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Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m .
Most of the mathematical functions are defined in <math.h> (<cmath> header in C++). The functions that operate on integers, such as abs, labs, div, and ldiv, are instead defined in the <stdlib.h> header (<cstdlib> header in C++). Any functions that operate on angles use radians as the unit of angle. [1]
In computer science, the shunting yard algorithm is a method for parsing arithmetical or logical expressions, or a combination of both, specified in infix notation.It can produce either a postfix notation string, also known as reverse Polish notation (RPN), or an abstract syntax tree (AST). [1]
This method is generally slower than the extended Euclidean algorithm, but is sometimes used when an implementation for modular exponentiation is already available. Some disadvantages of this method include: The value () must be known and the most efficient known computation requires m 's factorization. Factorization is widely believed to be a ...
The basic interface is for C, but wrappers exist for other languages, including Ada, C++, C#, Julia, .NET, OCaml, Perl, PHP, Python, R, Ruby, and Rust. Prior to 2008, Kaffe, a Java virtual machine, used GMP to support Java built-in arbitrary precision arithmetic. [6] Shortly after, GMP support was added to GNU Classpath. [7]
Exponentiation and primality testing are primitive recursive. Given primitive recursive functions e {\displaystyle e} , f {\displaystyle f} , g {\displaystyle g} , and h {\displaystyle h} , a function that returns the value of g {\displaystyle g} when e ≤ f {\displaystyle e\leq f} and the value of h {\displaystyle h} otherwise is primitive ...
If exponentiation is indicated by stacked symbols using superscript notation, the usual rule is to work from the top down: [2] [7] a b c = a (b c) which typically is not equal to (a b) c. This convention is useful because there is a property of exponentiation that (a b) c = a bc, so it's unnecessary to use serial exponentiation for this.
The simplest method is the double-and-add method, [3] similar to square-and-multiply in modular exponentiation. The algorithm works as follows: The algorithm works as follows: To compute sP , start with the binary representation for s : s = s 0 + 2 s 1 + 2 2 s 2 + ⋯ + 2 n − 1 s n − 1 {\displaystyle s=s_{0}+2s_{1}+2^{2}s_{2}+\cdots +2 ...