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After each step k of the Euclidean algorithm, the norm of the remainder f(r k) is smaller than the norm of the preceding remainder, f(r k−1). Since the norm is a nonnegative integer and decreases with every step, the Euclidean algorithm for Gaussian integers ends in a finite number of steps. [ 144 ]
Given an integer a and a non-zero integer d, it can be shown that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < | d |. The number q is called the quotient, while r is called the remainder. (For a proof of this result, see Euclidean division. For algorithms describing how to calculate the remainder, see Division algorithm.)
If Z(x) is constructed to have t or fewer non-zero coefficients, then keys which share fewer than t bits are guaranteed to not collide. Z is a function of k, t, and n (the last of which is a divisor of 2 k − 1) and is constructed from the finite field GF(2 k). Knuth gives an example: taking (n,m,t) = (15,10,7) yields Z(x) = x 10 + x 8 + x 5 ...
00000000001110 100 1011 00000000000101 100 101 1 ----- 00000000000000 000 <--- remainder The following Python code outlines a function which will return the initial CRC remainder for a chosen input and polynomial, with either 1 or 0 as the initial padding. Note that this code works with string inputs rather than raw numbers:
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 ...
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems.. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations.
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.