Lucas–Lehmer primality test

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This article is about the Lucas–Lehmer test that only applies to Mersenne numbers. For the Lucas–Lehmer test that applies to a natural number n, see Lucas primality test. For the Lucas–Lehmer–Riesel test, see Lucas–Lehmer–Riesel test.

In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1856[1] and subsequently improved by Lucas in 1878 and Derrick Henry Lehmer in the 1930s.

The test[edit]

The Lucas–Lehmer test works as follows. Let Mp = 2p − 1 be the Mersenne number to test with p an odd prime. The primality of p can be efficiently checked with a simple algorithm like trial division since p is exponentially smaller than Mp. Define a sequence \{s_i\} for all i ≥ 0 by


  s_i=
   \begin{cases}
    4 & \text{if }i=0;
   \\
    s_{i-1}^2-2 & \text{otherwise.}
   \end{cases}

The first few terms of this sequence are 4, 14, 194, 37634, ... (sequence A003010 in OEIS). Then Mp is prime if and only if

s_{p-2} \equiv 0 \pmod{M_p}.

The number sp − 2 mod Mp is called the Lucas–Lehmer residue of p. (Some authors equivalently set s1 = 4 and test sp−1 mod Mp). In pseudocode, the test might be written as

// Determine if Mp = 2p − 1 is prime
Lucas–Lehmer(p)
    var s = 4
    var M = 2p − 1
    repeat p − 2 times:
        s = ((s × s) − 2) mod M
    if s = 0 return PRIME else return COMPOSITE

Performing the mod M at each iteration ensures that all intermediate results are at most p bits (otherwise the number of bits would double each iteration). The same strategy is used in modular exponentiation.

Time complexity[edit]

In the algorithm as written above, there are two expensive operations during each iteration: the multiplication s × s, and the mod M operation. The mod M operation can be made particularly efficient on standard binary computers by observing that

k \equiv (k\,\bmod\,2^n) + \lfloor k/2^n \rfloor \pmod{2^n - 1}.

This says that the least significant n bits of k plus the remaining bits of k are equivalent to k modulo 2n−1. This equivalence can be used repeatedly until at most n bits remain. In this way, the remainder after dividing k by the Mersenne number 2n−1 is computed without using division. For example,

916 mod 25−1 = 11100101002 mod 25−1
= 111002 + 101002 mod 25−1
= 1100002 mod 25−1
= 12 + 100002 mod 25−1
= 100012 mod 25−1
= 100012
= 17.

Moreover, since s × s will never exceed M2 < 22p, this simple technique converges in at most 1 p-bit addition (and possibly a carry from the pth bit to the 1st bit), which can be done in linear time. This algorithm has a small exceptional case. It will produce 2n−1 for a multiple of the modulus rather than the correct value of 0. However, this case is easy to detect and correct.

With the modulus out of the way, the asymptotic complexity of the algorithm only depends on the multiplication algorithm used to square s at each step. The simple "grade-school" algorithm for multiplication requires O(p2) bit-level or word-level operations to square a p-bit number. Since this happens O(p) times, the total time complexity is O(p3). A more efficient multiplication algorithm is the Schönhage–Strassen algorithm, which is based on the Fast Fourier transform. It only requires O(p log p log log p) time to square a p-bit number. This reduces the complexity to O(p2 log p log log p) or Õ(p2).[2] Currently the most efficient known multiplication algorithm, Fürer's algorithm, only needs p \log p\ 2^{O(\log^* p)} time to multiply two p-bit numbers.

By comparison, the most efficient randomized primality test for general integers, the Miller–Rabin primality test, requires O(k n2 log n log log n) bit operations using FFT multiplication for an n-digit number, where k is the number of iterations and is related to the error rate. For constant k, this is in the same complexity class as the Lucas-Lehmer test. In practice however, the cost of doing many iterations and other differences leads to worse performance for Miller–Rabin. The most efficient deterministic primality test for any n-digit number, the AKS primality test, requires Õ(n6) bit operations in its best known variant and is dramatically slower in practice.

Examples[edit]

The Mersenne number M3 = 7 is prime. The Lucas–Lehmer test verifies this as follows. Initially s is set to 4 and then is updated 3−2 = 1 time:

  • s ← ((4 × 4) − 2) mod 7 = 0.

Since the final value of s is 0, the conclusion is that M3 is prime.

On the other hand, M11 = 2047 = 23 × 89 is not prime. Again, s is set to 4 but is now updated 11−2 = 9 times:

  • s ← ((4 × 4) − 2) mod 2047 = 14
  • s ← ((14 × 14) − 2) mod 2047 = 194
  • s ← ((194 × 194) − 2) mod 2047 = 788
  • s ← ((788 × 788) − 2) mod 2047 = 701
  • s ← ((701 × 701) − 2) mod 2047 = 119
  • s ← ((119 × 119) − 2) mod 2047 = 1877
  • s ← ((1877 × 1877) − 2) mod 2047 = 240
  • s ← ((240 × 240) − 2) mod 2047 = 282
  • s ← ((282 × 282) − 2) mod 2047 = 1736

Since the final value of s is not 0, the conclusion is that M11 = 2047 is not prime. Although M11 = 2047 has nontrivial factors, the Lucas–Lehmer test gives no indication about what they might be.

Proof of correctness[edit]

The proof of correctness for this test presented here is simpler than the original proof given by Lehmer. Recall the definition


  s_i=
   \begin{cases}
    4 & \text{if }i=0;\\
    s_{i-1}^2-2 & \text{otherwise.}
   \end{cases}

The goal is to show that Mp is prime iff s_{p-2} \equiv 0 \pmod{M_p}.

The sequence {\langle}s_i{\rangle} is a recurrence relation with a closed-form solution. Let \omega = 2 + \sqrt{3} and \bar{\omega} = 2 - \sqrt{3}. It then follows by induction that s_i = \omega^{2^i} + \bar{\omega}^{2^i} for all i:

s_0 = \omega^{2^0} + \bar{\omega}^{2^0} = \left(2 + \sqrt{3}\right) + \left(2 - \sqrt{3}\right) = 4

and


\begin{align}
 s_n
 &= s_{n-1}^2 - 2 \\
 &= \left(\omega^{2^{n-1}} + \bar{\omega}^{2^{n-1}}\right)^2 - 2 \\
 &= \omega^{2^n} + \bar{\omega}^{2^n} + 2(\omega\bar{\omega})^{2^{n-1}} - 2 \\
 &= \omega^{2^n} + \bar{\omega}^{2^n}.
\end{align}

The last step uses \omega\bar{\omega} = \left(2 + \sqrt{3}\right) \left(2 - \sqrt{3}\right) = 1. This closed form is used in both the proof of sufficiency and the proof of necessity.

Sufficiency[edit]

The goal is to show that s_{p-2} \equiv 0 \pmod{M_p} implies that M_p is prime. What follows is a straightforward proof exploiting elementary group theory given by J. W. Bruce[3] as related by Jason Wojciechowski.[4]

Suppose s_{p-2} \equiv 0 \pmod{M_p}. Then

\omega^{2^{p-2}} + \bar{\omega}^{2^{p-2}} = k M_p

for some integer k, so

\omega^{2^{p-2}} = k M_p - \bar{\omega}^{2^{p-2}}.

Multiplying by \omega^{2^{p - 2}} gives

\left(\omega^{2^{p-2}}\right)^2 = k M_p\omega^{2^{p-2}} - (\omega \bar{\omega})^{2^{p-2}}.

Thus,

\omega^{2^{p-1}} = k M_p\omega^{2^{p-2}} - 1.\qquad\qquad(1)

For a contradiction, suppose Mp is composite, and let q be the smallest prime factor of Mp. Mersenne numbers are odd, so q > 2. Informally,[note 1] let \mathbb{Z}_q be the integers modulo q, and let X = \left\{a + b \sqrt{3} \mid a, b \in \mathbb{Z}_q\right\}. Multiplication in X is defined as

\left(a + \sqrt{3} b\right) \left(c + \sqrt{3} d\right) = [(a c + 3 b d) \,\bmod\,q] + \sqrt{3} [(a d + b c) \,\bmod\,q].

Clearly, this multiplication is closed, i.e. the product of numbers from X is itself in X. The size of X is denoted by |X|.

Since q > 2, it follows that \omega and \bar{\omega} are in X.[note 2] The subset of elements in X with inverses forms a group, which is denoted by X* and has size |X^*|. One element in X that does not have an inverse is 0, so |X^*| \leq |X| - 1 = q^2 - 1.

Now M_p \equiv 0 \pmod{q} and \omega \in X, so

kM_p\omega^{2^{p-2}} = 0

in X. Then by equation (1),

\omega^{2^{p-1}} = -1

in X, and squaring both sides gives

\omega^{2^p} = 1.

Thus \omega lies in X* and has inverse \omega^{2^{p}-1}. Furthermore the order of \omega divides 2^p. However \omega^{2^{p-1}} \neq 1, so the order does not divide 2^{p-1}. Thus, the order is exactly 2^p.

The order of an element is at most the order (size) of the group, so

2^p \leq |X^*| \leq q^2 - 1 < q^2.

But q is the smallest prime factor of the composite M_p, so

q^2 \leq M_p = 2^p-1.

This yields the contradiction 2^p < 2^p-1. Therefore M_p is prime.

Necessity[edit]

In the other direction, the goal is to show that the primality of M_p implies s_{p-2} \equiv 0 \pmod{M_p}. The following simplified proof is by Öystein J. R. Ödseth.[5]

Since 2^p - 1 \equiv 7 \pmod{12} for odd p > 1, it follows from properties of the Legendre symbol that (3|M_p) = -1. This means that 3 is a quadratic nonresidue modulo M_p. By Euler's criterion, this is equivalent to

3^{\frac{M_p-1}{2}} \equiv -1 \pmod{M_p}.

In contrast, 2 is a quadratic residue modulo M_p since 2^p \equiv 1 \pmod{M_p} and so 2 \equiv 2^{p+1} = \left(2^{\frac{p+1}{2}}\right)^2 \pmod{M_p}. This time, Euler's criterion gives

2^{\frac{M_p-1}{2}} \equiv 1 \pmod{M_p}.

Combining these two equivalence relations yields

24^{\frac{M_p-1}{2}} \equiv \left(2^{\frac{M_p-1}{2}}\right)^3 \left(3^{\frac{M_p-1}{2}}\right) \equiv (1)^3(-1) \equiv -1 \pmod{M_p}.

Let \sigma = 2\sqrt{3}, and define X as before as the ring X = \{a + b\sqrt{3} \mid a, b \in \mathbb{Z}_{M_p}\}. Then in the ring X, it follows that


\begin{align}
 (6+\sigma)^{M_p}
 &= 6^{M_p} + \left(2^{M_p}\right) \left(\sqrt{3}^{M_p}\right) \\
 &= 6 + 2 \left(3^{\frac{M_p-1}{2}}\right) \sqrt{3} \\
 &= 6 + 2 (-1) \sqrt{3} \\
 &= 6 - \sigma,
\end{align}

where the first equality uses the Binomial Theorem in a finite field, which is

(x+y)^{M_p} \equiv x^{M_p} + y^{M_p} \pmod{M_p},

and the second equality uses Fermat's little theorem, which is

a^{M_p} \equiv a \pmod{M_p}

for any integer a. The value of \sigma was chosen so that \omega = \frac{(6+\sigma)^2}{24}. Consequently, this can be used to compute \omega^{\frac{M_p+1}{2}} in the ring X as


\begin{align}
 \omega^{\frac{M_p+1}{2}}
 &= \frac{(6 + \sigma)^{M_p+1}}{24^{\frac{M_p+1}{2}}} \\
 &= \frac{(6 + \sigma) (6 + \sigma)^{M_p}}{24 \cdot 24^{\frac{M_p-1}{2}}} \\
 &= \frac{(6 + \sigma) (6 - \sigma)}{-24} \\
 &= -1.
\end{align}

All that remains is to multiply both sides of this equation by \bar{\omega}^{\frac{M_p+1}{4}} and use \omega \bar{\omega} = 1, which gives


\begin{align}
 \omega^{\frac{M_p+1}{2}}     \bar{\omega}^{\frac{M_p+1}{4}}   &= -\bar{\omega}^{\frac{M_p+1}{4}} \\
 \omega^{\frac{M_p+1}{4}}   + \bar{\omega}^{\frac{M_p+1}{4}}   &= 0 \\
 \omega^{\frac{2^p-1+1}{4}} + \bar{\omega}^{\frac{2^p-1+1}{4}} &= 0 \\
 \omega^{2^{p-2}}           + \bar{\omega}^{2^{p-2}}           &= 0 \\
 s_{p-2}                                                       &= 0.
\end{align}

Since s_{p - 2} is 0 in X, it is also 0 modulo M_p.

Applications[edit]

The Lucas–Lehmer test is the primality test used by the Great Internet Mersenne Prime Search to locate large primes. This search has been successful in locating many of the largest primes known to date.[6] The test is considered valuable because it can provably test a large set of very large numbers for primality within an affordable amount of time. In contrast, the equivalently fast Pépin's test for any Fermat number can only be used on a much smaller set of very large numbers before reaching computational limits.

See also[edit]

Notes[edit]

  1. ^ Formally, let \mathbb{Z}_q = \mathbb{Z} / q \mathbb{Z} and X = \mathbb{Z}_q[T] / \langle T^2 - 3 \rangle.
  2. ^ Formally, \omega + \langle T^2 - 3 \rangle and \bar{\omega} + \langle T^2 - 3 \rangle are in X. By abuse of language \omega and \bar{\omega} are identified with their images in X under the natural ring homomorphism from \mathbb{Z}[\sqrt{3}] to X which sends \sqrt{3} to T.

References[edit]

  1. ^ The Largest Known Prime by Year: A Brief History
  2. ^ Colquitt, W. N.; Welsh, L., Jr. (1991), "A New Mersenne Prime", Mathematics of Computation 56 (194): 867–870, doi:10.2307/2008415, The use of the FFT speeds up the asymptotic time for the Lucas–Lehmer test for Mp from O(p3) to O(p2 log p log log p) bit operations. 
  3. ^ J. W. Bruce (1993). "A Really Trivial Proof of the Lucas–Lehmer Test". The American Mathematical Monthly 100 (4): 370–371. doi:10.2307/2324959. 
  4. ^ Jason Wojciechowski. Mersenne Primes, An Introduction and Overview. 2003.
  5. ^ Öystein J. R. Ödseth. A note on primality tests for N = h · 2n − 1. Department of Mathematics, University of Bergen.
  6. ^ The "Top Ten" Record Primes, The Prime Pages

External links[edit]