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In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation.
Outputs The modular exponent c where c = b e mod m. Initialise c = 1 and loop variable e′ = 0; While e′ < e do Increment e′ by 1; Calculate c = (b ⋅ c) mod m; Output c; Note that at the end of every iteration through the loop, the equation c ≡ b e′ (mod m) holds true. The algorithm ends when the loop has been executed e times.
In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain , with multiplication instead of addition, computes the desired exponent (instead of multiple) of the base .
The algorithm begins with a multiprecision integer T and reduces it one word at a time. First an appropriate multiple of N is added to make T divisible by B. Then a multiple of N is added to make T divisible by B 2, and so on. Eventually T is divisible by R, and after division by R the algorithm is in the same place as REDC was after the ...
The simplest method is the double-and-add method, [2] 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 ...
The properties of NAF make it useful in various algorithms, especially some in cryptography; e.g., for reducing the number of multiplications needed for performing an exponentiation. In the algorithm, exponentiation by squaring, the number of multiplications depends on the number of non-zero bits. If the exponent here is given in NAF form, a ...
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.