Fraktur

2015-01-03T13:13:45Z

2.10.56.45: /* Use */

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Landau's problems

2014-02-07T02:50:01Z

2.10.255.22: /* Legendre's conjecture */

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Normed algebra

2014-01-17T16:01:31Z

2.10.252.80:

In [[number theory]], the '''Pohlig–Hellman algorithm''' sometimes credited as the '''Silver–Pohlig–Hellman algorithm'''<ref name="Mollin06p344">[[#Mollin06|Mollin 2006]], pg. 344</ref> is a special-purpose [[algorithm]] for computing [[discrete logarithm]]s in a [[multiplicative group]] whose order is a [[smooth integer]].

The algorithm was discovered by Roland Silver, but first published by [[Stephen Pohlig]] and [[Martin Hellman]] (independent of Silver).

We will explain the algorithm as it applies to the group '''Z'''*''p'' consisting of all the elements of '''Z'''''p'' which are [[coprime]] to ''p'', and leave it to the advanced reader to extend the algorithm to other groups by using [[Lagrange's theorem (group theory)|Lagrange's theorem]].

:'''Input''' Integers ''p'', ''g'', ''e''.
:'''Output''' An Integer ''x'', such that ''e'' ≡ ''g''''x'' (mod ''p'') (if one exists).

:#Determine the prime factorization of the order of the group : <center><math>\varphi(p)= p_1\cdot p_2 \cdots p_n</math></center> (All the ''p''''i'' are considered small since the group order is smooth.)
:#From the [[Chinese remainder theorem]] it will be sufficient to determine the values of ''x'' modulo each prime power dividing the group order. Suppose for illustration that ''p''1 divides this order but ''p''12 does not. Then we need to determine ''x'' mod ''p''1, that is, we need to know the ending coefficient ''b''1 in the base-''p1'' expansion of ''x'', i.e. in the expansion ''x'' = ''a''1 ''p''1 + ''b''1. We can find the value of ''b1'' by examining all the possible values between 0 and ''p''1-1. (We may also use a faster algorithm such as [[baby-step giant-step]] when the order of the group is prime.<ref name="Menezes97p109">[[#Menezes97|Menezes, et. al 1997]], pg. 109</ref>) The key behind the examination is that: <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} \\
& \equiv (g^{\varphi(p)/p_1})^{b_1} \pmod{p}
\end{align}
</math></center> (using [[Euler's theorem]]). With everything else now known, we may try each value of ''b''1 to see which makes the equation be true; precisely one will work, and that ''b''1 is the value of ''x'' modulo ''p''1. (An exception arises if <math>g^{\varphi(p)/p_1} \equiv 1 \pmod{p}</math> since then the order of ''g'' is less than φ(''p''). The conclusion in this case depends on the value of <math>e^{\varphi(p)/p_1} \mod p</math> on the left: if this quantity is not 1, then no solution ''x'' exists; if instead this quantity is also equal to 1, there will be more than one solution for ''x'' less than φ(''p''), but since we are attempting to return only one solution ''x'', we may use ''b''1=0.)
:#The same operation is now performed for ''p''2 through ''pn''. A minor modification is needed where a prime number is repeated. Suppose we are seeing ''pi'' for the (''k'' + 1)st time. Then we already know ''ci'' in the equation ''x'' = ''a''''i'' ''p''''i''''k''+1 + ''b''''i'' ''p''''i''''k'' + ''c''''i'', and we find ''b''''i'' the same way as before.
:# With all the ''b''''i'' known, we have enough simultaneous [[congruence relation|congruence]]s to determine ''x'' using the [[Chinese remainder theorem]].

==Complexity==
The worst-case time complexity of the Pohlig–Hellman algorithm is <math>O(\sqrt n)</math> for a group of order ''n'', but it is more efficient if the order is smooth. Specifically, if <math>\prod_i p_i^{e_i}</math> is the prime factorization of ''n'', then the complexity can be stated as
<math>O\left(\sum_i {e_i(\log n+\sqrt p_i)}\right)</math>.<ref name="Menezes97p108">[[#Menezes97|Menezes, et. al 1997]], pg. 108</ref>

==Notes==
{{Reflist}}

==References==
*{{cite book|title=An Introduction To Cryptography|last=Mollin|first= Richard|date=2006-09-18|publisher=Chapman and Hall/CRC|edition=2nd|isbn=978-1-58488-618-1|page=344|ref=Mollin06}}
*{{cite journal | authors=S. Pohlig and [[Martin Hellman|M. Hellman]] | title=An Improved Algorithm for Computing Logarithms over GF(p) and its Cryptographic Significance | journal=[[IEEE]] Transactions on Information Theory | issue=24 | year=1978 | pages=106–110 | url=http://www-ee.stanford.edu/~hellman/publications/28.pdf}}
*{{cite book|first1=Alfred J.|last1=Menezes|authorlink1=Alfred Menezes|first2=Paul C.|last2=van Oorschot|authorlink2=Paul van Oorschot|first3=Scott A.|last3=Vanstone|authorlink3=Scott Vanstone|title=Handbook of Applied Cryptography|url=http://www.cacr.math.uwaterloo.ca/hac/|publisher=[[CRC Press]]|year=1997|pages=107–109|chapter=Number-Theoretic Reference Problems|chapterurl=http://www.cacr.math.uwaterloo.ca/hac/about/chap3.pdf|isbn=0-8493-8523-7|ref=Menezes97}}

{{Number-theoretic algorithms}}

{{DEFAULTSORT:Pohlig-Hellman algorithm}}
[[Category:Number theoretic algorithms]]

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