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Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567 with z=100. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and cyan denotes left shift. (A), (B) and (C) show recursion with z=10 to obtain intermediate values. The Karatsuba algorithm is a fast multiplication algorithm.
To use REDC to compute the product of 7 and 15 modulo 17, first convert to Montgomery form and multiply as integers to get 12 as above. Then apply REDC with R = 100, N = 17, N′ = 47, and T = 12. The first step sets m to 12 ⋅ 47 mod 100 = 64. The second step sets t to (12 + 64 ⋅ 17) / 100.
First multiply the quarters by 47, the result 94 is written into the first workspace. Next, multiply cwt 12*47 = (2 + 10)*47 but don't add up the partial results (94, 470) yet. Likewise multiply 23 by 47 yielding (141, 940). The quarters column is totaled and the result placed in the second workspace (a trivial move in this case).
In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that (). [ 1 ] In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n .
Compared to traditional long multiplication, the grid method differs in clearly breaking the multiplication and addition into two steps, and in being less dependent on place value. Whilst less efficient than the traditional method, grid multiplication is considered to be more reliable, in that children are less likely to make mistakes. Most ...
In any case, this algorithm will provide a way to multiply two positive integers, provided is chosen so that < +. Let n = D M {\displaystyle n=DM} be the number of bits in the signals a {\displaystyle a} and b {\displaystyle b} , where D = 2 k {\displaystyle D=2^{k}} is a power of two.
For this inductive step, we need the following lemma. Lemma. For any integers x and y and for any prime p, (x + y) p ≡ x p + y p (mod p). The lemma is a case of the freshman's dream. Leaving the proof for later on, we proceed with the induction. Proof. Assume k p ≡ k (mod p), and consider (k+1) p. By the lemma we have
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 ...