the informal name for modular arithmetic, and modular arithmetic sits right at the heart of RSA-style public-key cryptography.
For example, on a 12-hour clock:
9 + 5 = 2 (mod 12)
You "wrap around" after reaching 12. In number theory, mathematicians generalise this idea to enormous numbers — hundreds or thousands of digits long.
Clifford Cocks was fundamentally a number theorist, and modular arithmetic was one of the essential tools of his field. RSA encryption relies on properties of modular exponentiation - raising huge numbe's to huge powers while taking remainders modulo another huge number.
A tiny toy example looks like this:
7^4 = 1 (mod 20)
For ordinary people, that looks obscure. But for cryptographers, these modular relationships create mathematical "trapdoors." Some operations are easy to perform one way, but extraordinarily difficult to reverse unless you possess special information - namely the private key.
The brilliance of RSA is that it combines: Cocks already knew this mathematical territory intimately. So when James Ellis proposed the idea of
"non-secret encryption," Cocks immediately recognised that modular arithmetic with large primes might provide the missing mechanism.
What's remarkable is that the mathematics itself was not especially exotic by pure mathematics
standards. Much of it dated back to Euler in the 18th century and even to Fermat in the 17th.
The genius lay in realising that ancient number theory could solve a modern communications problem.
That's one of the lovely ironies of cryptography: ideas developed centuries ago for pure
intellectual curiosity became the foundation of secure internet commerce, banking, passwords,
and encrypted messaging.
The classic "Alice and Bob" demonstration of RSA encryption uses very small prime numbers so humans
can follow the arithmetic by hand. Real systems use primes hundreds of digits long, but the miniature
version shows the mechanism beautifully.
First, Alice chooses two prime numbers:
p = 5, q = 11
She multiplies them:
n = pq = 55
Then she computes Euler’s totient:
phi(n) = (5 - 1)(11 - 1) = 40
Now Alice chooses a public encryption exponent that has no common factor with 40.
A common toy choice is:
e = 3
Next she must find a private decryption exponent d such that:
ed = 1 (mod 40)
Since
3 x 27 = 81 = 1 (mod 40)
the private key is:
d = 27
So Alice publishes the public key:
(e,n) = (3,55)
but keeps d = 27 secret.
Now Bob wants to send Alice the message 17. He encrypts it using Alice's public key:
c = 17^3 mod 55
Calculate it:
17^3 = 4913
and dividing by 55 leaves remainder 18, so the encrypted ciphertext becomes:
c = 18
An eavesdropper sees only 18.
Alice now decrypts using her secret exponent:
m = 18^27 mod 55
After all the modular arithmetic, the result comes back to:
m = 17
- the original message.
What astonished cryptographers in the 1970s was not merely that this worked, but that Alice
and Bob never needed to exchange a secret key beforehand. Alice could publish her public key
openly for the entire world to see. Yet only she, possessing the private exponent d, could
efficiently decrypt the message.
That conceptual leap - separating the encryption key from the decryption key - was the revolutionary
part. Before that, cryptography was mostly symmetrical: both sides needed the same secret key in
advance. Public-key cryptography changed everything.
- prime numbers,
- modular arithmetic,
- Euler’s theorem,
- and the difficulty of factorising huge composite numbers. Alice and Bob