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Ring Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
which has a similar structure to that of ℤ[x]/〈xn−1〉. This observation provides an alternate construction of the former ring, which can be seen through the concept of a ring homomorphism. A ring homomorphism is a map between rings that respects both the additive structure and multiplicative structure of these rings.
Algebraic Structures and Applications
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
An isomorphism is a one-to-one, onto ring homomorphism. If f : R1 →R2 is an isomorphism, then we saythat R1 is isomorphic to R2 via f. In general, we write R1 ≅ R2 and read R1 is isomorphic to R2 if there exists an isomorphism from R1 to R2.
Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A ring homomorphism from ring R1 to ring R2 is a function ϕ : R1 → R2 such that ϕ(a + b) = ϕ(a) + ϕ(b) and ϕ(ab) = ϕ(a)ϕ(b) for all a, b ∊ R1.
A Framework for Filtering Step of Number Field Sieve and Function Field Sieve
Published in IETE Journal of Research, 2023
Rahul Janga, R. Padmavathy, S. K. Pal, S. Ravichandra
This phase is to collect relations popularly called as sieving phase. Based on the factor base constructed earlier this phase searches for smooth elements. Figure 1 represents the ring homomorphism. The first step in this phase is to fix the smoothness bound for both the sides, next is to generate relations in terms of (a,b) pairs, which are double smooth, i.e. smooth on both the sides together. Let us denote as the prime factors of the algebraic side and as the prime factors of the rational side. The roots for each prime at the algebraic side can be calculated as . At the rational side, we get unique roots. Relations are normally represented as where is the doubly smooth relation, which is factored into factors in the algebraic side and factored into factors in the rational side.
Linear Algebra on Parallel Structures Using Wiedemann Algorithm to Solve Discrete Logarithm Problem
Published in IETE Journal of Research, 2022
K S Spoorthi, R. Padmavathy, S K Pal, S Ravi Chandra
This phase is to collect relations popularly called as sieving phase. Based on the factor base constructed earlier this phase searches for smooth elements. Figure 1 represents ring homomorphism. The first step in this phase is to fix the smoothness bound for both the sides, next is to generate relations in terms of (a,b) pairs, which are double smooth, i.e. smooth on both the sides together. is denoted as the prime factors of the algebraic side and as the prime factors of rational side. The roots for each prime at algebraic side can be calculated as . The rational side has unique roots. Relations are normally represented as :
Fully homomorphic encryption: a general framework and implementations
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
As proved in Section 2.2 above, E is a ring homomorphism. Therefore if and are to be added or multiplied, then their images and are operated upon. The user computes or remotely. The same operations can be performed by anyone acting on behalf of the user. Note here that the mod m operation is not performed on the result of the sum or product. Both and are stored on the cloud for later use.