Hamming weight. Exponentiation by squaring can be viewed as a suboptimal addition-chain exponentiation algorithm: it computes the exponent by an addition...
22 KB (3,382 words) - 02:27, 29 June 2025
method and a more general principle called exponentiation by squaring (also known as binary exponentiation). First, it is required that the exponent e...
21 KB (2,759 words) - 02:20, 29 June 2025
{\displaystyle \sharp n+\lfloor \log _{2}n\rfloor -1,} by using exponentiation by squaring, where ♯ n {\displaystyle \sharp n} denotes the number of...
107 KB (13,693 words) - 20:30, 5 July 2025
inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using exponentiation by squaring, for example). This asymmetry...
17 KB (2,537 words) - 07:42, 7 July 2025
1{\pmod {p}}} , because the congruence relation is compatible with exponentiation. It also holds trivially for a ≡ − 1 ( mod p ) {\displaystyle a\equiv...
8 KB (1,125 words) - 02:50, 6 July 2025
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...
15 KB (1,990 words) - 03:13, 22 June 2025
Trachtenberg system (section Multiplying by 2)
allow one to perform arithmetic computations very quickly. It was developed by the Russian mathematician and engineer Jakow Trachtenberg in order to keep...
27 KB (6,358 words) - 13:45, 5 July 2025
from repeated multiplication, and eight multiplications with exponentiation by squaring: n2 = n × n n3 = n2 × n n6 = n3 × n3 n12 = n6 × n6 n24 = n12 ×...
9 KB (1,340 words) - 06:46, 28 April 2025
Fibonacci sequence (section Computation by rounding)
can be computed in O(log n) arithmetic operations, using the exponentiation by squaring method. Taking the determinant of both sides of this equation...
86 KB (13,080 words) - 14:15, 7 July 2025
+ 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....
5 KB (831 words) - 21:06, 30 September 2022
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...
13 KB (1,410 words) - 22:03, 16 April 2025
Examples of applications of the Hamming weight include: In modular exponentiation by squaring, the number of modular multiplications required for an exponent...
33 KB (3,163 words) - 07:42, 3 July 2025
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...
36 KB (4,739 words) - 03:56, 4 July 2025
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...
10 KB (1,518 words) - 08:52, 27 June 2025
The runtime bottleneck of Shor's algorithm is quantum modular exponentiation, which is by far slower than the quantum Fourier transform and classical...
40 KB (5,809 words) - 23:45, 1 July 2025
}}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...
22 KB (2,508 words) - 20:36, 19 June 2025
carry out these modular exponentiations, one could use a fast exponentiation algorithm like binary or addition-chain exponentiation). The algorithm can be...
6 KB (838 words) - 20:18, 14 March 2025
integers x and y such that a x + b y = gcd ( a , b ) . {\displaystyle ax+by=\gcd(a,b).} This is a certifying algorithm, because the gcd is the only number...
28 KB (4,467 words) - 20:39, 9 June 2025
test. Both the provable and probable primality tests rely on modular exponentiation. To further reduce the computational cost, the integers are first checked...
8 KB (1,158 words) - 16:41, 12 November 2024
Arithmetic (section Exponentiation and logarithm)
exponents is exponentiation by squaring. It breaks down the calculation into a number of squaring operations. For example, the exponentiation 3 65 {\displaystyle...
165 KB (16,396 words) - 04:14, 2 June 2025
+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...
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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...
27 KB (3,833 words) - 09:23, 3 May 2025
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...
38 KB (5,639 words) - 20:26, 3 May 2025
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"...
14 KB (1,911 words) - 17:11, 26 June 2025
of multiplications needed for performing an exponentiation. In the algorithm, exponentiation by squaring, the number of multiplications depends on the...
5 KB (616 words) - 05:59, 6 May 2023
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...
126 KB (15,349 words) - 16:35, 30 April 2025
Matrix multiplication (section Square matrices)
this may be very time consuming, one generally prefers using exponentiation by squaring, which requires less than 2 log2 k matrix multiplications, and...
41 KB (6,581 words) - 15:09, 5 July 2025
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...
25 KB (2,977 words) - 21:02, 19 June 2025
Order of operations (redirect from Serial exponentiation)
property of exponentiation that (ab)c = abc, so it's unnecessary to use serial exponentiation for this. However, when exponentiation is represented by an explicit...
48 KB (4,556 words) - 21:34, 26 June 2025
better efficiency is obtained by computing n! from its prime factorization, based on the principle that exponentiation by squaring is faster than expanding...
70 KB (8,432 words) - 06:19, 30 April 2025