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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.
Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567 with z=100. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and cyan denotes left shift. (A), (B) and (C) show recursion with z=10 to obtain intermediate values. The Karatsuba algorithm is a fast multiplication algorithm.
A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Numerous algorithms are known and there has been much research into the t
Let = be the number of bits in the signals and , where = is a power of two. Divide the signals a {\displaystyle a} and b {\displaystyle b} into D {\displaystyle D} blocks of M {\displaystyle M} bits each, storing the resulting blocks as arrays A , B {\displaystyle A,B} (whose entries we shall consider for simplicity as arbitrary precision ...
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.
When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, 3 5 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power.
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This is a consequence of the fact that, because gcd(R, N) = 1, multiplication by R is an isomorphism on the additive group Z/NZ. For example, (7 + 15) mod 17 = 5, which in Montgomery form becomes (3 + 4) mod 17 = 7. Multiplication in Montgomery form, however, is seemingly more complicated.