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CRGreathouse (talk | contribs) m Disambiguated: remainder theorem → Chinese remainder theorem using Dab solver |
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The algorithm was discovered by Roland Silver, but first published by [[Stephen Pohlig]] and [[Martin Hellman]] (independent of Silver).
We will explain the algorithm in terms of the group formed by taking all the elements of '''Z'''<sub>''p''</sub> which are [[coprime]] to ''p'', and leave it to the advanced reader to extend the algorithm to other groups by using [[Lagrange'
:'''Input''' Integers ''p'', ''g'', ''e''.
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:#Determine the prime factorization of the order of the group (the [[totient]] in this case) : <br><center><math>\varphi(p)= p_1\cdot p_2 \cdots p_n</math></center>
:#From the [[Chinese remainder theorem]] we know that ''x'' = ''a''<sub>1</sub> ''p''<sub>1</sub> + ''b''<sub>1</sub>. We now find the value of ''b<sub>1</sub>'' for which the following equation holds using a fast algorithm such as [[baby-step giant-step]] :<br> <center><math>
\begin{align}e^{\varphi(p)/p_1} & \equiv (g^x)^{\varphi(p)/p_1} \pmod{p} \\
& \equiv (g^{\varphi(p)})^{a_1}g^{b_1\varphi(p)/p_1} \pmod{p} \\
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