my question is, is there any alternative mathematical method of generating public keys from private keys apart from Elliptic curve secp256k1?
Sure there is. The fact is that secp256k1 is just one of the many curves defined on a Galois field (greetings to alexeyneu), so they all share the same basic fundamental properties of Galois fields, namely:
- All curves are cyclic and it is possible to generate any point on the curve by repeatively multiplying it by itself (taking its exponent to some random integer power)
- (k + l) * G = k * G + l * G , where G is a random point and k and l represent large numbers (the private key) - this demonstrates that if the sum of two private keys is greater than the curve order (maximum private key - 1) then addition will just wrap around like modular arithmetic.
- Likewise, modular inverse is also possible on all curves, because it is just raising to the power of negative values, which is equivalent to cycling through the points backwards.
There are already some cryptos that use alternative curves such as Monero (Edwards25519) and
Ontology (uses NIST P-256), but the vast majority of cryptocurrencies are using either secp256k1 or Edwards25519.