MPQS with Three Large Primes

1
MPQS with Three Large Primes

• Paul Leyland
• Arjen Lenstra
• Bruce Dodson
• Alec Muffett
• Sam Wagstaff

2
TMPQS Overview

• Given N, select prime bounds B1 and B2
• Sieve for quadratic residues (relations) which
  factor over primes lt B1 and at most 3 primes lt B2
• Form cycles to eliminate primes gt B1
• Linear algebra to eliminate primes lt B1 and find
  x2 y2 (mod N )
• A factor is gcd (xy, N )

3
Early Experiments with TMPQS

• 1085 lt N lt 10110 , B1 106 , B2 108
• Not significantly worse than PPMPQS
• Perhaps up to 10 faster for larger N
• Crossover at about 10100 ?
• (PPMPQS crossover at about 1080 and about twice
  as fast as PMPQS at 10100)

4
This Work

• N 2803 2402 1, aka 2,1606L.c135
• B1 17157953 (550000 primes)
• B2 230
• Sieving at 5 sites, Jan Aug 2001
• 13441627 relations
• N p66 p69
• Approx 8000 MIPS-years
• (Estimate GNFS 1300 MIPS-years)

5
Performance comparison

• Extrapolation from RSA129 on identical hardware
  gt 1.7 times
• Relation generation rate with different sievers,
  extrapolated to completion for PPMPQS gt 1.75
  times
• Estimate GNFS 6 times faster still

6
Cycle Behaviour

• PMPQS Quadratic growth (birthday paradox)
• PPMPQS Initially quadratic, finally
  approximately quartic (RSA-129)
• TMPQS Superpolynomial region

7
ln(cycles) vs ln(relations)
8
Large Prime Relations

• par p
• ppr p q
• r
• tpr p q

9
Cycle Types

• S
• D
  etc
• T
  etc

10
ln(S-cycles) vs ln(relations)
11
ln(D-cycles) vs ln(relations)
12
ln(T-cycles) vs ln(relations)
13
Pruning
1
2
3
4
5
6
14
Pruning stages vs relations
15
Cross-linking
16
The Chemical Potential Functions At Phase
Transitions (PW Atkins)

• The changes in thermodynamic properties for a
  schematic first-order transition.

Vm
T
?
Sm
T
T
17
Polymer folding (Doye et al.)
18
Q vs T (Doye et al)
1.0
Q
0.5
0.0
Temperature
19
ln(T-cycles) vs ln(relations)
20
Corrected ln(T-cycles) vs ln(rels)
21
Cv vs T (Doye et al)
4000
Cv
2000
0
Temperature
22
Pruning stages vs relations
23
Conclusions

• Despite predictions to the contrary, TMPQS is
  faster than PPMPQS for large enough N
• Apparently power-law growth in D-cycles and
  (initially) T-cycles
• Enhancement arises from a phase transition in
  T-cycles
• Insight from physics chemistry

24
Further work

• Explain polynomial regions of D-cycle and T-cycle
  growth are they really 7/2 and 11/2 power-laws?
• Better characterize the phase transition in the
  T-cycles.
• Optimize sieving parameters to bring forward the
  onset of the phase transition perhaps by
  altering par/ppr/tpr ratio
• Similar phenomenon seen in NFS