Search results
Results from the WOW.Com Content Network
Many algorithms for exponentiation do not provide defence against side-channel attacks. Namely, an attacker observing the sequence of squarings and multiplications can (partially) recover the exponent involved in the computation. This is a problem if the exponent should remain secret, as with many public-key cryptosystems.
For example, given b = 5, e = 3 and m = 13, dividing 5 3 = 125 by 13 leaves a remainder of c = 8. Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm. That is: c = b e mod m = d −e mod m, where e < 0 and b ⋅ d ≡ 1 (mod m).
Arbitrary precision arithmetic is also used to compute fundamental mathematical constants such as π to millions or more digits and to analyze the properties of the digit strings [8] or more generally to investigate the precise behaviour of functions such as the Riemann zeta function where certain questions are difficult to explore via ...
To convert, the program reads each symbol in order and does something based on that symbol. The result for the above examples would be (in reverse Polish notation) "3 4 +" and "3 4 2 1 − × +", respectively. The shunting yard algorithm will correctly parse all valid infix expressions, but does not reject all invalid expressions.
The simplest method is the double-and-add method, [3] similar to square-and-multiply in modular exponentiation. The algorithm works as follows: The algorithm works as follows: To compute sP , start with the binary representation for s : s = s 0 + 2 s 1 + 2 2 s 2 + ⋯ + 2 n − 1 s n − 1 {\displaystyle s=s_{0}+2s_{1}+2^{2}s_{2}+\cdots +2 ...
[8] GMP is part of the GNU project (although its website being off gnu.org may cause confusion), and is distributed under the GNU Lesser General Public License (LGPL). GMP is used for integer arithmetic in many computer algebra systems such as Mathematica [9] and Maple. [10] It is also used in the Computational Geometry Algorithms Library (CGAL).
The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term; [15] so a is the base, b is the exponent (or hyperexponent), [12] and n is the rank (or grade), [6] and moreover, (,) is read as "the bth n-ation of a", e.g. (,) is read as "the 9th tetration of 7", and (,) is read as "the 789th 123 ...
Exponentiation with Montgomery reduction O ( M ( n ) k ) {\displaystyle O(M(n)\,k)} On stronger computational models, specifically a pointer machine and consequently also a unit-cost random-access machine it is possible to multiply two n -bit numbers in time O ( n ).