I could not find a description of your algorithm, which makes it hard to give you feedback. However, here are a few questions that come to my mind from a cryptography/ZK perspective.
1) elliptic-curve cryptography cares about 256-bit multiplication (and perhaps 384 or 521 bits for the truly paranoid). Is your algorithm better than alternatives in that regime?
2) cryptography/ZK cares about multiplication mod p, and not about multiplication per se. Of course you can perform the multiplication and then reduce mod p, but other techniques exist (e.g. Montgomery multiplication) that interleave the multiplication and the reduction for better performance. It is hard to combine Montgomery and Karatsuba. Can your technique be combined with Montgomery?
3) ZK also cares about binary fields GF(2^k). Does your technique work in those fields?
I could not find a description of your algorithm, which makes it hard to give you feedback. However, here are a few questions that come to my mind from a cryptography/ZK perspective.
1) elliptic-curve cryptography cares about 256-bit multiplication (and perhaps 384 or 521 bits for the truly paranoid). Is your algorithm better than alternatives in that regime?
2) cryptography/ZK cares about multiplication mod p, and not about multiplication per se. Of course you can perform the multiplication and then reduce mod p, but other techniques exist (e.g. Montgomery multiplication) that interleave the multiplication and the reduction for better performance. It is hard to combine Montgomery and Karatsuba. Can your technique be combined with Montgomery?
3) ZK also cares about binary fields GF(2^k). Does your technique work in those fields?