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Some calculators have a mod() function button, and many programming languages have a similar function, expressed as mod(a, n), for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as a % n or a mod n. For environments lacking a similar function, any of the three definitions above can ...
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.
A primitive root modulo m exists if and only if m is equal to 2, 4, p k or 2p k, where p is an odd prime number and k is a positive integer. If a primitive root modulo m exists, then there are exactly φ ( φ ( m )) such primitive roots, where φ is the Euler's totient function.
If a instead is one, the variable base (containing the value b 2 i mod m of the original base) is simply multiplied in. In this example, the base b is raised to the exponent e = 13. The exponent is 1101 in binary. There are four binary digits, so the loop executes four times, with values a 0 = 1, a 1 = 0, a 2 = 1, and a 3 = 1.
An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. Arithmetic functions are often extremely irregular (see table ), but some of them have series expansions in terms of Ramanujan's sum .
However, the linear congruence 4x ≡ 6 (mod 10) has two solutions, namely, x = 4 and x = 9. The gcd(4, 10) = 2 and 2 does not divide 5, but does divide 6. Since gcd(3, 10) = 1, the linear congruence 3x ≡ 1 (mod 10) will have solutions, that is, modular multiplicative inverses of 3 modulo 10 will exist. In fact, 7 satisfies this congruence (i ...
This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or U(Z n). As a consequence of Lagrange's theorem, the order of a (mod n) always divides φ(n). If the order of a is actually equal to φ(n), and therefore as large as possible, then a is called a primitive root modulo n.
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 ...