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In mathematics, exponentiation, denoted b n, is an operation involving two numbers: the base, b, and the exponent or power, n. [1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [1] = ⏟.
This is a problem if the exponent should remain secret, as with many public-key cryptosystems. A technique called "Montgomery's ladder" [2] addresses this concern. Given the binary expansion of a positive, non-zero integer n = (n k−1...n 0) 2 with n k−1 = 1, we can compute x n as follows:
The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab
It is not known whether n q is rational for any positive integer n and positive non-integer rational q. [21] For example, it is not known whether the positive root of the equation 4 x = 2 is a rational number. [citation needed] It is not known whether e π or π e (defined using Kneser's extension) are rationals or not.
As one special case, it can be used to prove that if n is a positive integer then 4 divides () if and only if n is not a power of 2. It follows from Legendre's formula that the p -adic exponential function has radius of convergence p − 1 / ( p − 1 ) {\displaystyle p^{-1/(p-1)}} .
Inputs An integer b (base), integer e (exponent), and a positive integer m (modulus) 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 ...