Pollard's rho algorithm

The algorithm is used to factorize a number $n\; =\; pq$, where $p$ is a non-trivial factor. A polynomial modulo $n$, called $g(x)$ (e.g., $g(x)\; =\; (x^2\; +\; 1)\; \backslash bmod\; n$), is used to generate a pseudorandom sequence. It is important to note that $g(x)$ must be a polynomial. A starting value, say 2, is chosen, and the sequence continues as $x\_1\; =\; g(2)$, $x\_2\; =\; g(g(2))$, $x\_3\; =\; g(g(g(2)))$, etc. The sequence is related to another sequence $\backslash \{x\_k\; \backslash bmod\; p\backslash \}$. Since $p$ is not known beforehand, this sequence cannot be explicitly computed in the algorithm. Yet in it lies the core idea of the algorithm.

Because the number of possible values for these sequences is finite, both the $\backslash \{x\_k\backslash \}$ sequence, which is mod $n$, and $\backslash \{x\_k\; \backslash bmod\; p\backslash \}$ sequence will eventually repeat, even though these values are unknown. If the sequences were to behave like random numbers, the birthday paradox implies that the number of $x\_k$ before a repetition occurs would be expected to be $O(\backslash sqrt\; N)$, where $N$ is the number of possible values. So the sequence $\backslash \{x\_k\; \backslash bmod\; p\backslash \}$ will likely repeat much earlier than the sequence $\backslash \{x\_k\backslash \}$. When one has found a $k\_1,k\_2$ such that $x\_\{k\_1\}\backslash neq\; x\_\{k\_2\}$ but $x\_\{k\_1\}\backslash equiv\; x\_\{k\_2\}\backslash bmod\; p$, the number $|x\_\{k\_1\}-x\_\{k\_2\}|$ is a multiple of $p$, so a non-trivial divisor has been found.

Once a sequence has a repeated value, the sequence will cycle, because each value depends only on the one before it. This structure of eventual cycling gives rise to the name "rho algorithm", owing to similarity to the shape of the Greek letter ρ when the values $x\_1\; \backslash bmod\; p$, $x\_2\; \backslash bmod\; p$, etc. are represented as nodes in a directed graph.

File:Pollard rho cycle.svg

This is detected by Floyd's cycle-finding algorithm: two nodes $i$ and $j$ (i.e., $x\_i$ and $x\_j$) are kept. In each step, one moves to the next node in the sequence and the other moves forward by two nodes. After that, it is checked whether $\backslash gcd(x\_i\; -\; x\_j,\; n)\; \backslash ne\; 1$. If it is not 1, then this implies that there is a repetition in the $\backslash \{x\_k\; \backslash bmod\; p\backslash \}$ sequence (i.e. $x\_i\; \backslash bmod\; p\; =\; x\_j\; \backslash bmod\; p)$. This works because if the $x\_i\; \backslash bmod\; p$ is the same as $x\_j\; \backslash bmod\; p$, the difference between $x\_i$ and $x\_j$ is necessarily a multiple of $p$. Although this always happens eventually, the resulting greatest common divisor (GCD) is a divisor of $n$ other than 1. This may be $n$ itself, since the two sequences might repeat at the same time. In this (uncommon) case the algorithm fails, it can be repeated with a different parameter.

The algorithm takes as its inputs {{mvar|n}}, the integer to be factored; and {{tmath|g(x)}}, a polynomial in {{mvar|x}} computed modulo {{mvar|n}}. In the original algorithm, $g(x)\; =\; (x^2\; -\; 1)\; \backslash bmod\; n$, but nowadays it is more common to use $g(x)\; =\; (x^2\; +\; 1)\; \backslash bmod\; n$. The output is either a non-trivial factor of {{mvar|n}}, or failure.

It performs the following steps:{{cite book |last1=Cormen |first1=Thomas H. |authorlink=Thomas H. Cormen |last2=Leiserson |first2=Charles E. |authorlink2=Charles E. Leiserson |last3=Rivest |first3=Ronald L. |authorlink3=Ronald L. Rivest |last4=Stein |first4=Clifford |authorlink4=Clifford Stein |name-list-style=amp |chapter=Section 31.9: Integer factorization |title=Introduction to Algorithms |year=2009 |edition=third |publisher=MIT Press |location=Cambridge, MA |pages=975–980|isbn=978-0-262-03384-8 }} (this section discusses only Pollard's rho algorithm).

Pseudocode for Pollard's rho algorithm

x ← 2 // starting value

y ← x

d ← 1

while d = 1:

x ← g(x)

y ← g(g(y))

d ← gcd(|x - y|, n)

if d = n:

return failure

else:

return d

Here {{mvar|x}} and {{mvar|y}} corresponds to {{tmath|x_i}} and {{tmath|x_j}} in the previous section. Note that this algorithm may fail to find a nontrivial factor even when {{mvar|n}} is composite. In that case, the method can be tried again, using a starting value of x other than 2 () or a different {{tmath|g(x)}}, $g(x)\; =\; (x^2\; +\; b)\; \backslash bmod\; n$, with .

Let $n\; =\; 8051$ and $g(x)\; =\; (x^2\; +\; 1)\; \backslash bmod\; 8051$.

File:Rho-example-animated.gif.]]

Now 97 is a non-trivial factor of 8051. Starting values other than {{math|1=x = y = 2}} may give the cofactor (83) instead of 97. One extra iteration is shown above to make it clear that {{mvar|y}} moves twice as fast as {{mvar|x}}. Note that even after a repetition, the GCD can return to 1.

In 1980, Richard Brent published a faster variant of the rho algorithm. He used the same core ideas as Pollard but a different method of cycle detection, replacing Floyd's cycle-finding algorithm with the related Brent's cycle finding method.{{cite journal |last=Brent |first=Richard P. |authorlink=Richard Brent (scientist) |year=1980 |title=An Improved Monte Carlo Factorization Algorithm |journal=BIT |volume=20 |issue=2 |pages=176–184 |url=https://maths-people.anu.edu.au/~brent/pub/pub051.html |doi=10.1007/BF01933190|s2cid=17181286 }} CLRS gives a heuristic analysis and failure conditions (the trivial divisor $n$ is found).

A further improvement was made by Pollard and Brent. They observed that if $\backslash gcd(a,n)\; >\; 1$, then also $\backslash gcd(ab,n)\; >\; 1$ for any positive integer {{tmath|b}}. In particular, instead of computing $\backslash gcd\; (|x-y|,n)$ at every step, it suffices to define {{tmath|z}} as the product of 100 consecutive $|x-y|$ terms modulo {{tmath|n}}, and then compute a single $\backslash gcd(z,n)$. A major speed up results as 100 {{math|gcd}} steps are replaced with 99 multiplications modulo {{tmath|n}} and a single {{math|gcd}}. Occasionally it may cause the algorithm to fail by introducing a repeated factor, for instance when {{tmath|n}} is a square. But it then suffices to go back to the previous {{math|gcd}} term, where $\backslash gcd(z,n)=1$, and use the regular ρ algorithm from there.Exercise 31.9-4 in CLRS

The algorithm is very fast for numbers with small factors, but slower in cases where all factors are large. The ρ algorithm's most remarkable success was the 1980 factorization of the Fermat number {{math|F₈}} = 1238926361552897 × 93461639715357977769163558199606896584051237541638188580280321.{{cite journal |last1=Brent |first1=R.P. |last2=Pollard |first2=J. M. |year=1981 |title=Factorization of the Eighth Fermat Number |journal=Mathematics of Computation |volume=36 |issue=154 |pages=627–630 |doi=10.2307/2007666|jstor=2007666 |doi-access=free }} The ρ algorithm was a good choice for {{math|F₈}} because the prime factor {{mvar|p}} = 1238926361552897 is much smaller than the other factor. The factorization took 2 hours on a UNIVAC 1100/42.

The following table shows numbers produced by the algorithm, starting with $x=2$ and using the polynomial $g(x)\; =\; (x^2\; +\; 1)\; \backslash bmod\; 10403$.

The third and fourth columns of the table contain additional information not known by the algorithm.

They are included to show how the algorithm works.

The first repetition modulo 101 is 97 which occurs in step 17. The repetition is not detected until step 23, when $x\; \backslash equiv\; y\; \backslash pmod\{101\}$. This causes $\backslash gcd\; (x\; -\; y,\; n)\; =\; \backslash gcd\; (2799\; -\; 9970,\; n)$ to be $p\; =\; 101$, and a factor is found.

If the pseudorandom number $x\; =\; g(x)$ occurring in the Pollard ρ algorithm were an actual random number, it would follow that success would be achieved half the time, by the birthday paradox in $O(\backslash sqrt\; p)\backslash le\; O(n^\{1/4\})$ iterations. It is believed that the same analysis applies as well to the actual rho algorithm, but this is a heuristic claim, and rigorous analysis of the algorithm remains open.{{cite book|title=Mathematics of Public Key Cryptography|first=Steven D.|last=Galbraith|publisher=Cambridge University Press|year=2012|isbn=9781107013926|contribution=14.2.5 Towards a rigorous analysis of Pollard rho|pages=272–273|url=https://books.google.com/books?id=owd76BElvosC&pg=PA272}}.

== See also ==

• Pollard's rho algorithm for logarithms
• Pollard's kangaroo algorithm

{{reflist|group=note}}

{{reflist}}

• {{cite conference |first1=Shi |last1=Bai |first2=Richard P. |last2=Brent |authorlink2=Richard P. Brent |title=On the Efficiency of Pollard's Rho Method for Discrete Logarithms |conference=The Australasian Theory Symposium (CATS2008) |location=Wollongong |date=January 2008 |book-title=Conferences in Research and Practice in Information Technology, Vol. 77 |pages=125–131 |url=https://maths-people.anu.edu.au/~brent/pub/pub231.html}} Describes the improvements available from different iteration functions and cycle-finding algorithms.
• {{cite book |last1=Katz |first1=Jonathan |last2=Lindell |first2=Yehuda |chapter=Chapter 8 |title=Introduction to Modern Cryptography | year=2007 |publisher=CRC Press}}
• {{cite book | author =Samuel S. Wagstaff, Jr. | title=The Joy of Factoring | publisher=American Mathematical Society | location=Providence, RI | year=2013 | isbn=978-1-4704-1048-3 |url=https://www.ams.org/bookpages/stml-68 |author-link=Samuel S. Wagstaff, Jr. |pages= 135–138 }}

• [https://sofosband.wixsite.com/pversusnp/single-post/2018/08/27/factorising-part-2-pollards-rho-algorithm Comprehensive article on Pollard's Rho algorithm aimed at an introductory-level audience]
• {{MathWorld|title=Pollard rho Factorization Method|id=PollardRhoFactorizationMethod}}

• [https://introcs.cs.princeton.edu/java/99crypto/PollardRho.java.html Java Implementation]
• [https://forthmath.blogspot.com/2020/01/about-pollard-rho.html About Pollard rho]

{{number theoretic algorithms}}

{{DEFAULTSORT:Pollard's Rho Algorithm}}

Category:Integer factorization algorithms