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  2. Exponentiation by squaring - Wikipedia

    en.wikipedia.org/wiki/Exponentiation_by_squaring

    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.

  3. Modular exponentiation - Wikipedia

    en.wikipedia.org/wiki/Modular_exponentiation

    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 .

  4. Montgomery modular multiplication - Wikipedia

    en.wikipedia.org/wiki/Montgomery_modular...

    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.

  5. Karatsuba algorithm - Wikipedia

    en.wikipedia.org/wiki/Karatsuba_algorithm

    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 ...

  6. Modular multiplicative inverse - Wikipedia

    en.wikipedia.org/wiki/Modular_multiplicative_inverse

    This method is generally slower than the extended Euclidean algorithm, but is sometimes used when an implementation for modular exponentiation is already available. Some disadvantages of this method include: The value () must be known and the most efficient known computation requires m 's factorization. Factorization is widely believed to be a ...

  7. These videos are bringing joy to the internet: Inside 'The ...

    www.aol.com/videos-bringing-joy-internet-inside...

    Other guests include Damon Wayans & Damon Wayans Jr., "Squid Games" lead actor Lee Jung-jae, actress Angela Bassett, singer Gwen Stefani, and rapper GloRilla.

  8. Floating-point arithmetic - Wikipedia

    en.wikipedia.org/wiki/Floating-point_arithmetic

    Conversely to floating-point arithmetic, in a logarithmic number system multiplication, division and exponentiation are simple to implement, but addition and subtraction are complex. The level-index arithmetic (LI and SLI) of Charles Clenshaw, Frank Olver and Peter Turner is a scheme based on a generalized logarithm representation.

  9. 7 fastest-growing small businesses in food, restaurants, and ...

    www.aol.com/7-fastest-growing-small-businesses...

    From eco-friendly practices to functional foods, the U.S. food and beverage industry is feeding customers' hungry demands for wellness, convenience, and flavor. And for entrepreneurs, these ...