Common idea in Karatsuba, Gauss and Strassen multiplication The identities used in multiplication algorithms by


Karatsuba (integers)
Gauss (complex numbers)
Strassen (matrices)


seem very closely related. Is there a common abstract framework/generalization?
A partial answer to your question is the group-theoretic approach first developed by Cohn and Umans and further developed by Cohn, Kleinberg, Szegedy and Umans. It can "sort of" capture Strassen and Coppersmith-Winograd for matrix multiplication.