Logical bit shifting may be useful for multiplying or dividing unsigned integers by powers of two. For example, if the value "0001" or "1" is shifted left, it becomes "0010" or "2," shifted to the left again it becomes "0100," or "4." Shifting to the right has an opposite effect of dividing the value by two per shift.
One simple and good way to encrypt data is through rotation of bits or sometimes called bit shifting.
The bitwise shift operators are the right-shift operator ( >> ), which moves the bits of an integer or enumeration type expression to the right, and the left-shift operator ( << ), which moves the bits to the left.
Bit-shifting is still faster, but for non-power-of-two mul/div by the time you do all your shifts and add the results it's slower again.
I still write code for systems that do not have floating point support in hardware. In these systems you need bit-shifting for nearly all your arithmetic.
Also you need shifts to generate hashes. Polynomial arithmetic (CRC, Reed-Solomon Codes are the mainstream applications) or uses shifts as well.
However, shifts are just used because they are handy and express exactly what the writer intended. You can emulate all bit-shifts with multiplication if you want to, but that would be harder to write, less readable and sometimes slower.
The compilers detect cases where multiplication can be reduced to a shift.
Yes, I've used them a lot of times. Bit twiddling is important on embedded hardware where bit-masks are very common. It's also important in games programming, when you need every last bit of performance.
Edit: Also, I use them a lot for manipulating bitmaps, for example changing the colour depth, or converting RGB <-> BGR.
And I cannot think of many cases when they are being used. It's usually other way around - there is some specific problem, and it turns out that employing bit operations will yield the best results (usually in term of performance - time and/or space).
One place I use them all the time is when transposing the endian-ness of integers for cross-platform applications. They also sometimes come in handy (along with other bit-manipulation operators) when blitting 2D graphics.
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