Elliptic Curve Signatures
Let's dig into when, where, and how Bitcoin uses Elliptic curve signatures in transactions. This chapter covers the importance of signatures to transactions, the three purposes these signatures serve, and how they are applied.
Digital Signatures (ECDSA)
So far, we have not delved into any detail about "digital signatures". In this section we look at how digital signatures work and how they can present proof of ownership of a private key without revealing that private key.
The digital signature algorithm used in bitcoin is the Elliptic Curve Digital Signature Algorithm, or ECDSA. ECDSA is the algorithm used for digital signatures based on elliptic curve private/public key pairs, as described in [elliptic_curve]. ECDSA is used by the script functions OP_CHECKSIG, OP_CHECKSIGVERIFY, OP_CHECKMULTISIG, and OP_CHECKMULTISIGVERIFY. Any time you see those in a locking script, the unlocking script must contain an ECDSA signature.
A digital signature serves three purposes in bitcoin. First, the signature proves that the owner of the private key, who is by implication the owner of the funds, has authorized the spending of those funds. Secondly, the proof of authorization is undeniable (nonrepudiation). Thirdly, the signature proves that the transaction (or specific parts of the transaction) have not and cannot be modified by anyone after it has been signed.
Note that each transaction input is signed independently. This is critical, as neither the signatures nor the inputs have to belong to or be applied by the same "owners". In fact, a specific transaction scheme called "CoinJoin" uses this fact to create multi-party transactions for privacy.
Note: Each transaction input and any signature it may contain is completely independent of any other input or signature. Multiple parties can collaborate to construct transactions and sign only one input each.
Wikipedia's Definition of a "Digital Signature": A digital signature is a mathematical scheme for demonstrating the authenticity of a digital message or documents. A valid digital signature gives a recipient reason to believe that the message was created by a known sender (authentication), that the sender cannot deny having sent the message (nonrepudiation), and that the message was not altered in transit (integrity).
Source: Andreas M. Antonopoulos, https://github.com/bitcoinbook/bitcoinbook/blob/develop/ch06.asciidoc
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