Enhanced Euclid Algorithm for Modular Multiplicative Inverse and Its Application in Cryptographic Protocols

Abstract

Numerous cryptographic algorithms (ElGamal, Rabin, RSA, NTRU etc) require multiple computations of modulo multiplicative inverses. This paper describes and validates a new algorithm, called the Enhanced Euclid Algorithm, for modular multiplicative inverse (MMI). Analysis of the proposed algorithm shows that it is more efficient than the Extended Euclid algorithm (XEA). In addition, if a MMI does not exist, then it is not necessary to use the Backtracking procedure in the proposed algorithm; this case requires fewer operations on every step (divisions, multiplications, additions, assignments and push operations on stack), than the XEA. Overall, XEA uses more multiplications, additions, assignments and twice as many variables than the proposed algorithm.

A general method for to decompose modular multiplicative inverse operators over Group of units

“…For example, Wei in their original paper [33] has introduced the so-called "algorithm of sequential modular multiplication", based, inter alia, on residue signed-digit(SD). In the same context, other algorithms were obtained notably by Verkhovsky's [30], AL-Matari et.al. [1], and Hars [15], who, have been studied a modular inverse algorithm without multiplications for cryptographic applications.…”

Cubic Root Extractors of Gaussian Integers and Their Application in Fast Encryption for Time-Constrained Secure Communication

“…Therefore in the RSA extension it would have been necessary to compute a multiplicative inverse d of e modulo c. Yet, in the algorithm described above the encryption key e = 3. Hence, the decryption key d cannot be computed as a modular multiplicative inverse, since   gcd 3, 3 z  , which implies that such an inverse does not exist [14].…”

A Note on the Computation of the Modular Inverse for Cryptography

“…As mentioned, cryptographic algorithms rely on multiple computations of modulo multiplicative inverses. Examples are the RSA cryptographic algorithm by [8,9], RSA with digital signature [10], ElGamal cryptocol [11]; encryption and decryption schemes based on extraction of square roots [12], NTRU cryptosystem [13], modular multiplicative inverse (MMI) for cryptanalysis of public-key cryptographic protocols [14], etc. Recently, Boolean functions have gained attraction because of some interesting properties from a cryptographic point of view such as "nonlinearity, propagation criterion, resiliency, and balance" [15].…”