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1 %%%% why-RSA-works/conclusions.tex
2 %%%% Copyright 2012 Peter Franusic.
3 %%%% All rights reserved.
4 %%%%
5
6 So why does RSA work?
7 Why is it that we can take some message $m$,
8 run it through two modex operations, and come out with the same $m$?
9 There are several reasons.
10 First of all, RSA computations are done in a commutative ring
11 and the multiplicative association property holds in commutative rings.
12 This property tells us that
13 the two exponentiations $(m^e)^d$ are the same as the one exponentiation $m^{ed}$.
14
15 A second reason is that exponents $e$ and $d$ are chosen
16 such that they satisfy the multiples-plus-one condition $ed = k\lambda + 1$.
17 This insures that $ed$ is one of the identity columns
18 in the exponential table of ring $\mathcal{R}_n$.
19
20 A third reason is that the exponential table contains
21 repeating blocks of columns where $m^a=m^{k\lambda+a}$.
22 This is the wallpaper pattern that we saw in Table \ref{modex-33}.
23 This pattern is the reason for the multiples-plus-one condition.
24
25 Finally, RSA works because it relies on the intractability of the factoring problem.
26 A huge RSA modulus $n$ cannot be factored expeditiously.
27 Given that $n$ is the product of two distinct huge random primes,
28 it is virtually impossible to factor $n$ in any reasonable amount of time,
29 even if the factoring effort is distributed across thousands of computers.
30
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