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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. 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.
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Hamming weight. Exponentiationbysquaring can be viewed as a suboptimal addition-chain exponentiation algorithm: it computes the exponent by an addition...
method and a more general principle called exponentiationbysquaring (also known as binary exponentiation). First, it is required that the exponent e...
In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as bn, where b is the...
inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using exponentiationbysquaring, for example). This asymmetry...
1{\pmod {p}}} , because the congruence relation is compatible with exponentiation. It also holds trivially for a ≡ − 1 ( mod p ) {\displaystyle a\equiv...
integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root...
carry out these modular exponentiations, one could use a fast exponentiation algorithm like binary or addition-chain exponentiation). The algorithm can be...
exponents is exponentiationbysquaring. It breaks down the calculation into a number of squaring operations. For example, the exponentiation 3 65 {\displaystyle...
degree at most n, the exponentiation hq mod g can be done with O(log(q)) polynomial products, using exponentiationbysquaring method, that is O(n2log(q))...
+a_{5}x)+(a_{6}+a_{7}x)x^{2})x^{4}.\end{aligned}}} Combined byExponentiationbysquaring, this allows parallelizing the computation. Arbitrary polynomials...
any prime whose square does not exceed n divides it without a remainder, then n is not prime. Below is a version in C++ (without squaring f) template <class...
allow one to perform arithmetic computations very quickly. It was developed by the Russian engineer Jakow Trachtenberg in order to keep his mind occupied...
}}i=1,\ldots ,k} using a fast algorithm for modular exponentiation such as exponentiationbysquaring. A number g for which these k results are all different...
Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It is a divide-and-conquer...
trial division: checking if the number is divisible by prime numbers 2, 3, 5, and so on, up to the square root of n. For larger numbers, especially when using...
test. Both the provable and probable primality tests rely on modular exponentiation. To further reduce the computational cost, the integers are first checked...
this may be very time consuming, one generally prefers using exponentiationbysquaring, which requires less than 2 log2 k matrix multiplications, and...
by counting up from the square of the prime in increments of p (or 2p for odd primes). The generation must be initiated only when the prime's square is...
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