Hamming weight. Exponentiation by squaring can be viewed as a suboptimal addition-chain exponentiation algorithm: it computes the exponent by an addition...

method and a more general principle called exponentiation by squaring (also known as binary exponentiation). First, it is required that the exponent e...

{\displaystyle \sharp n+\lfloor \log _{2}n\rfloor -1,} by using exponentiation by squaring, where ♯ n {\displaystyle \sharp n} denotes the number of...

inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using exponentiation by squaring, 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...

mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as...

from repeated multiplication, and eight multiplications with exponentiation by squaring: n2 = n × n n3 = n2 × n n6 = n3 × n3 n12 = n6 × n6 n24 = n12 ×...

f_{z}(x)} by squaring the current polynomial and taking remainder modulo f z ( x ) {\displaystyle f_{z}(x)} , Using exponentiation by squaring and polynomials...

can be computed in O(log n) arithmetic operations, using the exponentiation by squaring method. Taking the determinant of both sides of this equation...

numbers is just "exponentiation in the additive monoid", this multiplication method can also be recognised as a special case of the Square and multiply algorithm...

+ 1 contains only small factors. It uses Lucas sequences to perform exponentiation in a quadratic field. It is analogous to Pollard's p − 1 algorithm....

carry out these modular exponentiations, one could use a fast exponentiation algorithm like binary or addition-chain exponentiation). The algorithm can be...

test. Both the provable and probable primality tests rely on modular exponentiation. To further reduce the computational cost, the integers are first checked...

allow one to perform arithmetic computations very quickly. It was developed by the Russian engineer Jakow Trachtenberg in order to keep his mind occupied...

The runtime bottleneck of Shor's algorithm is quantum modular exponentiation, which is by far slower than the quantum Fourier transform and classical...

}}i=1,\ldots ,k} using a fast algorithm for modular exponentiation such as exponentiation by squaring. A number g for which these k results are all different...

Examples of applications of the Hamming weight include: In modular exponentiation by squaring, the number of modular multiplications required for an exponent...

efficient method for raising a number to a power (mod n) such as binary exponentiation, we compute: a(n−1)/2 mod n = 47110 mod 221 = −1 mod 221 ( a n ) mod...

of multiplications needed for performing an exponentiation. In the algorithm, exponentiation by squaring, the number of multiplications depends on the...

+a_{5}x)+(a_{6}+a_{7}x)x^{2})x^{4}.\end{aligned}}} Combined by Exponentiation by squaring, this allows parallelizing the computation. Arbitrary polynomials...

selection here is not imperative) compute g = gcd(aM − 1, n) (note: exponentiation can be done modulo n) if 1 < g < n then return g if g = 1 then select...

eventually terminates. The Ferguson–Forcade algorithm was published in 1979 by Helaman Ferguson and R.W. Forcade. Although the paper treats general n, it...

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...

using the value obtained for the previous value of r {\displaystyle r} by squaring under the modulus of n {\displaystyle n} . The idea beneath this test...

by this method too. Each r is a norm of a − r1b and hence that the product of the corresponding factors a − r1b is a square in Z[r1], with a "square root"...

this may be very time consuming, one generally prefers using exponentiation by squaring, which requires less than 2 log2 k matrix multiplications, and...

exponents is exponentiation by squaring. 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 exponentiation by squaring method, that is O(n2log(q))...

comparison sort algorithms Searching in balanced binary search trees Exponentiation by squaring Longest increasing subsequence Binary logarithms also occur in...

divisibility by 2 and by just the odd numbers between 3 and n {\displaystyle {\sqrt {n}}} , since divisibility by an even number implies divisibility by 2. This...