Carry-free multiplication operation
Your operation is multiplication of polynomials over $GF(2)$, i.e., multiplication in the polynomial ring $GF(2)[x]$.

For instance, if $p=101$ and $q=1101$, you can represent them as $p(x)=x^2+1$, $q(x)=x^3+x^2+1$, and their product as polynomials is $p(x) \times q(x) = x^5+x^4+x^3+1$, so $p \otimes q = 111001$.

If $p,q$ are $r$ bits long, this polynomial multiplication operation can be computed in $O(r \lg r)$ time using FFT techniques, but in practice this may not be a win unless your polynomial is extremely large. There is also a Karatsuba-style algorithm, whose complexity is something like $O(r^{1.6})$, as well as other options.  The situation is somewhat analogous to integer multiplication, in that many of the same fast algorithms can be applied, but not identical.

See, e.g.,


Wikipedia on Fast multiplication algorithms for large inputs.  These methods are described in terms of multiplying two integers, but they can be transposed to apply to polynomial multiplication.
Polynomial Multiplication, Karatsuba and Fast Fourier Transform
When is FFT multiplication of arbitrary-precision polynomials practical?.  Richard Fateman, March 14, 2006.
Can you save time in multiplying polynomials by encoding them as integers?.  Richard Fateman, January 15, 2005.