Lookup Structures for Multiplicative Inverses Modulo 2 k
نویسندگان
چکیده
We introduce an inheritance property and related table lookup structures applicable to simplified evaluation of the modular operations “multiplicative inverse”, “discrete log”, and “exponential residue” in the particular modulus 2. Regarding applications, we describe an integer representation system of Benschop for transforming integer multiplications into additions which benefits from our table lookup function evaluation procedures. We focus herein on the multiplicative inverse modulo 2 to exhibit simplifications in hardware implementations realized from the inheritance property. A table lookup structure given by a bit string that can be interpreted with reference to a binary tree is described and analyzed. Using observed symmetries, the lookup structure size is reduced allowing a novel direct lookup process for multiplicative inverses for all 16-bit odd integers to be obtained from a table of size less than two KBytes. The 16-bit multiplicative inverse operation is also applicable for providing a seed inverse for obtaining 32/64-bit multiplicative inverses by one/two iterations of a known quadratic refinement algorithm.
منابع مشابه
Modulo Operation Free Reverse Conversion in the {2^(2n+1)-1,2^n,2^2n-1} Moduli Set
This paper proposes a fast Mixed Radix Conversion based reverse converter for the recently proposed moduli set ?{2?^(2n+1)-1,2^n,2^2n-1}. It shows that the computation of multiplicative inverses could be eliminated from the conversion process and presents a low complexity and modulo operation free implementation. Theoretical analysis shows that the proposed scheme outperforms all state of the a...
متن کاملOn Inversion in Z_{2^n-1}
In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z2n−1, and we described the binary weights of the inverses of the Gold and Kasami exponents. We studied the function Invd (n), which for a fixed positive integer d maps integers n ≥ 1 to the least positive residue of the inverse of d modulo 2n − 1, if it exists. In particular, we showe...
متن کاملOn Newton-Raphson iteration for multiplicative inverses modulo prime powers
We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over p-adic numbers gives a recurrence relation computing modular inverse modulo p, that is logarithmic in m. We solve the recurrence to obtain an explicit formula for the inverse. Then we study different implementation variants of this iteration and show that our explicit formula is interesting for small...
متن کاملEnhanced Euclid Algorithm for Modular Multiplicative Inverse and Its Application in Cryptographic Protocols
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 doe...
متن کاملThree Novel Theorems for Applied Cryptography
With advancements in computing capabilities public key cryptosystems are going to be more complex yet vulnerable over the modern day‟s computer networks and associated security mechanism, especially those based on novel approaches of applied mathematics. This paper explores three novel theorems derived while studying and implementing RSA algorithm, one of the strongest public key cryptosystem. ...
متن کامل