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Computer Science
The Math of Cryptocurrency Wallets and Signatures
Public-private keys, elliptic curves, and seed phrases
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A cryptocurrency wallet does not actually store coins the way a leather wallet stores cash. It stores secret mathematical information that lets you prove ownership of funds recorded on a public blockchain. The most important secret is a private key, which can create digital signatures that authorize transactions. This matters because there is usually no bank, support desk, or password reset that can restore control if the key is lost.
Key Facts
- A private key is a large secret number, often chosen from about 2^256 possible values in many systems.
- A public key is computed from the private key using elliptic curve multiplication: Q = dG.
- The security comes from the elliptic curve discrete logarithm problem: given Q and G, finding d is computationally infeasible.
- A digital signature proves control of the private key without revealing it.
- A simplified signature check asks whether a value computed from the message, signature, and public key matches an expected curve point.
- A seed phrase encodes enough randomness to recreate wallet keys, so anyone with the seed can control the funds.
Vocabulary
- Private key
- A private key is a secret number used to create signatures that prove ownership of cryptocurrency funds.
- Public key
- A public key is a value derived from the private key that others can use to verify signatures.
- Elliptic curve cryptography
- Elliptic curve cryptography is a method that uses points on special curves to create key pairs that are easy to use but extremely hard to reverse.
- Digital signature
- A digital signature is mathematical proof that a message or transaction was approved by someone who holds the matching private key.
- Seed phrase
- A seed phrase is a list of words that can regenerate the private keys for a wallet.
Common Mistakes to Avoid
- Thinking a wallet stores cryptocurrency coins. The coins are recorded on the blockchain, while the wallet stores keys that control spending rights.
- Sharing a seed phrase to get help. This is wrong because the seed phrase can recreate the private keys, so anyone who sees it can take the funds.
- Believing a public key can easily be reversed into a private key. Elliptic curve systems are designed so the forward calculation is easy but the reverse problem is infeasible with current computers.
- Using the same wallet security habits for hot and cold wallets. A hot wallet is connected to the internet and faces more attack paths, while a cold wallet keeps keys offline for stronger protection.
Practice Questions
- 1 A private key is chosen from 2^256 possibilities. Estimate the number of possible keys using 2^10 approximately equal to 10^3.
- 2 A wallet seed phrase has 12 words, and each word is chosen from a list of 2048 words. How many possible seed phrases are there, and what power of 2 is this equal to?
- 3 Explain why a blockchain network can verify that you authorized a transaction without learning your private key.