Search results
Results from the WOW.Com Content Network
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
Although all of the preceding text is written in terms of divisibility by the generator polynomial, any fixed remainder () may be used and will perform just as well as a zero remainder. Most commonly, the all-ones polynomial ( x n + 1 ) / ( x + 1 ) {\displaystyle (x^{n}+1)/(x+1)} is used, but, for example, the asynchronous transfer mode header ...
Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of have the form , where p is a prime number).
The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and multiplication can be defined on these m objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the ...
Tonelli–Shanks cannot be used for composite moduli: finding square roots modulo composite numbers is a computational problem equivalent to integer factorization. [1] An equivalent, but slightly more redundant version of this algorithm was developed by Alberto Tonelli [2] [3] in 1891.
If g is a primitive root modulo p k, then g is also a primitive root modulo all smaller powers of p. If g is a primitive root modulo p k, then either g or g + p k (whichever one is odd) is a primitive root modulo 2 p k. [14] Finding primitive roots modulo p is also equivalent to finding the roots of the (p − 1)st cyclotomic polynomial modulo p.
One could use a calculator to compute 4 13; this comes out to 67,108,864. Taking this value modulo 497, the answer c is determined to be 445. Note that b is only one digit in length and that e is only two digits in length, but the value b e is 8 digits in length. In strong cryptography, b is often at least 1024 bits. [1]
The algorithm takes as its inputs n, the integer to be factored; and , a polynomial in x computed modulo n.In the original algorithm, () = (), but nowadays it is more common to use () = (+).