SNFS versus (G)NFS

and the feasibility of

factoring a 1024-bit number with SNFS

Arjen K. Lenstra

Citibank, New YorkTechnische Universiteit Eindhoven

Factoring algorithms (to find factor p of n)

Special purpose methods

General purpose methods

Take advantage of special properties of p

Cannot take advantage of any properties of p

All based on the same approach

Relevant for RSA

Examples:

Examples:

Trial division, Pollard- (find tiny p, up to 10 or 20 digits)Pollard-p1 (finds p such that p1 has small factors)Elliptic curve method (ECM) (finds p up to 60? digits)

CFRAC, Dixon’s algorithmLinear sieve, Quadratic sieveNumber field sieve (NFS)Variant: SNFS, takes advantage of special form of n

, but possibly of n

th

is ta

lk

SNFS and NFS factorizations when # bits what how

199006 512 F9 = 2512+1 SNFS199406 534 (121511)/11 SNFS199407 384 p(11887) NFS199411 392 p(13171) NFS199604 429 RSA-130d NFS199809 615 12167+1 SNFS199902 462 RSA-140d NFS199904 698 (10211 1)/9 SNFS199908 512 RSA-155d NFS200011 773 2773+1 SNFS200201 522 c158d of 2953+1 NFS200301 809 M809 SNFS200303 529 RSA-160d NFS200312 576 RSA-576 NFS20?? 768 ?? NFS20?? 1024 ?? SNFS/NFS

1995 2000 2005 2010

600

700

800

900

1000

Special Number Field Sieve

Least squares prediction:1024-bit SNFS factorization by 2012

1995 2000 2005 2010

400

500

600

700

Number Field Sieve

Least squares predictions: 768-bit NFS factorization by 20151024-bit NFS factorization by 2028

Goal of this workshop:

Make sure that these predictions are

• too pessimistic from a factoring point of view

• too optimistic from a cryptographic point of view

Thus, we should be able to complete a

• 1024-bit SNFS factorization well before 2012

• 768-bit NFS factorization well before 2015

by 2005?

by 2010?

• 1024-bit NFS factorization well before 2028 ?

Problem: since 1989 nothing seems to be happening!Examples of NFS related things that did (or will) not happen:

• 1994, integers can quickly be factored on a quantum computerno one knows how to build one yet

• 1999, TWINKLE opto-electronic device to factor 512-bit moduli estimates too optimistic

• 2001, Bernstein’s factoring circuits:1536 bits for cost of 512 bits new interpretation of the cost function

• 200308, TWIRL hardware siever: 1024 bits in a year for US$10Mdoes not include research and development cost

• 2004, TWIRL hardware siever: 1024 bits in a year for < US$1M

For the moment:• stuck with existing algorithms and hardware ((G)NFS & PCs)• see if we can push them even further

How do general purpose factoring methods work?To factor n, attempt to find integers x, y, x y such that

x2 y2 mod n

If n divides x2 y2, then n divides (x y)(x + y), so

n = gcd(x y, n) gcd(x + y, n) may be a non-trivial factorization

Finding such x, y based on two-step Morrison-Brillhart approach:

1. Collect data

2. Combine data

, Relation collection

, Matrix step

: allows ‘obvious’ parallelization (internet) : often centralized (Cray, broadband network)

How to solve x2 y2 mod n?

1. Relation collection: collect integers v such that

v2 mod n factors into primes < B (i.e., is B-smooth)

Need to efficiently test many integers for smoothness

2. Matrix step: select a subset of the v’s such that primes < B in

corresponding (v2 mod n)’s occur an even number of times

Need to find elements of null space of (B)(B) matrix

Matrix step not further discussed: based on reported ‘overcapacity’ assume that current parallelized block Lanczos oncurrent (and future) small broadband networks will suffice

How to find v’s such that v2mod n is smooth?

Examples

Dixon’s method:• pick v at random in {0,1,…, n1} • test v2 mod n {0,1,…, n1} for B-smoothness• repeat until > (B) different v’s have been found

Speed depends on B-smoothness probability ofnumbers of size comparable to n

Quadratic sieve:• test (v + [n])2 n for B-smoothness for small v• repeat until > (B) different v’s have been found ( v < S(B))

Speed depends on B-smoothness probability ofnumbers of size comparable to 2S(B)n

no way to take advantage of special properties of p or n

Smaller |v2mod n|: higher smoothness probabilityQuadratic sieve:

• test (v + [n])2 n for B-smoothness for small v• repeat until > (B) different v’s have been found ( v < S(B))

Speed depends on B-smoothness probability ofnumbers of size comparable to 2S(B)n (as opposed to n)

Number field sieve:• select d; select m close to n1/(d+1)

and f(X) Z[X] of degree d with f(m) 0 mod n• look at S = S(Br,Ba) integer pairs (a,b) to find co-prime ones

such that |a bm| is Br-smooth and |bdf(a/b)| is Ba-smooth• S such that: expect to find > (Br) + (Ba) ‘good’ (a,b) pairs

Speed depends on simultaneous smoothness probability ofnumbers of sizes comparable to n1/(d+1)S and fSd/2

for some n there may be an m and f with f exceptionally small

‘Good’ cases for Number Field Sieve• select d; select m close to n1/(d+1)

and f(X) Z[X] of degree d with f(m) 0 mod n• look at S = S(Br,Ba) integer pairs (a,b) to find co-prime ones

such that |a bm| is Br-smooth and |bdf(a/b)| is Ba-smooth• S such that: expect to find > (Br) + (Ba) ‘good’ (a,b) pairs

Speed depends on simultaneous smoothness probability ofnumbers of sizes comparable to n1/(d+1)S and fSd/2

for some n there may be an m and f with f exceptionally small

For those n for which f is bounded by a constant: SNFS applies to n

Example: n = 2512+1• n divides 2515+8• m = 2103 and f(X) = X5+8, then f(m) 0 mod n

In general, f cannot be expected to be bounded by a constant,f will be of size comparable to m (i.e., n1/(d+1)): NFS applies to n

SNFS versus NFS

SNFS: speed depends on simultaneous smoothness probability ofnumbers of sizes comparable to n1/(d+1)S and Sd/2

NFS: speed depends on simultaneous smoothness probability ofnumbers of sizes comparable to n1/(d+1)S and n1/(d+1)Sd/2

SNFS overall heuristic asymptotic expected runtime is

exp((1.53+o(1))(log n)1/3(loglogn)2/3), n

NFS overall heuristic asymptotic expected runtime is

exp((1.92+o(1))(log n)1/3(loglogn)2/3), n

for 1024-bit n and d = 6, difference n1/(d+1) is 147-bit number (45 digit)S = 1020: smoothness of pairs of sizes (55d,60d) versus (55d,105d)

Determining Br, Ba, and S(Br, Ba) for nTraditionally based on combination of

• guesswork (‘extrapolation’)• experience• experiments

Alternative approach for TWIRL analysis (Asiacrypt 2003):• Let P(x,B) denote probability that |x| is B-smooth and

E(Br,Ba,A,B,m,f,t) = 0.6|a| A0<bBP(abm,Br)P(bdf(a/b)/t,Ba)

(‘expected yield’, approximated using numerical integration)• For several degrees d:

• Find ‘ok-ish’ m, dth degree f (with correction t), skewness s• For several Br and Ba determine S(Br,Ba) as least S such that

E(Br,Ba,A,B,m,f,t) ((Br) + (Ba))/c

for B = (S/2s), A = sB, and ‘reasonable’ c (say, 20)• Pick d for which ‘best’ feasible Br and Ba were found

for 1024-bit n: possibly unreliable unavailable (?) infeasible

Results

a

b

product of smoothness probabilities

• Rectangular region is not at all optimal: crown shaped regions

• Realistic estimates for Br and Ba and upper bounds for factoring effort

-41012 -21012 0 21012 410120

2109

4109

6109

8109

11010

Example of non-rectangular region

crown contains points with smoothness probability E16

Resulting parameter choices

1024-bit SNFS (pessimistic estimate):Br 6.7E7, Ba 1.3E8, (Br) + (Ba) 1.2E7, S 6.4E17

1024-bit NFS:Br 3.5E9, Ba 2.6E10, (Br) + (Ba) 1.7E9, S 3E23

Comparing 1024-bit SNFS and 1024-bit NFS:

Factor base sizes: about 140 times larger

Sieving: about 5E5 times harder

Matrix: about 140 times more rows

Potential feasibility of 1024-bit SNFS does not implyfeasibility of 1024-bit NFS

Feasibility of 1024-bit SNFS

512-bit NFS:Br 1.7E6, Ba 1.7E6, (Br) + (Ba) 2.1E6, S E15

1024-bit SNFS (pessimistic estimate):Br 6.7E7, Ba 1.3E8, (Br) + (Ba) 1.2E7, S 6.4E17

Comparing 512-bit NFS and 1024-bit SNFS

Factor base sizes: about 6 times larger

Sieving: about 700 times harder

Matrix: about 6 times more rows

512-bit NFS was (very) feasible in 1999 based on Moore’s law 1024-bit SNFS feasible by 2005

Feasibility of 768-bit NFS

1024-bit SNFS:Br 6.7E7, Ba 1.3E8, (Br) + (Ba) 1.2E7, S 6.4E17

768-bit NFSBr E8, Ba E9, (Br) + (Ba) 5.6E7, S 3E20

Comparing 1024-bit SNFS and 768-bit NFS

Factor base sizes: about 5 times larger

Sieving: about 500 times harder

Matrix: about 5 times more rows

If 1024-bit SNFS is feasible, then based on Moore’s law 768-bit NFS should be feasible about 5 years later

Comparing 768-bit NFS and 1024-bit NFS

768-bit NFSBr E8, Ba E9, (Br) + (Ba) 5.6E7, S 3E20

1024-bit NFS:Br 3.5E9, Ba 2.6E10, (Br) + (Ba) 1.7E9, S 3E23

Comparing 768-bit NFS and 1024-bit NFS

Factor base sizes: about 30 times larger

Sieving: at least 1000 times harder

Matrix: about 30 times more rows

Once 768-bit NFS is feasible it will be a while (7 years?)before 1024-bit NFS is feasible

(unless someone builds TWIRL)

Summary of 512, 768, 1024 estimates 512 NFS

1024 SNFS

768 NFS 1024 NFS

6 factor base size700 effort

30 factor base size1000 effort

140 factor base size 5E5 effort

5 factor base size500 effort

(suboptimal choices: much smaller effort with larger factor bases)

Conclusion

• Factoring 1024-bit ‘special’ numbers is within reach

We should prove it is

• Factoring 768-bit RSA moduli will soon be feasibleusing tomorrow’s hardware

We should get ready

• Factoring 1024-bit RSA moduli still looks infeasibleusing currently available hardware

but it may be expected before 2020