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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.
MATLAB (an abbreviation of "MATrix LABoratory" [22]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
If a ≡ b (mod m), then it is generally false that k a ≡ k b (mod m). However, the following is true: If c ≡ d (mod φ(m)), where φ is Euler's totient function, then a c ≡ a d (mod m) —provided that a is coprime with m. For cancellation of common terms, we have the following rules: If a + k ≡ b + k (mod m), where k is any integer ...
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 ...
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. On the other hand, computing the modular discrete logarithm – that is, finding the exponent e when given b , c , and m – is believed to be difficult.
output: Integer S in the range [0, N − 1] such that S ≡ TR −1 mod N m ← ((T mod R)N′) mod R t ← (T + mN) / R if t ≥ N then return t − N else return t end if end function To see that this algorithm is correct, first observe that m is chosen precisely so that T + mN is divisible by R .
Most commonly, the modulus is chosen as a prime number, making the choice of a coprime seed trivial (any 0 < X 0 < m will do). This produces the best-quality output, but introduces some implementation complexity, and the range of the output is unlikely to match the desired application; converting to the desired range requires an additional multiplication.