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I know that if $(n,e)$ is the public key in RSA and we also know $d$ the private key, then there is a probabilistic algorithm to factor $n$. I'm reading the proof from Fact 1 on here.

I understand the proof until it says

"A straightforward argument shows that if $g$ is chosen at random from $\mathbb{Z}^*_N$ then with probability at least $1/2$ (over the choice of g) one of the elements in the sequence $g^{k/2}, g^{k/4}, \cdots ,g^{k/2^t} \mod N$ is a square root of unity that reveals the factorization of N."

Why is that last paragraph true?

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  • $\begingroup$ Does this help you Is knowing the private key of RSA equivalent to the factorization of N? $\endgroup$ – kelalaka Mar 20 at 20:23
  • $\begingroup$ @kelalaka: I wouldn't expect it would help - an answer states the algorithm, but doesn't say why it works with probability $\ge 1/2$ $\endgroup$ – poncho Mar 20 at 20:26
  • $\begingroup$ @poncho last sentence of the 111's answer? $\endgroup$ – kelalaka Mar 20 at 20:29
  • $\begingroup$ @kelalaka: I missed that. However, that might not satisfy Eparoh; at the very least, I see no immediate justification for the claim that "with equal probability since x is random because g is random". In particular, if N is prime, this is not true (hence any proof would need to rely on the compositeness of N) $\endgroup$ – poncho Mar 20 at 20:31
  • $\begingroup$ Yes, I don't really get it from that last sentence :/ $\endgroup$ – Eparoh Mar 20 at 20:34
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Ok, lets see if we can go through one line of reasoning, and see if that makes sense.

If we have the sequence:

$$g^{k/2^t} \bmod N, g^{k/2^{t-1}} \bmod N, …, g^{k/2} \bmod N, g^k \bmod N$$

We know that the last entry evaluates to 1, and we are looking at the probability (for random $g$) that the entry immediately before the first 1 is a nontrivial squareroot of 1.

Now, we can consider the sequences modulo $p$ and $q$, namely

$$g^{k/2^t} \bmod p, g^{k/2^{t-1}} \bmod p, …, g^{k/2} \bmod p, g^k \bmod p$$

$$g^{k/2^t} \bmod q, g^{k/2^{t-1}} \bmod q, …, g^{k/2} \bmod q, g^k \bmod q$$

Now, we might not be able to compute these values before we know the factorization, but that's OK, we're just considering them.

Both sequences end in 1, and consist of a (possibly empty) sequence of non-1 values, followed by all 1 values. In addition, $g^{k/2^i} \bmod N$ is a nontrivial squareroot of 1 if $g^{k/2^i} \bmod p$ is 1 and g^{k/2^i} \bmod q$ is the last non-1 value (or versa-vica). In other words, we'll have a nontrivial squareroot if the two sequences turn into a 1 value at different positions in their sequences.

So, the question is: what is the probability bounds on that?

Now, if $g$ is a random (equiprobable) value in $\mathbb{Z}_N^*$, then $g \bmod p$ and $g \bmod q$ are equiprobable and independently distributed values in $\mathbb{Z}_p^*$ and $\mathbb{Z}_q^*$; hence both sequences are selected from random starting values.

And, half of the values of $g$ are quadratic residues modulo $p$ and half are quadratic nonresidues modulo $p$. For a quadratic nonresidue, the mod-p sequence will become a one at step $t_p$ (where $p-1 = 2^{t_p} z_p$ for odd $z_p$; for a quadratic residue, it will be come one at an earlier step. Hence, the maximum probability for any particular step will be 0.5.

The same logic holds for $q$.

Now, $g \bmod p$ and $g \bmod q$ are independent variables, and so the maximum coincidental probability will occur if both chains have two stages with probability 0.5 each; in that case, the probability that they are difficult will be $1 - (0.5 \cdot 0.5 + 0.5 \cdot 0.5) = 0.5$

This argument could be tightened up a bit; however it should be enough to give you some insight...

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  • $\begingroup$ I don't get the end of the reasoning. I don't know why for quadratic nonresudues modulo p the sequence becomes a one at that $t_p$. I also don't understand what do you mean by the maximum probability for any particular step will be 0.5. $\endgroup$ – Eparoh Mar 20 at 22:23
  • $\begingroup$ @Eparoh: if the sequence became 1 earlier than step $t_p$, then we have $(g^k)^{2^{t_p-1}} = 1$, that is, $h^{(p-1)/2} = 1$ (for $h = g^{k2^{t_p}/(p-1)}$ (and $k2^{t_p}/(p-1)$ is an odd integer). If this is the case, then $h$ is a nonQR, and if that's the case, then $g$ is one as well. Similarly, we can show that if $g$ is a QR, then it will become 1 at a step previous to the last one. $\endgroup$ – poncho Mar 21 at 3:16
  • $\begingroup$ @Eparoh: as for the maximum probability of a step being 0.5, well, I believe that the probability of the last step is 0.5; that implies that the maximum probability of any other step must be, at most, 0.5 (as the probabilities must sum to 1); hence we have 0.5 as the max probability of any step $\endgroup$ – poncho Mar 21 at 3:19
  • $\begingroup$ But then, all the sequences with $g$ a quadratic residue turn into $1 \mod p$ at the same point $t_p-1$, and if $g$ is a nonquadratic residue, it turns at point $t_p$. The same goes for $q$. Now, to get a non trivial square root of one we need that $t_p=t_q+1$ or $t_p=t_q-1$ but why should these values be related that way in the first place? $\endgroup$ – Eparoh Mar 21 at 10:42
  • $\begingroup$ @Eparoh: no, to get a nontrivial square root, we need that the two sides turn into 1 at different steps. For a random $g \bmod p$, the $p$ side becomes 1 at step $t_p$ with probability 0.5 (because $g \bmod p$ is a nonQR with that probability), and it becomes 1 at some earlier step with probability 0.5. Hence, the probability that the $p$ side turns into 1 at some specific step is, at most, 0.5. Now, the $q$ side will become 1 at some specific step (which is independent of the $p$ side) No matter what step that happens to be, the $p$ side will happen to be that value with prob at most 0.5 $\endgroup$ – poncho Mar 21 at 19:52

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