Circular shift
In combinatorial mathematics, a circular shift is the operation of rearranging the entries in a tuple, either by moving the final entry to the first position, while shifting all other entries to the next position, or by performing the inverse operation. A circular shift is a special kind of cyclic permutation, which in turn is a special kind of permutation. Formally, a circular shift is a permutation σ of the n entries in the tuple such that either
or
The result of repeatedly applying circular shifts to a given tuple are also called the circular shifts of the tuple.
For example, repeatedly applying circular shifts to the four-tuple (a, b, c, d) successively gives
and then the sequence repeats; this four-tuple therefore has four distinct circular shifts. However, not all n-tuples have n distinct circular shifts. For instance, the 4-tuple (a, b, a, b) only has 2 distinct circular shifts. In general the number of circular shifts of an n-tuple could be any divisor of n, depending on the entries of the tuple.
In computer programming, a circular shift (or bitwise rotation) is a shift operator that shifts all bits of its operand. Unlike an arithmetic shift, a circular shift does not preserve a number's sign bit or distinguish a number's exponent from its significand (sometimes referred to as the mantissa). Unlike a logical shift, the vacant bit positions are not filled in with zeros but are filled in with the bits that are shifted out of the sequence.
Circular shifts are used often in cryptography in order to permute bit sequences. Unfortunately, many programming languages, including C, do not have operators or standard functions for circular shifting, even though virtually all processors have bitwise operation instructions for it (e.g. Intel x86 has ROL and ROR). However, some compilers may provide access to the processor instructions by means of intrinsic functions. In addition, it is possible to write standard ANSI C code that compiles down to the "rotate" assembly language instruction (on CPUs that have such an instruction). Most C compilers recognize the following idiom, and compile it to a single 32-bit rotate instruction.
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