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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.
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.
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.
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.
When ax ≡ 1 (mod m) has a solution it is often denoted in this way − x ≡ a − 1 ( mod m ) , {\displaystyle x\equiv a^{-1}{\pmod {m}},} but this can be considered an abuse of notation since it could be misinterpreted as the reciprocal of a {\displaystyle a} (which, contrary to the modular multiplicative inverse, is not an integer except ...
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.
The constants R mod N and R 3 mod N can be generated as REDC(R 2 mod N) and as REDC((R 2 mod N)(R 2 mod N)). The fundamental operation is to compute REDC of a product. When standalone REDC is needed, it can be computed as REDC of a product with 1 mod N. The only place where a direct reduction modulo N is necessary is in the precomputation of R ...
If a and b are elements of K ×, the definition of a ≡ ∗ b (mod p ν) depends on what type of prime p is: [7] [8] if it is finite, then a ≡ ∗ b ( m o d p ν ) ⇔ o r d p ( a b − 1 ) ≥ ν {\displaystyle a\equiv ^{\ast }\!b\,(\mathrm {mod} \,\mathbf {p} ^{\nu })\Leftrightarrow \mathrm {ord} _{\mathbf {p} }\left({\frac {a}{b}}-1\right ...