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Use the extended Euclidean algorithm to compute k −1, the modular multiplicative inverse of k mod 2 w, where w is the number of bits in a word. This inverse will exist since the numbers are odd and the modulus has no odd factors. For each number in the list, multiply it by k −1 and take the least significant word of the result.
The last rule can be used to move modular arithmetic into division. If b divides a, then (a/b) mod m = (a mod b m) / b. The modular multiplicative inverse is defined by the following rules: Existence: There exists an integer denoted a −1 such that aa −1 ≡ 1 (mod m) if and only if a is coprime with m.
The modular inverse of aR mod N is REDC((aR mod N) −1 (R 3 mod N)). Modular exponentiation can be done using exponentiation by squaring by initializing the initial product to the Montgomery representation of 1, that is, to R mod N, and by replacing the multiply and square steps by Montgomery multiplies.
The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method. In particular, if either exp {\displaystyle \exp } or log {\displaystyle \log } in the complex domain can be computed with some complexity, then that complexity is ...
Similarity between two sets The Jaccard index formula measures the similarity between two sets based on the number of items that are present in both sets relative to the total number of items. It is commonly used in recommendation systems and social media analysis [ citation needed ] .
A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if M is the product of the moduli, there is, in an interval of length M, exactly one integer having any given set of modular values.
Let P and Q be two sets, each containing N points in .We want to find the transformation from Q to P.For simplicity, we will consider the three-dimensional case (=).The sets P and Q can each be represented by N × 3 matrices with the first row containing the coordinates of the first point, the second row containing the coordinates of the second point, and so on, as shown in this matrix:
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). Modular exponentiation is efficient to compute, even for very large integers.