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

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

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

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...

1{\pmod {p}}} , because the congruence relation is compatible with exponentiation. It also holds trivially for a ≡ − 1 ( mod p ) {\displaystyle a\equiv...

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

inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using exponentiation by squaring, for example). This asymmetry...

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

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))...

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

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....

46}).} Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA...

effective than either by itself. One tries various values of a, hoping that a 2 − N = b 2 {\displaystyle a^{2}-N=b^{2}} , a square. FermatFactor(N): //...

50 to 60 digits, as its running time is dominated by the size of the smallest factor p rather than by the size of the number n to be factored. Frequently...

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

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

+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...

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

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...

chain Addition-chain exponentiation Exponentiation by squaring Non-adjacent form de Rooij, Peter (1994). "Efficient exponentiation using procomputation...

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...

≤ π. It will be unique (and equal to π) if and only if μ < π. Exponentiation is given by | X | | Y | = | X Y | , {\displaystyle |X|^{|Y|}=\left|X^{Y}\right|...

exponents is exponentiation by squaring. It breaks down the calculation into a number of squaring operations. For example, the exponentiation 3 65 {\displaystyle...

computing the discrete logarithm or order of an element in a finite abelian group by Daniel Shanks. The discrete log problem is of fundamental importance to the...

}}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...

computing exponentiation by squaring, even though two modular exponentiations have to be computed. The reason is that these two modular exponentiations both...

squares, takes the difference of the results, and divides by four by shifting two bits to the right. For 8-bit integers the table of quarter squares will...

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

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...