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SNFS versus (G)NFS and the feasibility of factoring a 1024-bit number with SNFS Arjen K. Lenstra

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and the feasibility of

factoring a 1024-bit number with SNFS

Arjen K. Lenstra

Citibank, New York

Technische Universiteit Eindhoven

General purpose methods

Take advantage of special properties of p

Cannot take advantage of any properties of p

, but possibly of n

All based on the same approach

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)

Relevant for RSA

CFRAC, Dixon’s algorithm

Linear sieve, Quadratic sieve

Number field sieve (NFS)

this talk

Variant: SNFS, takes advantage of special form of n

Factoring algorithms (to find factor p of n)

when # bits what how

199006 512 F9 = 2512+1 SNFS

199406 534 (121511)/11 SNFS

199407 384 p(11887) NFS

199411 392 p(13171) NFS

199604 429 RSA-130d NFS

199809 615 12167+1 SNFS

199902 462 RSA-140d NFS

199904 698 (10211 1)/9 SNFS

199908 512 RSA-155d NFS

200011 773 2773+1 SNFS

200201 522 c158d of 2953+1 NFS

200301 809 M809 SNFS

200303 529 RSA-160d NFS

200312 576 RSA-576 NFS

20?? 768 ?? NFS

20?? 1024 ?? SNFS/NFS

Least squares predictions:

768-bit NFS factorization by 2015

1024-bit NFS factorization by 2028

- 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 computer
- no 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$10M
- does 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

To factor n, attempt to find integers x, y, x y such that

- x2 y2 mod n

If n divides x2y2, then n divides (xy)(x + y), so

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

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

, Relation collection

, Matrix step

: allows ‘obvious’

parallelization (internet)

: often centralized

(Cray, broadband network)

How do general purpose factoring methods work?

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

How to solve x2 y2 mod n?

- Matrix step not further discussed: based on reported ‘overcapacity’
- assume that current parallelized block Lanczos on
- current (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 of
- numbers 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 of
- numbers of size comparable to 2S(B)n

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

Smaller |v2mod n|: higher smoothness probability

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

- Traditionally based on combination of
- guesswork (‘extrapolation’)
- experience
- experiments

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

- 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 determineS(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

product of smoothness probabilities

a

b

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

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

Results

Example of non-rectangular region

crown contains points with smoothness probability E16

- 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 imply
- feasibility of 1024-bit NFS

- 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

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

700 effort

5 factor base size

500 effort

140 factor base size

5E5 effort

30 factor base size

1000 effort

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

- Factoring 1024-bit ‘special’ numbers is within reach
- We should prove it is
- Factoring 768-bit RSA moduli will soon be feasible
- using tomorrow’s hardware
- We should get ready
- Factoring 1024-bit RSA moduli still looks infeasible
- using currently available hardware

- but it may be expected before 2020

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