Lucas–Lehmer–Riesel test

In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form N = k ⋅ 2ⁿ − 1, with 2ⁿ > k. The test was developed by Hans Riesel and it is based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form. For numbers of the form N = k ⋅ 2ⁿ + 1 (Proth numbers), either application of Proth's theorem (a Las Vegas algorithm) or one of the deterministic proofs described in Brillhart-Lehmer-Selfridge 1975^[1] are used.

The algorithm

The algorithm is very similar to the Lucas–Lehmer test, but with a variable starting point depending on the value of k.

Define a sequence {u_i} for all i > 0 by:

u_{i}=u_{{i-1}}^{2}-2.\,

Then N is prime if and only if it divides u_n−2.

Finding the starting value

The starting value u₀ is determined as follows.

If k = 1: if n is odd, then we can take u₀ = 4. If n ≡ 3 (mod 4), then we can take u₀ = 3. Note that if n is prime, these are Mersenne numbers.
If k = 3: if n ≡ 0 or 3 (mod 4), then u₀ = 5778.
If k ≡ 1 or 5 (mod 6): if 3 does not divide N, then we take $u_{0}=(2+{\sqrt {3}})^{k}+(2-{\sqrt {3}})^{k}$ . See below for how to calculate this using a Lucas(4,1) sequence.
Otherwise, we are in the case where k is a multiple of 3, and it is more difficult to select the right value of u₀.

An alternative method for finding the starting value u₀ is given in Rödseth 1994.^[2] The selection method is much easier than that used by Riesel for the 3 divides k case. First find a P value that satisfies the following equalities of Jacobi symbols:

\left({\frac {P-2}{N}}\right)=1\quad {\text{and}}\quad \left({\frac {P+2}{N}}\right)=-1

In practice, only a few P values need be checked before one is found (5, 8, 9, or 11 work in about 85% of trials).

To find the starting value u₀ from the P value we can use a Lucas(P,1) sequence, as shown in ^[2] as well as page 124 of.^[3] The latter explains that when 3 ∤ k, P=4 may be used, hence the earlier search is not necessary. The starting value u₀ is then equal to the modular Lucas sequence V_k(P,Q) mod N. This process takes very little time compared to the main test.

How the test works

The Lucas–Lehmer–Riesel test is a particular case of group-order primality testing; we demonstrate that some number is prime by showing that some group has the order that it would have were that number prime, and we do this by finding an element of that group of precisely the right order.

For Lucas-style tests on a number N, we work in the multiplicative group of a quadratic extension of the integers modulo N; if N is prime, the order of this multiplicative group is N² − 1, it has a subgroup of order N + 1, and we try to find a generator for that subgroup.

We start off by trying to find a non-iterative expression for the $u_{i}$ . Following the model of the Lucas–Lehmer test, put $u_{i}=a^{{2^{i}}}+a^{{-2^{i}}}$ , and by induction we have $u_{i}=u_{{i-1}}^{2}-2$ .

So we can consider ourselves as looking at the 2ⁱth term of the sequence $v(i)=a^{i}+a^{{-i}}$ . If a satisfies a quadratic equation, this is a Lucas sequence, and has an expression of the form $v(i)=\alpha v(i-1)+\beta v(i-2)$ . Really, we're looking at the k ⋅ 2ⁱth term of a different sequence, but since decimations (take every kth term starting with the zeroth) of a Lucas sequence are themselves Lucas sequences, we can deal with the factor k by picking a different starting point.

LLR software

LLR is a program that can run the LLR tests. The program was developed by Jean Penné. Vincent Penné has modified the program so that it can obtain tests via the Internet. The software is both used by individual prime searchers and some distributed computing projects including Riesel Sieve and PrimeGrid.

References

↑ John Brillhart; Derrick Henry Lehmer; John Selfridge (April 1975). "New Primality Criteria and Factorizations of 2^m ± 1". Mathematics of Computation. 29 (130): 620–647. doi:10.1090/S0025-5718-1975-0384673-1.
1 2 Rödseth,Öystein J. (1994). "A note on primailty tests for N=h·2^n−1" (PDF). BIT Numerical Mathematics. 34 (3): 451–454. doi:10.1007/BF01935653.
↑ Hans Riesel (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. 126 (2nd ed.). Birkhäuser. pp. 107–121. ISBN 0-8176-3743-5.

Riesel, Hans (1969). "Lucasian Criteria for the Primality of N = h·2ⁿ − 1". Mathematics of Computation. American Mathematical Society. 23 (108): 869–875. JSTOR 2004975. doi:10.2307/2004975.

External links

Download Jean Penné's LLR
Math::Prime::Util::GMP - Part of Perl's ntheory module, this has basic implementations of LLR and Proth form testing, as well as some BLS75 proof methods.

Number-theoretic algorithms
Primality tests	AKS test APR test Baillie–PSW Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin
Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization
Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's
Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's
Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve
Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's
Modular square root	Cipolla Pocklington's Tonelli–Shanks
Other algorithms	Chakravala Cornacchia LLL Integer square root Modular exponentiation Schoof's
Italics indicate that algorithm is for numbers of special forms

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