Factorization of a 512-bits RSA key using the Number Field Sieve
On August 22, 1999, we found that the 512-bits number
RSA-155 =

can be written as the product of two 78-digit primes:

Primality of the factors was proved with the help of two different primality
proving codes. An Appendix gives the prime decompositions of p +- 1.
The number RSA-155 is taken from the RSA Challenge list 
(see http://www.rsa.com/rsalabs/html/factoring.html).

This factorization was found using the Number Field Sieve (NFS) factoring
algorithm, and beats the 140-digit record RSA-140 that was set on 
February 2, 1999, also with the help of NFS [RSA140].
The amount of computer time spent on this new factoring world record is
estimated to be equivalent to 8000 mips years.
For the old 140-digit NFS-record, this effort was estimated to be 
2000 mips years. Extrapolation using the asymptotic complexity formula
for NFS would predict approximately 14000 mips years for RSA-155. The gain 
is caused by an improved application of the polynomial search method used 
for RSA-140.
For information about NFS, see [LL]. For additional information,
implementations and previous large NFS factorizations, see [DL, E1, E2, GLM].
Polynomial selection
The following two polynomials

F_1(x,y) =                           11 93771 38320     x^5
                             - 80 16893 72849 97582 y  *x^4
                          - 66269 85223 41185 74445 y^2*x^3
                  + 1 18168 48430 07952 18803 56852 y^3*x^2
                + 745 96615 80071 78644 39197 43056 y^4*x
           - 40 67984 35423 62159 36191 37084 05064 y^5

F_2(x,y)  =  x - 3912 30797 21168 00077 13134 49081 y

were selected with the help of a polynomial search method developed by 
Peter Montgomery (Microsoft Research, USA and CWI) and Brian Murphy 
(The Australian National University, Canberra), which was applied already 
to RSA-140, and now, even more successfully, to RSA-155.

The polynomial F_1(x,y) was chosen to have a good combination of
two properties: being unusually small over its sieving region and
having unusually many roots modulo small primes (and prime powers).
The effect of the second property alone gives F_1(x,y) a smoothness
yield comparable to that of a polynomial chosen at random for an
integer of 137 decimal digits.
Measured in a different way: the pair (F_1, F_2) has a yield of relations 
approximately 13.5 times that of a random polynomial selection for 
RSA-155 (the corresponding figure for the polynomial selected for the
RSA-140 factorisation is 8).

The polynomial selection took approximately 100 MIPS years,
which is equivalent to 0.40 CPU years on a 250 MHz SGI Origin 2000 
processor (most of the searches were done on such processors).

The original polynomial selection code was ported by Arjen Lenstra
to use his multiple precision arithmetic package LIP.
Brian Murphy, Peter Montgomery, Arjen Lenstra and Bruce Dodson
ran the polynomial searches for RSA-155 with this code. The above 
polynomial emerged from Bruce Dodson's search.

Calendar time for the polynomial selection was approximately nine
The Sieving
Sieving was done on about 160 175-400 MHz SGI and Sun workstations,
on 8 300 MHz SGI Origin 2000 processors, on about 120 300-450 MHz 
Pentium II PCs, and on 4 500 MHz Digital/Compaq boxes.
The total amount of CPU-time spent on sieving was 35.7 CPU years
estimated to be equivalent to approximately 8000 mips years.
Calendar time for sieving was 3 1/2 months.

For the purpose of comparison, both lattice sieving and line sieving were used.
Lattice sieving was introduced by Pollard [P] and the code used is based
on the implementation described in [GLM, Cetal].

For the lattice sieve, a factor base bound of 16 777 216 (2^24) was chosen, 
both for the rational and for the algebraic side. Two large primes were 
allowed on both sides.
Most of the line sieve was carried out with two large primes on both the 
rational and the algebraic side. The rational factor base consisted of the 
primes < 44 000 000 and the algebraic factor base of the primes < 110 000 000.
Some line sieving allowed three large primes instead of two on the algebraic
side. In that case the rational factor base consisted of the primes < 8 000 000
and the algebraic factor base of the primes < 25 000 000.

For both sieves the large prime bound 1 000 000 000 was used both  
for the rational and for the algebraic primes.

A total of 124 722 179 relations were generated, 71% of them with lattice
sieving (L), 29% with line sieving (C). Among them, there were 39 187 441
duplicates, partially because of the simultaneous use of the two sievers.
Sieving was done at eleven different locations with the following contributions:

(L: using lattice sieving code from Arjen K. Lenstra
 C: using line sieving code from CWI)

20.1 % (3057 CPU days) Alec Muffett (L at Sun Microsystems Professional 
                                     Services, Camberley, UK)
17.5 % (2092 CPU days) Paul Leyland (L,C at Microsoft, Cambridge, UK)
14.6 % (1819) Peter L. Montgomery, Stefania Cavallar (C,L at CWI, Amsterdam)
13.6 % (2222) Bruce Dodson (L,C at Lehigh University, Bethlehem, PA, USA)
13.0 % (1801) Francois Morain and Gerard Guillerm 
              (L,C at Ecole Polytechnique, Palaiseau, France)
 6.4 % (576)  Joel Marchand (L,C at Ecole Polytechnique/CNRS, Palaiseau, France)
 5.0 % (737)  Arjen K. Lenstra (L at Citibank, Parsippany, NJ, USA
                                and Univ. of Sydney, Australia)
 4.5 % (252)  Paul Zimmermann (C at Inria Lorraine and Loria, Nancy, France)
 4.0 % (366)  Jeff Gilchrist (L at Entrust Technologies Ltd., Ottawa, Canada)
 0.65 % (62)  Karen Aardal (L at Utrecht University, The Netherlands)
 0.56 % (47)  Chris and Craig Putnam (L at ?)

Calendar time for the sieving was 3.7 months.
The relations were collected at CWI and required 3.7 Gbytes of disk space.

Filtering and linear algebra

The filtering of the data and the building of the matrix were carried out
at CWI and took one month.

The resulting matrix had 6 699 191 rows, 6 711 336 columns, and weight 
417 132 631 (62.27 nonzeros per row). 
With the help of Peter Montgomery's Cray implementation of the blocked 
Lanczos algorithm (cf. [M95]) it took 224 CPU hours and 2 Gbytes of central 
memory on the Cray C916 at the SARA Amsterdam Academic Computer Center to find 
64 dependencies among the rows of this matrix. 
Calendar time for this job was 9 1/2 days.

Square root 

On August 20-21, 1999, four different square root (cf. [M93]) jobs were 
started in parallel on four different 300 MHz processors of CWI's SGI Origin 
2000, each handling one dependency. 
One job found the factorisation after 39.4 CPU-hours, the other three jobs 
found the trivial factorization after 38.3, 41.9, and 61.6 CPU-hours (different
CPU times are due to the use of different parameters in the four jobs).

Herman te Riele, CWI, August 26, 1999
with the algorithmic and sieving contributors:

        Stefania Cavallar
        Bruce Dodson
        Arjen Lenstra
        Walter Lioen
        Peter L. Montgomery
        Brian Murphy

and the sieving contributors:

        Karen Aardal
        Jeff Gilchrist
        Gerard Guillerm
        Paul Leyland
        Joel Marchand
        Francois Morain
        Alec Muffett
        Craig and Chris Putnam
        Paul Zimmermann
Acknowledgements are due to the contributors of idle PC and workstation cycles, 
to the Dutch National Computing Facilities Foundation (NCF) for the use of 
the Cray-C916 supercomputer at SARA and (in alphabetical order) to

Centre Charles Hermite (Nancy, France),
Citibank (Parsippany, NJ, USA),
CWI (Amsterdam, The Netherlands), 
Ecole Polytechnique/CNRS (Palaiseau, France),
Entrust Technologies Ltd. (Ottawa, Canada),
Lehigh University (Bethlehem, PA, USA),
the Magma Group of John Cannon at the University of Sydney, 
the Medicis Center at Ecole Polytechnique (Palaiseau, France),
Microsoft Research (Cambridge, UK),
Sun Microsystems Professional Services (Camberley, UK), and
The Australian National University (Canberra, Australia)

for the use of their computing resources.


[RSA140]Stefania Cavallar, Bruce Dodson, Arjen Lenstra, Paul Leyland, 
        Walter Lioen, Peter L. Montgomery, Brian Murphy, Herman te Riele, 
        and Paul Zimmermann,
        Factorization of RSA-140 using the Number Field Sieve, to appear in: 
        Lam Kwok Yan, Eiji Okamoto, and Xing Chaoping, editors,
        Advances in Cryptology -- Asiacrypt '99, 
        Lecture Notes in Computer Science # xxxx,
        Springer--Verlag, Berlin, etc., 1999.

[Cetal] James Cowie, Bruce Dodson, R.-Marije Elkenbracht-Huizing,
        Arjen K. Lenstra, Peter L. Montgomery and Joerg Zayer,
        A world wide number field sieve factoring record: on to 512 bits,
        pp. 382-394 in: Kwangjo Kim and Tsutomu Matsumoto (editors),
        Advances in Cryptology - Asiacrypt '96, Lecture Notes in 
        Computer Science # 1163, Springer-Verlag, Berlin, 1996.

[DL]    B. Dodson, A.K. Lenstra, NFS with four large primes: an
        explosive experiment, Proceedings Crypto 95, Lecture Notes
        in Comput. Sci. 963, (1995) 372-385.
[E1]    R.-M. Elkenbracht-Huizing, Factoring integers with the 
        Number Field Sieve, Doctor's Thesis, Leiden University,

[E2]    R.-M. Elkenbracht-Huizing, An implementation of the number
        field sieve, Exp. Math. 5, (1996) 231-253.  
[GLM]   R. Golliver, A.K. Lenstra, K.S. McCurley, Lattice sieving
        and trial division, Algorithmic number theory symposium,
        Proceedings, Lecture Notes in Comput. Sci. 877, (1994) 18-27.
[LL]    A.K. Lenstra, H.W. Lenstra, Jr., The development of the
        number field sieve, Lecture Notes in Math. 1554, Springer-
        Verlag, Berlin, 1993
[M93]   Peter L. Montgomery, Square roots of products of algebraic
        numbers, in Proceedings of Symposia in Applied Mathematics,
        Mathematics of Computation 1943-1993, Vancouver, 1993,
        Walter Gautschi, ed.
[M95]   Peter L. Montgomery, A block Lanczos algorithm for finding
        dependencies over GF(2), Proceedings Eurocrypt 1995,
        Lecture Notes in Comput. Sci. 921, (1995) 106-120.
[P]     J.M. Pollard, The lattice sieve, pages 43-49 in [LL].


Here are the P+-1 factorizations of the factors: