Pohlig–Hellman algorithm: Difference between revisions

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{{Short description|Algorithm for computing logarithms}}

In mathematics, the '''Pohlig-Hellman algorithm''' is an [[algorithm]] for the computation of [[discrete logarithm]]s in a [[multiplicative group]] whose order is a [[smooth integer]]. The algorithm is based on the [[Chinese remainder theorem]] and runs in [[polynomial time]].

[[File:Pohlig-Hellman-Diagram.svg|thumb|350px|alt=Pohlig Hellman Algorithm|Steps of the Pohlig–Hellman algorithm.]]

In [[group 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 [[finite abelian group]] whose order is a [[smooth integer]].

The algorithm was introduced by Roland Silver, but first published by [[Stephen Pohlig]] and [[Martin Hellman]], who credit Silver with its earlier independent but unpublished discovery. Pohlig and Hellman also list Richard Schroeppel and H. Block as having found the same algorithm, later than Silver, but again without publishing it.{{sfn|Pohlig|Hellman|1978}}

We will explain the algorithm in terms of the group formed by taking 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.

== Groups of prime-power order ==

:'''Input''' Integers ''p'', ''g'', ''e''.

As an important special case, which is used as a subroutine in the general algorithm (see below), the Pohlig–Hellman algorithm applies to [[Group (mathematics)|groups]] whose order is a [[prime power]]. The basic idea of this algorithm is to iteratively compute the <math>p</math>-adic digits of the logarithm by repeatedly "shifting out" all but one unknown digit in the exponent, and computing that digit by elementary methods.

:'''Output''' Integer ''x'', such that ''e ≡ g^x (mod p)''.

(Note that for readability, the algorithm is stated for cyclic groups — in general, <math>G</math> must be replaced by the subgroup <math>\langle g\rangle</math> generated by <math>g</math>, which is always cyclic.)

:#Use [[Euler's totient function]] to determine the prime factorization of the order of the group : <br><center><math>\varphi(p)= p_0\cdot p_1 \cdot \ldots \cdot p_n</math></center>

:#From the remainder theorem we know that ''x = a₁ p₁ + b₁''. We now find the value of ''b₁'' for which the following equation holds using a fast algorithm such as [[Baby-step giant-step]] :<br> <center><math>

\begin{matrix}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{matrix}

</math> (using [[Euler's theorem]])</center><br>Note that if <math>g^{\varphi(p)/p_1} \equiv 1 \pmod{p}</math> then the order of ''g'' is less than φ(''p'') and <math>e^{\varphi(p)/p_1} \mod p</math> must be 1 for a solution to exist. In this case there will be more than one solution for ''x'' less than φ(''p''), but since we are not looking for the whole set, we can require that ''b''₁=0.<br><br>The same operation is now performed for ''p₂'' up to ''p_n''.<br><br>A minor modification is needed where a prime number is repeated. Suppose we are seeing ''p_i'' for the ''k+1''-th time. Then we already know ''c_i'' 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.

:#We end up with enough simultaneous [[congruence|congruences]] so that ''x'' can be solved using the [[Chinese remainder theorem]].

:'''Input.''' A cyclic group <math>G</math> of order <math>n=p^e</math> with generator <math>g</math> and an element <math>h\in G</math>.

{{Numtheory-stub}}

:'''Output.''' The unique integer <math>x\in\{0,\dots,n-1\}</math> such that <math>g^x=h</math>.

[[Category:Algorithms]]

:# Initialize <math>x_0:=0.</math>

[[Category:Number theory]]

:# Compute <math>\gamma:=g^{p^{e-1}}</math>. By [[Lagrange's theorem (group theory)|Lagrange's theorem]], this element has order <math>p</math>.

[[Category:Articles that use pidgincode]]

:# For all <math>k\in\{0,\dots,e-1\}</math>, do:

[[de: Pohlig-Hellman-Algorithmus]]

:## Compute <math>h_k:=(g^{-x_k}h)^{p^{e-1-k}}</math>. By construction, the order of this element must divide <math>p</math>, hence <math>h_k\in\langle\gamma\rangle</math>.

[[pl: Redukcja Pohliga-Hellmana]]

:## Using the [[Baby-step giant-step|baby-step giant-step algorithm]], compute <math>d_k\in\{0,\dots,p-1\}</math> such that <math>\gamma^{d_k}=h_k</math>. It takes time [[Big O notation|<math>O(\sqrt p)</math>]].

:## Set <math>x_{k+1}:=x_k+p^kd_k</math>.

:# Return <math>x_e</math>.

The algorithm computes discrete logarithms in time complexity [[Big O notation|<math>O(e\sqrt p)</math>]], far better than the [[Baby-step giant-step|baby-step giant-step algorithm's]] [[Big O notation|<math>O(\sqrt{p^e})</math>]] when <math>e</math> is large.

== The general algorithm ==

In this section, we present the general case of the Pohlig–Hellman algorithm. The core ingredients are the algorithm from the previous section (to compute a logarithm modulo each prime power in the group order) and the [[Chinese remainder theorem]] (to combine these to a logarithm in the full group).

(Again, we assume the group to be cyclic, with the understanding that a non-cyclic group must be replaced by the subgroup generated by the logarithm's base element.)

:'''Input.''' A cyclic group <math>G</math> of order <math>n</math> with generator <math>g</math>, an element <math>h\in G</math>, and a prime factorization <math display="inline">n=\prod_{i=1}^rp_i^{e_i}</math>.

:'''Output.''' The unique integer <math>x\in\{0,\dots,n-1\}</math> such that <math>g^x=h</math>.

:# For each <math>i\in\{1,\dots,r\}</math>, do:

:## Compute <math>g_i:=g^{n/p_i^{e_i}}</math>. By [[Lagrange's theorem (group theory)|Lagrange's theorem]], this element has order <math>p_i^{e_i}</math>.

:## Compute <math>h_i:=h^{n/p_i^{e_i}}</math>. By construction, <math>h_i\in\langle g_i\rangle</math>.

:## Using the algorithm above in the group <math>\langle g_i\rangle</math>, compute <math>x_i\in\{0,\dots,p_i^{e_i}-1\}</math> such that <math>g_i^{x_i}=h_i</math>.

:# Solve the simultaneous congruence <math display="block">x\equiv x_i\pmod{p_i^{e_i}}

\quad\forall i\in\{1,\dots,r\}

\text{.}</math>The [[Chinese remainder theorem]] guarantees there exists a unique solution <math>x\in\{0,\dots,n-1\}</math>.

:# Return <math>x</math>.

The correctness of this algorithm can be verified via the [[Abelian group#Classification|classification of finite abelian groups]]: Raising <math>g</math> and <math>h</math> to the power of <math>n/p_i^{e_i}</math> can be understood as the projection to the factor group of order <math>p_i^{e_i}</math>.

==Complexity==

The worst-case input for the Pohlig–Hellman algorithm is a group of prime order: In that case, it degrades to the [[Baby-step giant-step|baby-step giant-step algorithm]], hence the worst-case time complexity is <math>\mathcal O(\sqrt n)</math>. However, it is much more efficient if the order is smooth: Specifically, if <math>\prod_i p_i^{e_i}</math> is the prime factorization of <math>n</math>, then the algorithm's complexity is <math display="block">\mathcal O\left(\sum_i {e_i(\log n+\sqrt {p_i})}\right)</math> group operations.<ref name="Menezes97p108">[[#Menezes97|Menezes, et al. 1997]], pg. 108</ref>

==Notes==

{{Reflist}}

==References==

*{{cite book|title=An Introduction To Cryptography|url=https://archive.org/details/An_Introduction_to_Cryptography_Second_Edition|last=Mollin|first= Richard|date=2006-09-18|publisher=Chapman and Hall/CRC|edition=2nd|isbn=978-1-58488-618-1|page=[https://archive.org/details/An_Introduction_to_Cryptography_Second_Edition/page/n353 344]|ref=Mollin06}}

*{{cite journal |first1=S.|last1=Pohlig |author2-link=Martin Hellman|first2=M. |last2=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 | doi=10.1109/TIT.1978.1055817 | url=http://www-ee.stanford.edu/~hellman/publications/28.pdf}}

*{{cite book|first1=Alfred J.|last1=Menezes|author-link1=Alfred Menezes|first2=Paul C.|last2=van Oorschot|author-link2=Paul van Oorschot|first3=Scott A.|last3=Vanstone|author-link3=Scott Vanstone|title=Handbook of Applied Cryptography|url=https://archive.org/details/handbookofapplie0000mene/page/107|publisher=[[CRC Press]]|year=1997|pages=[https://archive.org/details/handbookofapplie0000mene/page/107 107–109]|chapter=Number-Theoretic Reference Problems|chapter-url=http://www.cacr.math.uwaterloo.ca/hac/about/chap3.pdf|isbn=0-8493-8523-7|ref=Menezes97|url-access=registration}}

{{Number-theoretic algorithms}}

{{DEFAULTSORT:Pohlig-Hellman algorithm}}

[[Category:Number theorytheoretic algorithms]]