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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.
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 Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. [ 1 ] [ 2 ] [ 3 ] It is a divide-and-conquer algorithm that reduces the multiplication of two n -digit numbers to three multiplications of n /2-digit numbers and, by repeating this reduction, to at most n log 2 3 ...
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 ).
Bargain Hunt announced the launch of a "massive going-out-of-business sale" following the discount store chain's bankruptcy filing this week. The sale will be held at all 92 Bargain Hunt locations ...
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 .
Here are 5 of the easiest ways you can catch up (and fast) President Trump has promised to protect Social Security, but has also floated the idea of cutting taxes on Social Security benefits. This ...
Country music star Colt Ford is switching up his diet after he suffered a heart attack last April.. The "Slow Ride" singer shared that he ditched certain food and drinks, including fried chicken ...