Finding Elliptical curve points and encoding text using them

Question

I recently got into learning Elliptical curve cryptography and are currently building a project in C#.

Everything is working well so far, I can encode and decode points, and thanks to this forum I can do point addition and doubling really fast. Now I want to take the next step and start using really big numbers, so far I've been working off different ECC samples and papers normally using really small primes and numbers for clarity. The end goal is to being able to encrypt and decrypt a text, this will later be used for digital signing of this text. So what I must figure out now is:

1. What standard curves could I use? I know curves like p192 etc exists, am I allowed to use these?
2. How can I find valid points to use for messages and generator on these huge curves? So far I've done it using brute force on smaller curves with a small prime. I can't find any using my previous brute force approach since the numbers are too large.
3. Whats a good way for encoding/decoding text using ECC? Should I simply map each valid character to a point on the curve and perform encryption/decryption once for each character and translate between the character and point using a lookup table?

Thank you!

Answer 1

1. What standard curves could I use? I know curves like p192 etc exists, am I allowed to use these?

Yes, of course you may use those, for more curves - including safer ones - check the safecurves website.

2. How can I find valid points to use for messages and generator on these huge curves? So far I've done it using brute force on smaller curves with a small prime. I can't find any using my previous brute force approach since the numbers are too large.

This is part of key generation procedure. For Elliptic Curve the private key is a vector (usually of order N) and the public key is calculated for this particular vector. So this is basically asking how EC key generation takes place (for which there are tutorials available online, I won't completely explain it here).

3. Whats a good way for encoding/decoding text using ECC? Should I simply map each valid character to a point on the curve and perform encryption/decryption once for each character and translate between the character and point using a lookup table?

No, you shouldn't directly encrypt anything with EC crypto, try and look up ECIES instead, it is basically ECDH key agreement coupled with symmetric cryptography on the derived key. Start with ECDH and then implement ECIES. Progress on with ECDSA if you want to have signatures.

EC is different than RSA in this regard, you cannot just alter the padding scheme (as there is no padding scheme).