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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.
To convert, the program reads each symbol in order and does something based on that symbol. The result for the above examples would be (in reverse Polish notation) "3 4 +" and "3 4 2 1 − × +", respectively. The shunting yard algorithm will correctly parse all valid infix expressions, but does not reject all invalid expressions.
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 .
Exponentiation with Montgomery reduction O ( M ( n ) k ) {\displaystyle O(M(n)\,k)} On stronger computational models, specifically a pointer machine and consequently also a unit-cost random-access machine it is possible to multiply two n -bit numbers in time O ( n ).
Modular exponentiation can be done using exponentiation by squaring by initializing the initial product to the Montgomery representation of 1, that is, to R mod N, and by replacing the multiply and square steps by Montgomery multiplies. Performing these operations requires knowing at least N′ and R 2 mod N.
The divide-and-conquer paradigm often helps in the discovery of efficient algorithms. It was the key, for example, to Karatsuba's fast multiplication method, the quicksort and mergesort algorithms, the Strassen algorithm for matrix multiplication, and fast Fourier transforms.
This can be accomplished via modular exponentiation, which is the slowest part of the algorithm. The gate thus defined satisfies U r = I {\displaystyle U^{r}=I} , which immediately implies that its eigenvalues are the r {\displaystyle r} -th roots of unity ω r k = e 2 π i k / r {\displaystyle \omega _{r}^{k}=e^{2\pi ik/r}} .
The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen in 1971. [1] It works by recursively applying fast Fourier transform (FFT) over the integers modulo 2 n + 1 {\displaystyle 2^{n}+1} .