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For simplicity, the modulus calculation is omitted here. This example shows how to compute using left to right binary exponentiation. The exponent is 1101 in binary; there are 4 bits, so there are 4 iterations. Initialize the result to 1: (=).
Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For semigroups for which additive notation is commonly used, like elliptic curves used in cryptography , this method is also referred to as double-and-add .
The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and ...
Right-associative operations include the following: Exponentiation of real numbers in superscript notation = Exponentiation is commonly used with brackets or right-associatively because a repeated left-associative exponentiation operation is of little use.
Calculators generally perform operations with the same precedence from left to right, [1] but some programming languages and calculators adopt different conventions. For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation .
In algebra, the terms left and right denote the order of a binary operation (usually, but not always, called "multiplication") in non-commutative algebraic structures. A binary operation ∗ is usually written in the infix form: s ∗ t. The argument s is placed on the left side, and the argument t is on the right side.
So, the binary representation of π is calculated from left-to-right as follows: (=) = (+ + + + + +) where p is the precision ( 24 in this example), n is the position of the bit of the significand from the left (starting at 0 and finishing at 23 here) and e is the exponent ( 1 in this example).
To find the M-th element V of the sequence characterized by B, we proceed in a manner similar to left-to-right exponentiation: x := B y := (B ^ 2 − 2) mod N for each bit of M to the right of the most significant bit do if the bit is 1 then x := (x × y − B) mod N y := (y ^ 2 − 2) mod N else y := (x × y − B) mod N x := (x ^ 2 − 2) mod ...