Table of costs of operations in elliptic curves: Difference between revisions

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== Intuitive Proof ==
The algorithm is based on a [[Fermat's little theorem]]
: <math> a^{(p-1)} \equiv 1 \pmod{p}</math>
p prime, a & p coprime.
 
So, if we choose ''d'' such that
: <math>e d \equiv 1 \pmod{p-1}</math>
i.e.
: <math>e d - 1 = k(p-1)</math>
Then for a message m
: <math>m^{e^d} \equiv m^{e d} \equiv m^{(e d - 1)} \cdot m^1 \equiv m^{k(p-1)}m \equiv 1^km \equiv m \pmod{p}</math>
 
So a message that is encrypted by raising ''m'' to ''e'' can be decrypted by raising the result to ''d''.
 
In order to provide security, RSA actually calculates
: <math> n = p q</math>
(''p'', ''q'' prime) and then ''d'' such that
: <math>e d \equiv 1\pmod{(p-1)(q-1)}</math>
so
: <math>e d - 1 = k(p-1)(q-1)</math>
 
We can then continue to calculate
: <math>m^{e^d} \equiv m^{e d} \equiv m^{(e d - 1)}m \equiv m^{k(p-1)(q-1)}m \equiv 1^{k(q-1)}m\equiv m \pmod{p}</math>
 
And likewise for q
: <math>m^{e^d} \equiv m^{e d} \equiv m^{(e d - 1)}m \equiv m^{k(p-1)(q-1)}m \equiv 1^{k(p-1)}m\equiv m \pmod{q}</math>
 
Now if <math>a\equiv b \pmod{p}</math> and <math>a\equiv b \pmod{q}</math> then <math>a\equiv b \pmod{pq}</math>, ''p'', ''q'' coprime.
 
so
: <math>m^{e^d} \equiv m \pmod{pq}</math>
 
An attacker would need to factor ''n'' into ''p'' and ''q'' in order to determine ''d'', and this is a hard problem.
 
Note that while an attacker could easily calculate ''f'' as
: <math>e f \equiv 1 \pmod{n-1}</math>
that
: <math>m^{e^f} \equiv m^{k(n-1)}m \neq m \pmod{n}</math>
 
because ''n'' is not prime.
 
{{DEFAULTSORT:RSA Intuitive}}
[[Category:Public-key cryptography]]

Latest revision as of 13:55, 2 December 2014

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