factoring

Factoring
Factoring
 Security of RSA algorithm depends on
(presumed) difficulty of factoring
o Given N = pq, find p or q and RSA is broken
o Rabin cipher also based on factoring
 Factoring like “exhaustive search” for RSA
 Lots of interest/research in factoring
 What are best factoring methods?
o How does RSA “key size” compare to symmetric
cipher key size?

Factoring
Factoring Methods
 Trial division
o Obvious method but not practical
 Dixon’s algorithm
o Less obvious and much faster
 Quadratic sieve
o Refinement of Dixon’s algorithm
o Best algorithm up to about 110 decimal digits
 Number field sieve
o Best for numbers greater than 100 digits
o We only briefly mention this algorithm

Factoring
Trial Division
 Given N, try to divide N by each of
2,3,5,7,9,11,…,sqrt(N)
 As soon as a factor found, we are done
o So, expected work is about sqrt(N)/2
 Improvement: try only prime numbers
 Work is then on order of π(N)
o Where π(N) ≈ N/ln(N) is number of primes up to N

Factoring
Congruence of Squares
 We want to factor N = pq
 Suppose we find x,y such that N = x2
− y2
 Then N = (x − y)(x + y), have factored N
 More generally, congruence of squares…
 Suppose x2
= y2
(mod N)
 Then x2
− y2
= kN for some k
 Which implies (x − y)(x + y) = kN

Factoring
 Suppose x2
= y2
(mod N)
 Then (x − y)(x + y) = kN
 Implies (x − y) or (x + y) is factor of N
o Or x − y = k and x + y = N (or vice versa)
 With probability at least 1/2, we obtain a
factor of N
o If so, gcd(N, x − y) or gcd(N, x + y) factors N
o And the gcd is easy to compute

Factoring
 For example 102
= 32
(mod 91)
 That is, (10 − 3)(10 + 3) = 91
o Factors of 91 are, in fact, 7 and 13
 Also, 342
= 82
(mod 91)
o Then 26⋅42 = 0 (mod 91) and we have
gcd(26,91) = 13 and gcd(42,91) = 7
 In general, gcd is necessary

Factoring
 Find congruence of squares: x2
= y2
(mod N)
and we can likely factor N
 How to find congruence of squares?
 Consider, for example,
412
= 32 (mod 1649) and 432
= 200 (mod 1649)
 Neither 32 nor 200 is a square
 But 32⋅200 = 6400 = 802
 Therefore, (41⋅43)2
= 802
(mod 1649)

Factoring
 Can combine non-squares to obtain a
square, for example
32 = 25
⋅50
and 200 = 23
⋅52
 And 32⋅200 = 28
⋅52
= (24
⋅51
)2
 We obtain a perfect square provided each
exponent in product is even
 Only concerned with exponents and only
need consider even or odd, i.e., mod 2

Factoring
 Number has an exponent vector
 For example, first element of vector is
power of 2 and second power of 5
 Then
 And

Factoring
 Mod 2 exponent vector of product 200⋅32
is all zero, so perfect square
 Also, this vector is sum (mod 2) of vectors
for 200 and 32
 Any set of exponent vectors that sum to
all-zero, mod 2, gives us a square
 We need to keep vectors small
o Only allow numbers with “small” prime factors

Factoring
 Choose bound B and primes less than B
o This is our factor base
o For technical reasons, include “−1” in factor base
 A number that factors completely over the
factor base is B-smooth
 Smooth relations factor over factor base
 Restrict our attention to B-smooth relations
o Good: Exponent vectors are small
o Bad: Harder to find relations

Factoring
Example
 Want to factor N = 1829
 Choose bound B = 13
 Choose factor base −1,2,3,5,7,11,13
 Look at values in −N/2 to N/2
 To be systematic, we choose sqrt(kN) and
sqrt(kN) for k = 1,2,3,4
 And test each for B-smoothness

Factoring
Example
 Compute
 All are B-smooth except 602
and 752

Factoring
Example
 Obtain
exponent
vectors

Factoring
Example
 Find collection of exponent vectors that
sum, mod 2, to zero vector
 Vectors corresponding to 422
, 432
, 612
and
852
work:

Factoring
Example
 Implies that
(42⋅43⋅61⋅85)2
= (2⋅3⋅5⋅ 7⋅13)2
(mod 1829)
 Simplifies to 14592
= 9012
(mod 1829)
 Since 1459 − 901 = 558, we find factor
of 1829 by gcd(558,1829) = 31
 Easily verified 1829 = 59⋅31

Factoring
Example
 A systematic way to find set of
vectors that sum to zero vector…
 In this example, want x0,x1,…,x5
 This is a basic linear algebra problem

Factoring
Linear Algebra
 Suppose n elements in factor base
o Factor base includes “−1”
o Then matrix on previous slide has n rows
o Seek linearly dependent set of columns
 Theorem: If matrix has n rows and n + 1 or
more columns then a linearly dependent set
of columns exists
 Therefore, if we find n + 1 or more smooth
relations, we can solve the linear equations

Factoring
Dixon’s Algorithm
1. To factor N: select bound B and factor
base with n−1 primes less than B and “−1”
2. Select r, compute y = r2
(mod N)
Number r can be selected at random
3. If y factors completely over factor base,
save mod 2 exponent vector
4. Repeat steps 2 and 3 to obtain n+1 vectors
5. Solve linear system and compute gcd

Factoring
Dixon’s Algorithm
 If factor base is large, easier to find
B-smooth relations
o But linear algebra problem is harder
 Relation finding phase parallelizable
o Linear algebra part is not
 Next, quadratic sieve
o An improved version of Dixon’s algorithm

Factoring
Quadratic Sieve
 Quadratic sieve (QS) algorithm
o Dixon’s algorithm “on steroids”
 Finding B-smooth relations beefed up
 As in Dixon’s algorithm
o Choose bound B and factor base of
primes less than B
o Must find lots of B-smooth relations

Factoring
Quadratic Sieve
 Define quadratic polynomial
Q(x) = (sqrt(N) + x)2
− N
 The is the “quadratic” in QS
 Use Q(x) to find B-smooth values
o For each x ∈ [−M,M] compute y = Q(x)
o Mod N, we have y = z2
, where z = sqrt(N) + x
o Test y for B-smoothness
o If y is smooth, save mod 2 exponent vector

Factoring
Quadratic Sieve
 Advantage of QS over Dixon’s is that
by using Q(x) we can sieve
 What is sieving? Glad you asked…
 First, consider sieve of Eratosthenes
o Used to sieve for prime numbers
 Then modify it for B-smooth numbers

Factoring
Sieve of Eratosthenes
 To find prime numbers less than M
 List all numbers 2,3,4,…,M−1
 Cross out all numbers with factor of 2,
other than 2
 Cross out all numbers with factor of 3,
other than 3, and so on
 Number that “fall thru” sieve are prime

Factoring
 To find prime numbers less than 31…
2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
— — — —
—————
— — — — —
⁄ ⁄
⁄ ⁄ ⁄
⁄ ⁄ ⁄ ⁄

|
| ||| ∩
 Find that primes less than 31 are
2,3,5,7,11,13,17,19,23 and 29

Factoring
 This sieve gives us primes
 But also provides info on non-primes
 For example, 24 marked with “ ” and
“ ” so it is divisible by 2 and 3
 Note: we only find that 24 is divisible
by 2, not by 4 or 8
—
⁄

Factoring
Sieving for Smooth Numbers
 Instead of crossing out, we divide by
the prime (including prime itself)
2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
1 1 3
2 5 3
7 4 9 5
 All 1s represent 7-smooth numbers
 Some non-1s also 7-smooth
o Must divide by highest powers of primes
1 1
1 2
5 1
1
1
1 2

Factoring
Quadratic Sieve
 QS uses similar sieving strategy as on
previous slide
o And some computational refinements
 Suppose p in factor base divides Q(x)
o Then p divides Q(x + kp) for all k ≠ 0 (homework)
o That is, p divides Q of …,x−2p,x−p,x,x+p,x+2p,…
o No need to test these for divisibility by p
 This observation allows us to sieve

Factoring
Quadratic Sieve
 One trick to speed up sieving
 If Q(x) divisible by p, then Q(x) = 0 (mod p)
 Defn of Q implies (sqrt(N) + x)2
= N (mod p)
 Square roots of N (mod p), say, sp and p − sp
o Let x0 = sp − sqrt(N) and x1 = p − sp − sqrt(N)
o Then Q(x0) and Q(x1) divisible by p
o Implies Q(x0 + kp) and Q(x1 + kp) divisible by p
 Efficient algorithm for these square roots

Factoring
Quadratic Sieve
 How to sieve for B-smooth relations
 Array: Q(x) for x = −M,−M+1,…, M−1,M
 For first prime p in factor base
o Generate all x ∈ [−M,M] for which p divides Q(x)
(as described on previous slide)
o For each, divide by highest power of p
o For each, store power, mod 2, in vector for x
 Repeat for all primes in factor base
 Numbers reduced to 1 are B-smooth

Factoring
Quadratic Sieve
 Linear algebra phase same as Dixon’s
 Sieving is the dominant work
 Lots of tricks used to speed up sieving
o For example, “logarithms” to avoid division
 Multiple Polynomial QS (MPQS)
o Multiple polynomials of form (ax+b)2
− N
o Can then use smaller interval [−M,M]
o Yields much faster parallel implementations

Factoring
Sieving Conclusions
 QS/MPQS attack has two phases
 Distributed relation finding phase
o Could recruit volunteers on Internet
 Linear equation solving phase
o For big problems, requires a supercomputer
 Number field sieve better than QS
o Requires 2 phases, like QS
o Number field sieve uses advanced math

Factoring
Factoring Algorithms
 Work to factor N = 2x
 Last column measures “bits” of work
 Symmetric cipher exhaustive key
search: x bit key is x−1 bits of work

Factoring
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Factoring Algorithms
 Comparison of work factors
 QS best to
390 bit N
o 117 digits
 390-bit N is
as secure as
60-bit key

Factoring
Factoring Conclusions
 Work for factoring is subexponential
o Better than exponential time but worse
than polynomial time
o Exhaustive key search is exponential
 Factoring is active area of research
o Expect to see incremental improvement
 Next, discrete log algorithms…

factoring

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factoring