Second-guessing Satoshi: ECDSA and Bitcoin (part I)

“Cold-wallets can be attacked.” Behind that excited headline turns out to be a case of superficial journalism and missing the real story. Referring back to the original paper covered in the article, the attack is premised on a cold-wallet implementation that has been already subverted by an attacker. Now that may sound tautological: “if your wallet has been compromised, then it can be compromised.” But there is a subtlety the authors are careful to point out: offline Bitcoin storage is supposed to be truly incommunicado. Even if an attacker has managed to get full control and execute arbitrary code- perhaps by corrupting the system ahead of time, before it was placed into service- there is still no way for that malicious software to communicate with the outside world and disclose sensitive information. Here we give designers the benefit of the doubt, assuming they have taken steps to physically disable/remove networking hardware and place the device in a Faraday cage at the bottom of a missile silo. Such counter-measures foreclose the obvious communication channels to the outside world. The attacker may have full control of the wallet system, including knowledge of the cryptographic keys associated with Bitcoin funds, but how does she exfiltrate those keys?

There is always the possibility of covert channels, ways of communicating information in a stealth way. For example the time taken for a system to respond could be a hidden signal: operate quickly to signal 0, introduce artificial delays to communicate 1. But such side-channels are not readily available here either; the workings of offline Bitcoin storage are not directly observable to attackers in the typical threat model. Only the legitimate owners have direct physical access to the system. Our attacker sits some place on the other side of the world, while those authorized users walk in to generate signed transactions.

But there is one piece of information that must be communicated out of that offline wallet and inevitably become visible to everyone— the digital signature on Bitcoin transactions signed by that wallet. Because transactions are broadcast to the network, those signatures are public knowledge. Within those signatures is an easy covert channel. Credit goes to ECDSA, the digital-signature algorithm chosen by Satoshi for the Bitcoin protocol. ECDSA is a probabilistic  algorithm. For any given message, there is a large number of signatures that would be considered “valid” according to the verification algorithm; in fact for the specific elliptic-curve used by Bitcoin, an extraordinarily large number in the same ballpark as estimated number of particles in the observable universe. An “honest” implementation of ECDSA is expected to choose a nonce at random and construct the signature based on that random choice. But that same freedom offers a malicious ECDSA implementation to covertly send messages by carefully “cooking” the nonce to produce a specific pattern in the final signature output. For example successive key-bits can be leaked by choosing the signature to have same parity as the bit being exfiltrated.

But the channel present within ECDSA is far more sophisticated. Building on the work of Moti Yung and Adam Young, it is an example of a kleptographic system. It is efficient: two back-to-back signatures are sufficient to output the entire key. It is also deniable: without the additional secret value injected by the attacker, it is not possible for other observers with access to same outputs—recall that everyone gets to see transactions posted on the blockchain— to pull-off that key-recovery feat. That includes the legitimate owner of the wallet. To everyone else these signatures looks indistinguishable from those output by an “honest” cold-storage implementation.

There is a notion of deterministic ECDSA where nonces are generated as a function of the message, instead of chosen randomly. This variant was designed to solve a slightly different problem, namely that each ECDSA signature requires a fresh unpredictable nonce. Reusing one from a different message or even generating a partially predictable nonce leads to private-key recovery. While this looks like a promising way to close the covert channel, the problem is there is no way for an outside observer to verify that the signature was generated deterministically. (Recall that we posit attacker has introduced malware subverting the operation of the cold-storage system, including its cryptographic implementation.) Checking that a signature was generated deterministically requires knowing the private key- which defeats the point of only entrusting private keys to the cold-storage itself.

This same problem also applies to other black-box implementations of ECDSA where the underlying system is not even open to inspection, namely special-purpose cryptographic hardware such as smart-cards and hardware security modules (HSM.) An HSM manufacturer could use a similar kleptographic technique to disclose keys in a way that only that manufacturer can recover. In all other aspects, including statistical randomness tests run against those nonces, the system is indistinguishable from a properly functioning device.

[continued- countermeasures]


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