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...

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...

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...

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...

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

+ 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....

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 )...

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...

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

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

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

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"...

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...

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

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))...

collects relations among the discrete logarithms of small primes, computes them by a linear algebra procedure and finally expresses the desired discrete logarithm...

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

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

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

}}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...

would double each iteration). The same strategy is used in modular exponentiation. Starting values s0 other than 4 are possible, for instance 10, 52,...

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...

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...

order is a smooth integer. The algorithm was introduced by Roland Silver, but first published by Stephen Pohlig and Martin Hellman, who credit Silver with...

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...

of r by squaring under the modulus of n. The idea beneath this test is that when n is an odd prime, it passes the test because of two facts: by Fermat's...

its contemporary form was first published by the physicist and programmer Josef Stein in 1967, it was known by the 2nd century BCE, in ancient China. The...

example, if N = 84923, (by starting at 292, the first number greater than √N and counting up) the 5052 mod 84923 is 256, the square of 16. So (505 − 16)(505...