Yes. RPF consistently outperforms Karatsuba and other classic methods (including Toom-Cook and even some FFTs) in the 128–800 digit (approx. 400–2600 bit) range. Specifically, at 256, 384, and 521 bits, which are critical for elliptic curve cryptography, RPF is faster in squaring — both in raw integer mode and when enhanced with GMP.

We observed:

Lower execution times than Karatsuba starting from 150–180 bits onward.

Better scaling, meaning performance advantage widens as bit size increases.

2) Can your technique be combined with Montgomery multiplication?

Yes, and this is one of RPF’s strong points.

Karatsuba and Montgomery are hard to combine efficiently due to their recursive and carry-heavy nature. RPF, however, has a more linear and structured squaring layout, making it much easier to wrap inside Montgomery’s reduction loop.

In initial tests, we were able to:

Integrate RPF inside a Montgomery-style multiplication framework with minimal change.

Maintain a performance lead compared to Karatsuba-integrated approaches.

We’re exploring even more optimized coupling with Montgomery reductions for modular cryptography applications.

3) Does your technique work in binary fields GF(2^k)?

RPF can be adapted to binary fields.

Since GF(2^k) squaring operations have specific bit-level behaviors (like lack of carry propagation), the base version of RPF designed for integers won’t directly transfer. However, we are working on a variant of RPF optimized for GF(2^k) that leverages:

XOR-style operations instead of additions,

and efficient bit-splitting to emulate RPF’s structure in binary logic.