Achieving non-linearity in a block cipher using only, rotations, shifts, AND, OR, and XOR
The brief answer is: Yes, you can!

You may wish to study NORX, an AEAD scheme submitted to the CAESAR competition, whose essential thesis is that you can simultaneously attain high security, high software performance, and high hardware performance by the same design principles as ARX, but replacing $a + b$, which is slow in hardware, by the approximation $a \oplus b \oplus (a \wedge b) \ll 1$.