Shift-and-or multiplication operation
This is equivalent to an operation known as boolean convolution. See Models of Computation page 419 defn 9.6.3.

Dunne states (p5):


  Weiss 1984 and Blum 1984b obtained lower bounds for the n-point boolean convolution function which is closely related to integer multiplication.



Models of computation
Relationships between monotone and non-monotone network complexity, Dunne
Blum, N. (1984b) An $\Omega(n^{4/3})$ lower bound on the monotone network complexity of n-th degree convolution; Theoretical Computer Science, 36, 59-70