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...

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 ×...

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

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....

}}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...

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

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...

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

+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...

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

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...

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

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...

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"...

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

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): //...

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

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

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...

24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares...

area can be divided into a grid of: 1×1 squares, 2×2 squares, 3×3 squares, 4×4 squares, 6×6 squares or 12×12 squares. Therefore, 12 is the GCD of 24 and 60...

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

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...