"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."
The inverse of a number modulo another number.
Divisibility and Modular Arithmetic: Understanding the basic concepts of divisibility and modular arithmetic is crucial when learning about modular inverses. This includes the concepts of remainder, congruence, and modular arithmetic operations such as addition, subtraction, and multiplication.
Euclidean Algorithm: The Euclidean algorithm is an efficient method to find the greatest common divisor (GCD) of two numbers. This algorithm plays a key role in finding modular inverses, as the GCD can be used to determine if an inverse exists.
Extended Euclidean Algorithm: The extended Euclidean algorithm is a variation of the Euclidean algorithm that not only finds the GCD, but also the specific values of x and y that satisfy the equation ax + by = GCD(a, b). This is used to find modular inverses in a modular arithmetic system.
Euler's Totient Function: The Euler's totient function is a number-theoretic function that counts the number of positive integers less than or equal to n that are coprime to n. This function is useful in finding modular inverses in a prime modulus system.
Fermat's Little Theorem: Fermat's little theorem is a number-theoretic theorem that can be used to simplify certain exponentiation calculations in a modular arithmetic system. This theorem is also useful in finding modular inverses when the modulus is a prime number.
Chinese Remainder Theorem: The Chinese remainder theorem is a theorem that states that if the remainders of a division operation with a set of distinct moduli are given, then the original number can be uniquely determined. This theorem is useful to solve problems that involve finding a modular inverse in a system with different moduli.
Modular Exponentiation: Modular exponentiation is the process of computing the remainder when an integer is raised to a power in a modular arithmetic system. This computation involves finding the value of the modulo operation at each step of the exponentiation process.
Primitive Roots and Discrete Logarithms: Primitive roots and discrete logarithms are advanced topics in number theory that are used to solve certain problems related to modular arithmetic, including finding modular inverses. A primitive root is a number that generates the entire set of residues modulo a prime number, while discrete logarithms are used to find the power to which a primitive root must be raised to obtain a given residue.
Fermat's Little Theorem: This theorem states that if p is a prime number and a is any positive integer not divisible by p, then a raised to the power of p-1 is congruent to 1 modulo p. In other words, a has a modular inverse modulo p and it is given by a^(p-2) modulo p.
Extended Euclidean Algorithm: This algorithm is used to find the greatest common divisor (GCD) of two integers and their corresponding Bezout coefficients, which can then be used to find the modular inverse of one of the integers. Specifically, if a and m are integers with gcd(a,m)=1, then there exist integers x and y such that ax+my=1. Solving this equation for x gives the modular inverse of a mod m, which is x mod m.
Chinese Remainder Theorem: This theorem provides a way to solve a system of congruences of the form x congruent to a_i modulo m_i, where m_i are pairwise relatively prime. If we let M equal the product of all m_i, then we can solve each congruence modulo m_i and then combine the solutions using the Chinese Remainder Theorem. If a solution exists, then it is unique modulo M and is the modular inverse of a modulo M.
Inverse by Inspection: Sometimes it is possible to find the modular inverse of a by inspection, especially if the modulus is small. For example, in mod 10, the inverse of 3 is 7 since 3*7=21 is congruent to 1 mod 10.
Modular exponentiation: Given a and m, we can find a^(-1) mod m by calculating a^(phi(m)-1) mod m, where phi(m) is Euler's totient function that counts the positive integers less than or equal to m that are relatively prime to m. This method works if a and m are relatively prime.
Modular arithmetic properties: In some cases, the modular inverse can be found using basic modular arithmetic properties. For example, if a and m are odd integers, then a has a modular inverse mod m if and only if m has a modular inverse mod a.
"The congruence is written as: a * x ≡ 1 (mod m)."
"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."
"Any integer that is congruent to a has any element of x's congruence class as a modular multiplicative inverse."
"The modulo multiplicative inverse of the congruence class [a] is the congruence class [x] such that: [a] * [x] = [1]."
"The symbol ⋅m denotes the multiplication of equivalence classes modulo m."
"A fundamental use of this operation is in solving linear congruences of the form: ax ≡ b (mod m)."
"Finding modular multiplicative inverses also has practical applications in the field of cryptography, e.g., public-key cryptography and the RSA algorithm."
"The RSA algorithm."
"There exists a very fast algorithm (the extended Euclidean algorithm) that can be used for the calculation of modular multiplicative inverses."