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Let say we have $N=pq$ (Both $p$ and $q$ are safe primes, meaning $\gcd(p-1,q-1)=2$)

let's assume $e$ is an odd number which can be efficiently factored and assume that we know the $e$'th root of 1 modulo $N$ - call it $r$ - as well es $e$ and its prime-factorization. There is a claim that knowing the $e$'th root of 1 modulo $N$ we can find such $y>1$ and an odd prime $p_0$ such that $y^{p_0} = 1 \bmod N$.

How can we calculate $y$ and $p_0$?

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  • $\begingroup$ brute force search using c code? $\endgroup$ – super Feb 11 at 18:02
  • $\begingroup$ I suppose there must be a mathematical way and algorithm to calculate such values, and no brute force search ;) $\endgroup$ – arman haghighi Feb 11 at 18:03
  • $\begingroup$ Do we know $e$? If we do, and there's a non-trivial root of unity, then one of the prime factors $f$ of $e$ if $(p-1)/2$ or $(q-1)/2$, hence we can factor $N$, which makes the problem easier. $\endgroup$ – fgrieu Feb 11 at 18:39
  • $\begingroup$ @fgrieu: if we don't know $e$, then it might not be doable (e.g. if $e = (p-1)(q-1)$, then all values r.p. to $N$ are root of unity (although one could claim it fails the 'easy to factor' criteria). On the other hand, once we factor $N$, it's straight-forward from there... $\endgroup$ – poncho Feb 11 at 18:47
  • $\begingroup$ we know e and its factors $\endgroup$ – arman haghighi Feb 11 at 19:01
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If we know $e$, then it's easy.

If don't have to know the factors, but it does make it easier - as fgrieu pointed out, either $(p-1)/2$ or $(q-1)/2$ will appear as one of the prime factors, that will allow us to immediately recover $p, q$

Once we have that, then we know that we want:

$$y^{p_0} \equiv 1 \pmod p$$ $$y^{p_0} \equiv 1 \pmod q$$

If we have $y \equiv 4 \pmod p$ (or any other quadratic residue) and $p_0 = (p-1)/2$ (which is prime), then we satisfy the first equation.

If we have $y \equiv 1 \pmod q$, then we satisfy the second equation.

It's easy to find a $y$ that meets both modulii (as $p, q$ are relatively prime) using the Chinese Remainder Theorem, and so that's our answer.

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We know by assumption that $$r^e\equiv 1\pmod N$$ then if $e$ is an odd prime, we're done and have $y=r, p_0=e$.
However if $e$ is a composite, it must only have odd prime factors - otherwise its factorization would contain $2$ which would make it even, so we arbitrarily pick any prime factor of $e$ and call it $p_0$ such that $e=e'p_0$. Then we're done with $y=r^{e'},p_0=p_0$ as $$r^{e}=r^{e'p_0}=\left(r^{e'}\right)^{p_0}\equiv 1\pmod N.$$

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