Achieving non-linearity in a block cipher using only, rotations, shifts, AND, OR, and XOR It is supposedly difficult to introduce enough non-linearity in block ciphers based on modular addition, data-independent bitwise rotations, and XOR to make them secure against certain kinds of cryptanalysis even though addition is used because it introduces non-linearity into the design that XOR does not have. But what if we also use bitwise AND, bitwise OR, and bitwise shifts (a left bitwise shift fills a fixed-length bit string with zeroes on its left-hand side)? And let's toss modular addition into the flames since that actually does not run in constant time on some processors.

I figure that bitwise AND and bitwise OR can be used to introduce more non-linearity than modular addition. Additionally, though I'm not familiar with the details rotational cryptanalysis, I figured that bitwise shifts can help to thwart it since the for some inputs, the outputs of a function using rotations might be made equal with a rotation applied to one of them. Bitwise shifts would not suffer from this problem. As an example, consider that for inputs $x_0 = 0101$ and $x_1 = 1010$ the function $f(x) = (x \ggg 1) \oplus x$ will return $111$ in both instances. But, if $f(x) = (x \gg 1) \oplus x$, then the outputs will be $0111$ and $1111$, respectively.
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 $a \oplus b \oplus (a \wedge b) \ll 1$.