# Modular Inverse Calculator (Extended Euclidean Algorithm) a⁻¹ mod m via extended Euclid with Bézout coefficients, an optional step table, and an a × a⁻¹ ≡ 1 (mod m) verification; RSA-sized inputs supported. > Canonical page: https://elysiatools.com/en/tools/modular-inverse-calculator - **Category:** Math & Numbers - **Keywords:** modular inverse, extended euclidean, bezout identity, bezout coefficients, rsa, diophantine, modular arithmetic, number theory, multiplicative inverse ## Overview The Modular Inverse Calculator computes the multiplicative inverse a⁻¹ mod m using the Extended Euclidean Algorithm, supporting standard integers as well as large RSA-scale values up to 10⁵¹². It automatically derives Bézout coefficients, confirms whether gcd(a, m) equals 1, validates the identity a × a⁻¹ ≡ 1 (mod m), and optionally displays the full step-by-step division and reduction table. ## Inputs - **Value a** (text): The value to invert; negative values are reduced mod m first. - **Modulus m** (text): Modulus, m ≥ 2. Inputs up to 10⁵¹² are accepted (RSA-sized). - **Output style** (select) ## When to use - Generating or verifying private key exponents in RSA key generation routines. - Solving linear congruences, Diophantine equations, or finding decryption keys in classical affine ciphers. - Debugging number theory assignments and modular arithmetic proofs that require step-by-step Bézout identity tracking. ## How it works - Enter the value to invert (a) and the modulus (m, where m ≥ 2); negative values for a are automatically reduced modulo m. - Select your preferred output style: a concise result with verification or a full Extended Euclidean step-by-step table. - The algorithm calculates gcd(a, m) and computes Bézout coefficients s and t such that a·s + m·t = gcd(a, m). - If gcd(a, m) = 1, the calculator outputs the normalized modular inverse along with the identity verification a × a⁻¹ ≡ 1 (mod m); otherwise, it reports that no inverse exists. ## Use cases - Cryptographic key pair calculation where the private exponent d satisfies d ≡ e⁻¹ (mod φ(n)). - Finding decryption multipliers for classical modular ciphers such as the affine cipher. - Academic problem-solving in discrete mathematics and computational number theory. ## Frequently asked questions ### When does a modular inverse exist? A modular inverse a⁻¹ mod m exists if and only if a and m are coprime, meaning gcd(a, m) = 1. ### How are negative values for a handled? Negative values of a are automatically reduced modulo m to their positive canonical equivalent before computing the inverse. ### What is the maximum supported number size? The calculator supports large integers up to RSA dimensions (10⁵¹²). ### What are Bézout coefficients in the output? They are the integers s and t satisfying Bézout's identity, a·s + m·t = gcd(a, m), where s reduced modulo m yields the modular inverse. ### What does the verification step check? 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