Answered: Prove that logg(h1h2) = logg

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Prove that log_g(h1h2) = log_g(h1) + log_g(h2) for all h1, h2 ∈ F∗p

Expert Solution

Step 1: Step 1:

To prove that

$log g left parenthesis h 1 h 2 right parenthesis space equals space log g left parenthesis h 1 right parenthesis space plus space log g left parenthesis h 2 right parenthesis$

for all $h 1 comma space h 2 space element of space F cross times p$ (where $F cross times p$ is the multiplicative group of non-zero elements in the finite field Fp), we can use the properties of logarithms in modular arithmetic.

First, let's define some terms:

g is a generator of the multiplicative group $F cross times p$ , meaning that for every element h in $F cross times p$ , there exists an integer x such that $g to the power of x identical to h left parenthesis m o d p right parenthesis$ .
logg(h) represents the discrete logarithm of h with respect to the base g in $F cross times p$ , which is the smallest non-negative integer x such that $g to the power of x identical to h left parenthesis m o d p right parenthesis$ .