About: Modular multiplicative inverse

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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. In the standard notation of modular arithmetic this congruence is written as where the symbol denotes the multiplication of equivalence classes modulo m.Written in this way, the analogy with the usual concept of a multiplicative inverse in the set of rational or real numbers is clearly represented, replacing the numbers by congruence classes and altering the binary operation appropriately.

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rdfs:label	Modular multiplicative inverse (en)
rdfs:comment	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. In the standard notation of modular arithmetic this congruence is written as where the symbol denotes the multiplication of equivalence classes modulo m.Written in this way, the analogy with the usual concept of a multiplicative inverse in the set of rational or real numbers is clearly represented, replacing the numbers by congruence classes and altering the binary operation appropriately. (en)
sameAs	Modular multiplicative inverse Modular multiplicative inverse
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_book dbt:Main dbt:Math dbt:MathWorld dbt:Mvar dbt:Reflist dbt:Wikibooks dbt:Abs dbt:Su
Subject	Binary operations Modular arithmetic
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Modular_multiplicative_inverse?oldid=1064723123&ns=0
Wikipage page ID	9815338 (xsd:integer)
page length (characters) of wiki page	23876 (xsd:nonNegativeInteger)
Wikipage revision ID	1064723123 (xsd:integer)
Link from a Wikipage to another Wikipage	Greatest common divisor Group (mathematics) Unit (ring theory) Well-defined expression Abuse of notation Arithmetic Big O notation Binary operation Binary relation Montgomery modular multiplication Bézout's identity Binary operations Modular arithmetic Chinese remainder theorem Cryptography Curve25519 Cyclic group Prefix sum Prime number Public-key cryptography Congruence relation Coprime integers Order (group theory) Parallel computing Equivalence relation Finite field If and only if Exponentiation by squaring Extended Euclidean algorithm Factorization Integer Inversive congruential generator Modular arithmetic Modular exponentiation Euclidean algorithm Euler's theorem Euler's totient function Isomorphism Kloosterman sum Long division Least common multiple Mathematics Inverse second RSA (cryptosystem) Rational number Rational reconstruction (mathematics) Real number Reduced residue system Ring (mathematics) Side-channel attack
has abstract	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. In the standard notation of modular arithmetic this congruence is written as which is the shorthand way of writing the statement that m divides (evenly) the quantity ax − 1, or, put another way, the remainder after dividing ax by the integer m is 1. If a does have an inverse modulo m, then there are an infinite number of solutions of this congruence, which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to a (i.e., in a's congruence class) has any element of x's congruence class as a modular multiplicative inverse. Using the notation of to indicate the congruence class containing w, this can be expressed by saying that the modulo multiplicative inverse of the congruence class is the congruence class such that: where the symbol denotes the multiplication of equivalence classes modulo m.Written in this way, the analogy with the usual concept of a multiplicative inverse in the set of rational or real numbers is clearly represented, replacing the numbers by congruence classes and altering the binary operation appropriately. As with the analogous operation on the real numbers, a fundamental use of this operation is in solving, when possible, linear congruences of the form Finding modular multiplicative inverses also has practical applications in the field of cryptography, i.e. public-key cryptography and the RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm) that can be used for the calculation of modular multiplicative inverses. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Modular_multiplicative_inverse
is Wikipage redirect of	Multiplicative modular inverse Modular inverse Modular reciprocal Discrete inverse
is Link from a Wikipage to another Wikipage of	Affine cipher Montgomery modular multiplication James A. Morrow Division (mathematics) Hill cipher Lenstra elliptic-curve factorization Merkle–Hellman knapsack cryptosystem Method of successive substitution Coprime integers ElGamal encryption Finite field arithmetic Kaprekar number Extended Euclidean algorithm Fermat's little theorem Inversive congruential generator Modular arithmetic Modular exponentiation Modulo operation Euclidean algorithm Euclidean division Multiplicative group of integers modulo n Inverse second Multiplicative modular inverse Modular inverse

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