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Notes, Chapter 15

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15 Factoring and computing discrete logarithms
----------------------------------------------

In this chapter we survey algorithms for factoring large numbers, in
particular products of equal sized primes, and for computing discrete
logarithms. These algorithms or rather their runtime provide lower
bounds on the necessary key lengths for public key cryptography for it
relies on the difficulty of these problems.

For the sections about factoring we let N=pq be a product of two
roughly equal sized primes and n = O(log(N)) be the length of N in
binary. The task is to find p,q from N. 

For the sections about discrete logarithms let G be a cyclic group of
order q and g,y:G with g a generator. The task is to find x:Z_q such
that g^x=y. 

Trial division
- - - - - - - -

In order to factor N=pq, i.e. to find p or q given N, we can try out
divisors d=1...sqrt(N). The runtime of this is polynomial in sqrt(N),
hence takes time exponential in n. Once n>120, hence ||sqrt(N)||=~=60
this is no longer feasible.


Pollard's p-1 method
- - - - - - - - - - -

Pollard's p-1 method may help when p-1 is a product of powers of small
primes. We let p_1,...p_k be the first k primes and construct

              B_k = prod_{i=1..k}p_i^{n / ||p_i||} 

The exponent n/||p_i|| is chosen so that if some power of p_i divides
p-1 then that power is at most n/||p_i||. As a result, if p-1 is a
product of powers of primes from among the set {p_1..p_k} then p-1 |
B_k. If, in addition, q-1 does *not* divide B_k then we can find out p
as we show shortly. In order that q-1 does *not* divide B_k we can
repeatedly execute the method with increasing values for k hoping that
at some point one of the two prime factors "works" and the other does
not.

OK so, now let B any number so that p-1 | B and q-1 does not divide
B. For any x it holds that x^B mod p = 1 whereas with constant
probability x^B mod q =/= 1. The former holds since the order of Z_p^*
is p-1 and B is a multiple thereof. The latter holds since when x is a
generator of Z_q^* then x^B mod q is not 1 and the proportion of those
generators is about 50% (not hard to see or use Thm B.15 of KL).

Now let y = x^B-1 mod N. We have y mod p = 0 and y mod q =/= 0. Thus,
p divides y and q does not divide y so gcd(y,N)=p.

The resulting algorithm is now as follows:

for k=1,2,3,... do the following until either a factor of N is found
or computing resources are exhausted.
   Put B = B_k
   for a certain number of times do the following
        x<-Z_N
        d = gcd(x^B-1 mod N,N)
	if d=/=1 then a factor of N has been found.



Pollard's rho method
- - - - - - - - - - -

This algorithm is based on the observation that once we have two
distinct values x,y:Z_N such that x mod p = y mod p then p =
gcd(x-y,N). Thus, factoring amounts to finding a collision in the
"hash function" H(x) = x mod p. Notice that while we cannot compute
this function we can detect whether x,y constitute a collision by gcd
computation.

Now, by the birthday paradox, a random set of size S := 2*sqrt(p) =
O(N^{1/4}) contains such a collision with likelihood 50%. Finding it,
requires S^2 collision tests thus, when implemented naively, this
method is no better than trial division.

However, the constant space method for collision detection described
earlier only has a runtime of O(S), but of course, as already
mentioned, we cannot compute H(). However, we can replace it with any
function F which has the property that x = y (mod p) implies F(x)=F(y)
(mod p) and which is such that for random x0 the set
{x_0,F(x_0),F(F(x_0)),..., F^S(x_0)} is also sufficiently close to a
random set so that it is likely to contain a collision. The function
F(x)=x^2 + b for some b =/= 0, -2 mod N is believed to have this
property and at any rate behaves well in practice. Just as in the
earlier algorithm we have that F^i(x0) = F^j(x0) (mod p) implies
F^k(x0) = F^{2k}(x0) (mod p) for some k.

So, the set of the first S iterates of F() starting from random x0 is
likely to contain a collision. But since once x,y form a collision so
do F(x), F(y), it suffices to search for a collision among the pairs
F^k(x0), F^{2k}(x0) which leads to the following algorithm:

b <- Z_N \ {0,-2}
F(x) = x^2+b mod N

x <- Z_N
y = x
for k=1..2*sqrt(N) do
  x = F(x); y=F(F(x))
  if gcd(x-y,N) != 1 a factor is found. 

repeat the whole procedure until either a factor is found or resources
are exhausted.

Assuming that the function F() has the required properties (which has
not been proved to this date!) the runtime of this algorithm is
proportional to N^{1/4} which is still exponential in n but requires
one to double n if one wants to stay immune against this
method.


The quadratic sieve method
- - - - - - - - - - - - - -

The starting point for the most efficient factoring algorithms to this
date is the following observation made by Pierre de Fermat:

If we succeed in expressing N as the difference of two
squares N=x^2-y^2 then N=(x+y)(x-y) so we have factored N.

Fermat suggested to try out successive values for x starting at
sqrt(N) and for each of those checking whether x^2-N is a square.

The first improvement one can make on this observation is that it
suffices that N divides x^2 - y^2 for if kN = (x+y)(x-y)
then---assuming that x != (+/-)y mod N---the factors p and q of N
must straddle the product so that gcd(x-y,N) or gcd(x+y,N) will be
different from 1.

Thus, we seek x,y:Z_N such that x^2-y^2 = 0 (mod N) and x !=y, x!=-y mod N. 

Notice that by the Chinese Remainder Theorem each square has four
roots modulo N, so given x, two out of the four roots of x^2 are good.

Now, to find such x,y more efficiently we proceed as follows. We
select a set B of small primes and try out various values x>sqrt(N) so
that x^2 mod N can be factored (because after reduction mod N it is
sufficiently small) and its factors are in B. In this way, we find
numbers x_1 .. x_l such that each x_i^2 mod N is expressed as a
product of primes in B.

With each of these numbers x_i we associate a B-indexed 0,1 vector
whose entry at p:B is the exponent of p in the factorisation of x_i
modulo 2.

For example, if B={2,3,5,11} and x_1^2 mod N is 2^5*3^2*11^2 then the
associated vector is (1 0 0 0).

If l exceeds the size of k then there is a (mod 2) linear combination
of these vectors which results in the all zero vector. But multiplying
the corresponding x_i together then yields a square because all prime
factors have even exponents.

Continuing the example, if x_2^2 mod N is 2^15*11, i.e. (1 0 0 1), and
x_3^2 mod N is 11^13, i.e. (0 0 0 1) then, since 1000+1001+0001=0000,
we obtain the following identity mod N:

                      (x_1*x_2*x_3)^2 = (2^10 * 3 * 11^8)^2

and, heuristically, we may assume that there is a 50% chance for this
identity to be nontrivial in the sense that it yields a good pair of
square roots.

Summing up, we build up a large basis of values x whose squares mod N
can be completely factored, then compute a linear combination (mod 2)
of the zero vector from the corresponding B-indexed vectors and check
whether the resulting identity of squares mod N is nontrivial which
then yields the desired factor.

We now study the problem of finding discrete logarithms in a cyclic
group G as above. 


The Pohlig-Hellman algorithm
- - - - - - - - - - - - - - -

If we have a factorisation of the group order q as prod_{i=1}^n q^i
with the q_i relatively prime (gcd = 1) then, if we have values x_i
such that

                 (g^{q/q_i})^x_i = y^{q/q_i}

we can find x by solving (using the Chinese Remainder Theorem) the
system of modular equations

                          x mod q_i = x_i

Since g^{q/q_i} generates a subgroup of order q_i which is easier if
the q_i are small. It remains to see why x satisfies the above system
of equations. Indeed, if g^x=y then g^{q/q_i}^x = y^{q/q_i} so x mod
q_i solves the discrete log problem of order q_i.

It remains to see how to deal with groups whose order is a prime power
q=p^f. In this case, one can expand the unknown x as

     x = x_0 + x_1 * p + x_2 * p^2 + ... + x_{f-1} p^{f-1}

Now, by exhaustive search or using any other algorithm we can
determine x_0:Z_p such that g^{q/p}^{x_0} = y^{q/p} (mod p). Once we
know x_0 we can then recursively solve (g^p)^{x'} = y/g^{x_0} which
yields x_1,... The details are left as an exercise.


The giant-step baby-step algorithm
- - - - - - - - - - - - - - - - - - 

Here the idea is that the unknown x can be decomposed as x =
t*[sqrt(q)] + s for some s <sqrt(q).  Here [sqrt(q)] stands for the
greatest integer <= sqrt(q). We then precompute a table of the values
y_t := g^{t*[sqrt{q}]} for t=0..sqrt(q), this are the giant
steps. Now, for each s we look for (using binary search or similar)
for a t such that

         g^{t*[sqrt(q)]} = y*g^{-s}

Once we have found it we can output x = s*[sqrt(q)]+t. The runtime of
this method is O(sqrt(q)) which again essentially halves the "bits of
security".


The index calculus method
- - - - - - - - - - - - - 

Unlike the earlier methods the index calculus makes special
assumptions on the group G for which to solve the discrete logarithm
problem. Namely, it must be the group Z_p^* for some prime p.  It can
be generalised to the group of invertible elements in a finite field,
but does not work in the case of ellipic curves, for instance.

So, let us assume that G=Z_p^* and that g,y:Z_p^*. As before, we seek
x such that g^x=y. As before, we write q=p-1 for the group order. 

Let us write log_g(z) for the discrete logarithm of some value z:Z_p^*
so that g^{log_g(z)} = z. The idea is to precompute a table of such
logarithms. To that end, we choose values x_1..x_l such that g^{x_i}
can be factored modulo p into small primes from a previously fixed set
B={p_1,...,p_k}.From

        g^{x_j} = prod_{i=1}^k p_i^{e_{j,i}} (mod p)

we then obtain

          x_j = sum_{i=1}^k e_{j,i} * log_g(p_i) (mod q)

Now, assuming that l in relation to k was chosen large enough this
system can be olved for the unknowns log_g(p_i). We thus have obtained
the desired "table of logarithms".

It allows us to easily compute the discrete logarithm of any number
w:Z_p^* which can be factored into primes from B: if w = prod_{i=1}^k
p_i^{e_i} then log_g(w) = sum_{i=1}^k e_i * log(p_i) (mod q).

If y itself has that property we are obviously done. Otherwise, we can
search for a value With the latter at hand we can then --- with luck
--- take the logarithm of our y as follows: we choose some value z:Z_q
at random such that y*g^z can be so factored and thus taken a
logarithm of. This helps, since log_g(y) = log_g(y*g^z)-z.

Using advanced number-theoretic methods one can analyse the
probabilities that the various random choices in this method succeed
and thus derive an expected runtime that is subexponential (in
||q||). Indeed, the index calculus method is currently the best known
method for computing discrete logarithms.

We also remark that it is the possibility of factoring elements of G
into primes what makes this algorithm possible. For other groups, in
particular the multiplicative group of a finite field, analogous
notions of "primes" can be found, but for groups derived from elliptic
curves no such notion has been discovered and hence these groups are
immune against index calculus making them particularly promising for
cryptography. Having said that, taking discrete logarithms in a group
Z_p^* where p has 1000+ digits remains unfeasible.

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