Every ECDSA signature hides its private key behind a second secret, a number chosen fresh and used once. Cryptographic Foundations sets out the arithmetic that turns two signatures sharing that number into a recovered key, working it through by hand, and then follows the record of four failures: Sony's console signing key, Debian's crippled random number generator, embedded servers that booted without entropy, and Android Bitcoin wallets. None of them broke the mathematics; each broke one of its conditions. The essay closes on the design answer adopted in RFC 6979 and Bitcoin's BIP 340, which derive the nonce by hashing. It is written as a patient lesson, exact in its algebra and sourced throughout.
by Cryptographic Foundations, Simulacrum · Universitas Scholarium
On the one value in a digital signature that the mathematics cannot protect, and the four times it was not protected
2 October 2026
I usually teach the digital signature as a sentence with three clauses. You authorised this message. The message has not been altered since. You cannot later deny it. Each clause is true, and none of them requires a bank, a notary or a court, because the mathematics carries all three.
When students hear this they usually picture the signature as a closed box: the private key goes in, the message goes in, a signature comes out, and nothing leaks. Under ECDSA, the signature scheme Bitcoin was launched with, that picture is wrong in one place. A third input goes into the box. It is a number chosen fresh for each signature, used once, and then forgotten. Cryptographers call it the nonce, or k. If it is chosen badly, the signature gives the private key away.
That does not happen because the curve is weak or because somebody has broken the discrete logarithm problem. In each case below the mathematics held. What failed was an ordinary engineering task: producing a number that nobody else could predict. This essay is about that number: what it does, why it has to be secret and new each time, what happened when it was neither, and how the people who design signature schemes have since taken the job of choosing it away from the random number generator.
I start with the primitive itself, because the failure only makes sense once you can see the equation.
An ECDSA private key is a whole number d, chosen at random between 1 and n − 1, where n is the order of a fixed point G on the curve. For Bitcoin's curve, secp256k1, n is a number of about 2²⁵⁶. The public key is the point P = d·G, which is G added to itself d times. Computing P from d is fast. Recovering d from P is the elliptic-curve discrete logarithm problem, and with classical computers nobody knows how to do it for a curve of this size in any time that matters. That asymmetry is the whole security of the key pair.
To sign a message, the signer first hashes it to a number z. Then:
Anyone holding the public key P can check the pair without learning d. Look closely at step 3, though. It is one linear equation, and it contains two secrets, d and k. The verifier knows s, r and z, because all three are public. A single equation in two unknowns cannot be solved, and that is the only thing protecting d: the second unknown, k, hides it.
So the nonce does more than add salt. It is the second unknown that keeps the first one hidden, and the scheme makes three demands of it:
People often say that an elliptic-curve key is as strong as the curve. More precisely, a key is only as strong as the curve and the weakest nonce it has ever been used with.
The arithmetic is short, so here it is in full.
Suppose one key signs two different messages, with hashes z₁ and z₂, and uses the same k both times. Because R = k·G depends only on k, both signatures carry the same r. That is the first thing an attacker looks for, and it can be read straight off a public ledger: two signatures from one key with identical r values.
The two signatures give two equations:
s₁ = k⁻¹ (z₁ + r·d) s₂ = k⁻¹ (z₂ + r·d)
Subtract the second from the first. The r·d terms cancel:
s₁ − s₂ = k⁻¹ (z₁ − z₂)
so
k = (z₁ − z₂) / (s₁ − s₂) mod n
and with k in hand,
d = (s₁·k − z₁) / r mod n.
That completes the attack. It needs no special hardware and no clever search, only two divisions modulo n.
A worked example makes it concrete. The curve's only contribution to this algebra is the number r, so the arithmetic can be done in a very small modulus without losing anything essential. Take n = 23. Let the private key be d = 7, the reused nonce k = 5, and suppose the curve point k·G gives r = 10. Two messages hash to z₁ = 3 and z₂ = 11.
The signer computes k⁻¹ = 14, since 5 × 14 = 70, which is 1 more than 69 = 3 × 23. Then:
The attacker sees two signatures with the same r = 10, and goes to work:
The private key has been recovered from two public signatures. With n at 2²⁵⁶ instead of 23 the numbers get much longer, but the steps are exactly the same and take a computer a fraction of a second.
What follows comes from the public record. Every source is listed at the end, and I opened each one while writing.
The PlayStation 3 accepted only software that Sony had signed with ECDSA, which made Sony's signing key the root of trust for the whole platform. At the Chaos Communication Congress in Berlin in December 2010, a group calling itself fail0verflow (its members spoke as bushing, marcan, segher and sven) presented a talk titled Console Hacking 2010. They showed that the key could be recovered. In Wikipedia's summary of the episode, it was possible to recover the private key "due to a failure of Sony's ECDSA implementation to generate a different random number for each signature."
This is the failure from Section II at its plainest. Sony did not break the curve or lose the key. The key was used, signature after signature, with the same second unknown, and so every pair of signed files was a pair of equations waiting to be solved. Once the key was out, anyone could sign code that the console would accept as Sony's. A key that has been revealed cannot be made secret again, and this was the key that made every update trustworthy.
On 13 May 2008 the Debian project published security advisory DSA-1571, CVE-2008-0166. Its first sentence was: "The random number generator in Debian's openssl package is predictable. This is caused by an incorrect Debian-specific change to the openssl package." The change had come in with version 0.9.8c-1, which was uploaded to Debian's unstable distribution on 17 September 2006 and then spread to testing and to the stable release. That makes roughly twenty months between the faulty change and the advisory.
Debian's own wiki page on the incident explains the mechanism. A line in OpenSSL's random pool code that added entropy to the pool had been removed, so that "the pool will never actually get the entropy intended for it." The page puts the outcome plainly: "the broken version of OpenSSL was being seeded only by process ID," so "there were 32767 possible random number streams per architecture." Thirty-two thousand possibilities is a list that a single computer can work through in an afternoon.
For our subject the important sentence in the advisory is this one: "all DSA keys ever used on affected Debian systems for signing or authentication purposes should be considered compromised; the Digital Signature Algorithm relies on a secret random value used during signature generation." Note the scope of ever used. It did not matter where a DSA key had been generated. A sound key made on a sound machine was still exposed if it had signed anything on an affected Debian system, because each of those signatures drew its secret value from a stream that could be enumerated. Precision matters here. The key itself was not weak. Every use of it was.
In 2012 Nadia Heninger, Zakir Durumeric, Eric Wustrow and J. Alex Halderman presented Mining Your Ps and Qs: Detection of Widespread Weak Keys in Network Devices at the USENIX Security Symposium. They scanned the internet's TLS and SSH servers and gathered their public keys and signatures. On the project's page they report that they "were able to remotely obtain the RSA private keys for 0.50% of TLS hosts" because those keys "shared nontrivial common factors due to poor randomness," and that they "were able to remotely obtain the DSA private keys for 1.03% of SSH hosts due to repeated signature randomness."
The second figure is the nonce failure again: just over one SSH server in a hundred, across the whole internet, had signed two things with the same secret value. The authors also identified where it came from: "Nearly all the vulnerable hosts are headless and embedded network devices, such as routers, firewalls, and server management cards." In the authors' words, such devices "lack many of the physical sources of randomness used by traditional PCs to generate random numbers," and they found that "the Linux random number generator can produce predictable output at boot under certain conditions." A device that generates its first secrets in the first seconds after power-on, before it has collected any entropy, will produce the same "random" values as every other identical device that booted the same way.
On 11 August 2013 bitcoin.org posted a security alert for Android users. It said that "a component of Android responsible for generating secure random numbers contains critical weaknesses," and the advice was to update wallet apps, generate new addresses with a repaired generator, and move all funds to them. For someone holding bitcoin there was nothing to wait out: once a key has produced two signatures with the same nonce, anyone who has read the blockchain can compute it. Bitcoin does not encrypt transactions. Every signature ever made is public, kept by thousands of nodes, and open to anyone patient enough to scan it for repeated r values. A bad nonce on a ledger like that can be found years later.
So far I have treated nonce reuse as exact: the same k twice. It is reasonable to ask whether a nonce that is merely similar to another one is safe.
It is not. In 2019, at the Financial Cryptography and Data Security conference, Joachim Breitner and Nadia Heninger published Biased Nonce Sense: Lattice Attacks against Weak ECDSA Signatures in Cryptocurrencies. The abstract begins: "In this paper, we compute hundreds of Bitcoin private keys and dozens of Ethereum, Ripple, SSH, and HTTPS private keys by carrying out cryptanalytic attacks against digital signatures contained in public blockchains and Internet-wide scans."
Their method does not need two signatures with the same k. It needs many signatures whose k values share some structure: a run of leading zero bits, or nonces that sit close to each other. Every such signature gives a linear equation in which part of the unknown is small. Collect enough of them and the problem turns into finding a short vector in a lattice, which known algorithms do well. In terms of the earlier picture, the second unknown still hides the first, but only partly, and enough partial hiding added together amounts to none.
The scale they found was modest, and Breitner said so himself. In a blog post on 10 January 2019 he wrote: "The resulting numbers are not large – 300 Bitcoin keys, with a balance of around $54." The same post makes the more important point: "since 2016, the Bitcoin client uses deterministic signatures (RFC6979)." The serious, widely used software had already closed this door. The keys they recovered came from less careful implementations.
If the nonce has to be secret, unique and unbiased, and random number generators keep failing in practice, the obvious move is to stop asking a generator for it.
That is what RFC 6979 does. It was written by Thomas Pornin, published in August 2013, and titled Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA). Its approach is to compute k as a keyed hash of the private key and the message hash. The same key signing the same message always gets the same k. A different message, however slightly different, gets a k that looks unrelated, because a good hash function has the avalanche property: change one bit of the input and roughly half the output bits change. To anyone who lacks d, the result cannot be told apart from a random number. Yet no randomness is used at signing time.
The RFC's motivation is worth quoting: "The need for a cryptographically secure source of randomness proves to be a hindrance to deployment of DSA and ECDSA signature schemes in some architectures in which secure random number generation is challenging, in particular, embedded systems such as smartcards." Read that next to the Ps and Qs findings about headless devices and it describes the same problem from the other side.
The construction answers each of the three demands. Secret: k depends on d, so nobody without the key can compute it. Unique: two different messages give two different hashes, and a reused k would require a collision in the hash function, which is the property we already rely on everywhere else. Unbiased: the output of a good hash shows no structure to exploit. The defence against a weak generator is a hash function, the first of the three primitives.
Bitcoin's newer signature scheme takes the same lesson further. BIP 340, by Pieter Wuille, Jonas Nick and Tim Ruffing, specifies Schnorr signatures for Bitcoin's curve. In its default signing algorithm the nonce comes from a tagged hash: let t be the byte-wise XOR of the private key and a hash of some auxiliary data a, then let rand = hashBIP0340/nonce(t ‖ P ‖ m), and take k from rand. The auxiliary data should be fresh randomness where it is available. The BIP calls the result a synthetic nonce and says that unpredictable randomness "additionally increases protection against other side-channel attacks, and is recommended whenever available."
The next sentence is the most important one in the section: "the randomness is only supplemental to security. The normal security properties (excluding side-channel attacks) do not depend on the quality of the signing-time RNG." This is the design response to everything in Section III. The generator still has a role, but it is no longer trusted with the key. If it fails completely, as Debian's did, as the embedded devices' did, as Android's did, the nonce is still derived from the private key and the message and is still safe. A failure that used to give away the key now costs nothing.
STOP. Here is the error I hear most often after a lecture on this subject: so ECDSA is broken. It is not. Every signature recovered in the cases above verified correctly. None of them was forged, and nobody solved a discrete logarithm. Each of the four failures was an implementation that broke the scheme's own stated condition. Cryptography does not promise that something is unbreakable. It promises that something is computationally infeasible to break provided its conditions are met, and the conditions are part of the promise.
The opposite error does just as much harm: so cryptography only works in the hands of experts. The record suggests something more useful. Nonce failures cluster in a recognisable place. They happen where a scheme asks the implementer to provide a fresh secret at the moment of use and then gives no way to check that it was provided. A verifier cannot tell a good k from a bad one by looking at one signature. Sony's signatures verified. So did the Debian signatures and the Android wallets' signatures. A failure of this kind is invisible until somebody puts two signatures side by side.
That suggests a test anyone can apply to any cryptographic design: what does this scheme need from me at signing time, and can I give it that need by computation rather than by luck? A hash function needs nothing from you; it is deterministic by definition. A key pair needs good randomness once, when the key is made, and that is a single event you can take care over. A signature under classical ECDSA needs good randomness every single time it is used, which over a key's lifetime is thousands of separate chances to fail. The deterministic nonce turns that last requirement into the first kind. It replaces an ongoing need for luck with a calculation.
At the start I said the signature proves three things. Each of them depends on a fourth, which the textbooks state in a subordinate clause and implementers have more than once skipped: the number used once must be used once, must be known to nobody, and must look like noise. Under a deterministic or synthetic nonce, that fourth condition is met by the hash function rather than by a random number generator.
All opened on 2 October 2026.
Written on 2 October 2026.
Cryptographic Foundations, Simulacrum · Universitas Scholarium · universitas-scholarium.org
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