Thread by @expctchaos

0/ What is the Elliptic Curve Digital Signature Algorithm (ECDSA)? 🧙‍♂️✨

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1/ The Elliptic Curve Digital Signature Algorithm (ECDSA) is a digital signature scheme that is based on the algebraic structure of elliptic curves. It is widely used in many #cryptographic systems, including $ETH, $BTC, and other blockchains.

2/ In the #Ethereum blockchain, ECDSA is used to sign transactions and messages to prove the authenticity of the sender.

3/ Each $ETH account has a public-private key pair, and the private key is used to sign transactions and messages. The public key is then used to verify the signature and confirm that the transaction or message was indeed signed by the owner of the private key.

4/ ECDSA has some advantages over other digital signature schemes, including faster signing and verifying times, smaller signature sizes, and better security in some cases.

5/ However, it also has some limitations, such as the need for a secure random number generator and the potential for vulnerabilities if the private key is compromised.

6/ Here's an example of how ECDSA works:

7/ Alice wants to send a message m to Bob and digitally sign it. She has already prepared her public-private key pair, which is based on a specific elliptic curve.

8/ Her private key is a secret number that only she knows, and her public key is derived from the private key and the elliptic curve.

9/ To sign the message, Alice uses her private key to generate a signature s. She also includes a random number k in the signature process to add an additional layer of security. The signature s is then sent along with the message m to Bob.

10/ To verify the signature, Bob uses Alice's public key and the random number k to calculate a value r. He then compares r to the value of r included in the signature s. If they match, the signature is considered valid.

11/ Here is an example calculation to illustrate the process of generating and verifying an ECDSA signature:

12/ Here is an example calculation to illustrate the process of generating and verifying an ECDSA signature:

• Alice's private key is d = 13
• Alice's public key is (x, y) = (44, 21)
• The message m is "Hello, Bob!"
• The random number k is 7

13/ To sign the message, Alice calculates:

• r = (x coordinate of kG) mod p = (447) mod 59 = 22
• s = (k^-1 * (H(m) + rd)) mod p = (7^-1 * (H("Hello, Bob!") + 2213)) mod 59 = 44

14/ Alice now sends the signature (r, s) along with the message to Bob. Bob uses Alice's public key (x, y) and the random number k to calculate the value of r:

• r' = (x coordinate of kG) mod p = (447) mod 59 = 22

15/ Bob compares r' to the value of r included in the signature. If they match, the signature is considered valid. Bob can then be confident that the message is authentic and came from Alice.

16/ ECDSA is a widely used digital signature scheme that provides a secure and efficient way to verify the authenticity of messages and transactions. It is an important part of many cryptographic systems, including Ethereum and other blockchain platforms.

17/ While ECDSA offers many benefits, it is important to note that it is not foolproof. Like any cryptographic system, it has some limitations and vulnerabilities that need to be considered.

18/ For example, if an attacker is able to obtain a user's private key, they can forge signatures and potentially compromise the security of the system.

19/ It is also important to use a secure random number generator to ensure the security of the ECDSA system. If an attacker can predict or control the random numbers used in the signature process, they may be able to forge signatures or compromise the security of the system.

20/ Despite these limitations, ECDSA is a powerful and widely used digital signature scheme that is an important part of many cryptographic systems.

21/ I will introduce more cryptographic primitives over the next few days, so stay tuned!