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Modular multiplicative inverse. In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. [ 1] In the standard notation of modular arithmetic this congruence is written as.
Luhn algorithm. The Luhn algorithm or Luhn formula, also known as the " modulus 10" or "mod 10" algorithm, named after its creator, IBM scientist Hans Peter Luhn, is a simple check digit formula used to validate a variety of identification numbers. It is described in US patent 2950048A, granted on 23 August 1960. [ 1]
Luhn mod N algorithm. The Luhn mod N algorithm is an extension to the Luhn algorithm (also known as mod 10 algorithm) that allows it to work with sequences of values in any even-numbered base. This can be useful when a check digit is required to validate an identification string composed of letters, a combination of letters and digits or any ...
Using the extended Euclidean algorithm, compute −5 ⋅ 10 + 3 ⋅ 17 = 1, so N′ will be −3 mod 10 = 7. The Montgomery forms of 7 and 15 are 70 mod 17 = 2 and 150 mod 17 = 14 , respectively. Their product 28 is the input T to REDC, and since 28 < RN = 170 , the assumptions of REDC are satisfied.
The MixColumns operation performed by the Rijndael cipher or Advanced Encryption Standard is, along with the ShiftRows step, its primary source of diffusion . Each column of bytes is treated as a four-term polynomial , each byte representing an element in the Galois field . The coefficients are elements within the prime sub-field .
Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = be mod m. From the definition of division, it follows that 0 ≤ c < m . For example, given b = 5, e = 3 and m = 13, dividing 53 = 125 by 13 leaves a remainder of c = 8 .
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In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n .