Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography...

Algorithm (IDEA), and RC4. RSA and Diffie–Hellman use modular exponentiation. In computer algebra, modular arithmetic is commonly used to limit the size of...

square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For...

is known. The relative cost of exponentiation. Though it can be implemented more efficiently using modular exponentiation, when large values of m are involved...

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...

U 2 j {\displaystyle U^{2^{j}}} . This can be accomplished via modular exponentiation, which is the slowest part of the algorithm. The gate thus defined...

discrete logarithm problem. The computation of ga mod p is known as modular exponentiation and can be done efficiently even for large numbers. Note that g...

However, when performing many multiplications in a row, as in modular exponentiation, intermediate results can be left in Montgomery form. Then the initial...

Standard for digital signatures, based on the mathematical concept of modular exponentiation and the discrete logarithm problem. In a public-key cryptosystem...

2 {\displaystyle 2} through p − 2 {\displaystyle p-2} and uses modular exponentiation to check whether a ( p − 1 ) / 2 ± 1 {\displaystyle a^{(p-1)/2}\pm...

well-studied at the time. Moreover, like Diffie-Hellman, RSA is based on modular exponentiation. Ron Rivest, Adi Shamir, and Leonard Adleman at the Massachusetts...

provide equivalent security, compared to cryptosystems based on modular exponentiation in Galois fields, such as the RSA cryptosystem and ElGamal cryptosystem...

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

are coprime. With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly...

Euclidean Lehmer's Modular square root Cipolla Pocklington's Tonelli–Shanks Berlekamp Kunerth Other algorithms Chakravala Cornacchia Exponentiation by squaring...

is based on the assumption that this Rabin function is one-way. Modular exponentiation can be done in polynomial time. Inverting this function requires...

theorem, characterizing even perfect numbers Euler's theorem, on modular exponentiation Euler's partition theorem relating the product and series representations...

respectively, hence testing them adds no value. Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm...

Euclidean Lehmer's Modular square root Cipolla Pocklington's Tonelli–Shanks Berlekamp Kunerth Other algorithms Chakravala Cornacchia Exponentiation by squaring...

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

Regardless of the specific algorithm used, this operation is called modular exponentiation. For example, consider Z17×. To compute 34 in this group, compute...

return composite return probably prime Using fast algorithms for modular exponentiation, the running time of this algorithm is O(k·log3 n), where k is the...

Euclidean Lehmer's Modular square root Cipolla Pocklington's Tonelli–Shanks Berlekamp Kunerth Other algorithms Chakravala Cornacchia Exponentiation by squaring...

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

Gauss's introduction of modular arithmetic in 1801. Modulo (mathematics), general use of the term in mathematics Modular exponentiation Turn (angle) Mathematically...

Euclidean Lehmer's Modular square root Cipolla Pocklington's Tonelli–Shanks Berlekamp Kunerth Other algorithms Chakravala Cornacchia Exponentiation by squaring...

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

tree Dedekind sum Egyptian fraction quack Montgomery reduction Modular exponentiation Linear congruence theorem Method of successive substitution Chinese...

reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic...

Euclidean Lehmer's Modular square root Cipolla Pocklington's Tonelli–Shanks Berlekamp Kunerth Other algorithms Chakravala Cornacchia Exponentiation by squaring...