Mike: How are ECDSA pairs generated?


GPT: ECDSA (Elliptic Curve Digital Signature Algorithm) key pairs are generated using elliptic curve cryptography. The key pair consists of a private key and a corresponding public key. Here is a brief overview of the key pair generation process:

Choose an elliptic curve: A standard elliptic curve is chosen, such as one from the NIST family of curves.

Choose a base point: A base point, G, is selected on the curve. This point has a specific set of coordinates (x, y) that are used in the generation process.

Generate a private key: A random integer, k, is generated as the private key. This value should be a large random number that is difficult to guess.

Compute the public key: The public key is derived by performing scalar multiplication of the base point, G, by the private key, k. The result is a new point on the curve, which is the public key.

Encode the public key: The coordinates of the public key point are encoded as a string of bytes using a specified format, such as the uncompressed or compressed format.

Once the key pair is generated, the private key should be kept secure and not shared with anyone, while the public key can be shared with anyone who needs to verify signatures generated by the private key. When signing a message with ECDSA, the private key is used to generate a signature that can be verified with the corresponding public key.


Mike: Why use ECDSA when things like RSA exist? Do I really need ECDSA when I can encrypt an entire file with RSA?


GPT: Both ECDSA (Elliptic Curve Digital Signature Algorithm) and RSA (Rivest-Shamir-Adleman) are popular cryptographic algorithms used for digital signatures and encryption. However, they differ in their strengths and weaknesses, and the choice of which to use depends on the specific use case and security requirements.

ECDSA has a few advantages over RSA when it comes to digital signatures. First, it can provide the same level of security with smaller key sizes, which makes it more efficient in terms of both computation and storage. This is particularly important in applications where resources are limited, such as mobile devices and embedded systems. Second, ECDSA is resistant to certain types of attacks that can be used against RSA, such as the factoring and discrete logarithm attacks.

However, ECDSA is not typically used for encryption of data, as it is a digital signature algorithm rather than an encryption algorithm. RSA, on the other hand, can be used for both encryption and digital signatures. However, when using RSA for encryption, it is important to consider the size of the data to be encrypted, as RSA encryption can become inefficient for large amounts of data.

In summary, the choice of whether to use ECDSA or RSA depends on the specific security requirements of the application. If digital signatures are the primary concern and efficient computation and storage are important, ECDSA may be the better choice. If encryption is the primary concern, RSA may be a better choice, though other algorithms such as AES may be more efficient for large amounts of data.


Mike: Can you give an example of generated an ECDSA keypair (with values shown in hex) using the secp256k1 curve? I'm having a hard time understanding the mathematical side, so please add explanations along the way.


GPT: Sure, here's an example of how to generate an ECDSA keypair using the secp256k1 curve, with values shown in hexadecimal:

Choose the curve: In this example, we'll use the secp256k1 curve, which is commonly used in Bitcoin and other cryptocurrencies. This curve has the following equation:

y^2 = x^3 + 7 (mod p)
where p is a large prime number:
p = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F

2. Choose a base point: The secp256k1 curve has a base point G with the following coordinates in hexadecimal:
Gx = 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798
Gy = 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8


3. Generate a private key: Choose a random number k as the private key, where 1 <= k < n, and n is the order of the base point G. For secp256k1, the order n is:
n = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141

For this example, let's choose k = 1111111111111111111111111111111111111111111111111111111111111111 (in base 10), which is equivalent to:
k = 0x1111111111111111111111111111111111111111111111111111111111111111

4. Compute the public key: The public key is derived by performing scalar multiplication of the base point G by the private key k, which is represented as kG. In other words, we compute:
K = kG
To perform this multiplication, we can use the double-and-add algorithm, which involves repeatedly adding the point to itself in binary representation. Here's the step-by-step calculation using the given values:
a. Convert k to binary: 1000100010001000100010001000100010001000100010001000100010000001
b. Starting from the second bit, double the previous result and add G if the current bit is 1. Otherwise, just double the previous result. Here's the calculation:

   - 0 (skip the first bit)
   - 2G (double 0 and add G)
   - 4G (double 2G)
   - 8G (double 4G)
   - 9G (double 8G and add G)
   - 18G (double 9G)
   - 19G (double 18G and add G)
   - 38G (double 19G)
   - 76G (double 38G)
   - 152G (double 76G)
   - 305G (double 152G and add G)
   - 610G (double 305G)
   - 1220G (double 610G)
   - 1221G (double 1220G and add G)
   - 2442G (double 1221G)
   - 4884G (double 2442G)
   - 9769G (double 4884G and add G)


Mike: Why are digital signatures, such as ECDSA, used instead of fully encrypting files?


GPT: Digital signatures, such as ECDSA, are used for authentication and integrity verification of data, rather than for encryption. While encryption is used to protect the confidentiality of data by preventing unauthorized access, digital signatures are used to ensure the authenticity and integrity of data by providing a way to verify that the data has not been tampered with and that it comes from a trusted source.

Encryption and digital signatures serve different purposes and are often used together to provide end-to-end security for data. For example, data can be encrypted to protect its confidentiality, and a digital signature can be added to verify the authenticity and integrity of the data after it is decrypted. In this way, both the confidentiality and integrity of the data are protected.

Another reason why digital signatures are used instead of fully encrypting files is that digital signatures are typically faster and less computationally intensive than encryption. Digital signatures involve only a single computation, while encryption involves multiple computations to encrypt and decrypt the data. Additionally, digital signatures are often used to sign small pieces of data, such as messages, whereas encryption is used to protect the confidentiality of larger amounts of data, such as files.

In summary, digital signatures such as ECDSA are used for authentication and integrity verification of data, while encryption is used to protect the confidentiality of data. Both encryption and digital signatures are important components of secure communication and are often used together to provide end-to-end security.