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I'm trying to understand part of a paper. How does the difference of gcd in $\mathbb Z_p[x]$ and $\mathbb Z_q[x]$ relate to the gcd in $\mathbb Z_n[x]$? And why is the result of gcd in $\mathbb Z_n[x]$ a non-trivial non-invertible element?

For $b(x), c(x)\in\mathbb Z_n[x]$, let $\gcd_p(b(x), c(x))$ and $\gcd_q (b(x), c(x))$ be the greatest common divisor of the polynomials modulo $p$ and $q$, respectively. Where $n = p\cdot q$ and $p$ and $q$ are primes.

Proposition 1: Let $b(x), c(x)\in\mathbb Z_n[x]$. If $\deg(\gcd_p(b(x), c(x))) \neq \deg(\gcd_q (b(x), c(x)))$, then Euclid’s algorithm on $\mathbb Z_n[x]$ with input $b(x)$ and $c(x)$ yields a non-trivial non-invertible element of $\mathbb Z_n$.

Link to paper:

D. Aggarwal and U. Maurer. Breaking RSA Generically Is Equivalent to Factoring. EUROCRYPT 2009. http://eprint.iacr.org/2008/260.pdf

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  • $\begingroup$ This might be better suited to crypto.stackexchange.com (please avoid cross-posting though). $\endgroup$
    – cody
    Commented Apr 8 at 18:52

1 Answer 1

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To explain their proposition, let me recall Euclid's algorithm to compute the gcd of $b(x)$ and $c(x)$:

while c(x) ≠ 0:
     t(x) ← b(x) mod c(x)
     b(x) ← c(x)
     c(x) ← t(x)
return b(x)

To compute $b(x) \bmod c(x)$, you need to invert the leading coefficient of $c(x)$. And this element may be non-invertible (when computing in $ℤ_n[x]$), this is what the authors write (or hint more precisely).

If such a non-invertible leading coefficient does not appear during the algorithm run in $ℤ_n[x]$, Euclid's algorithm terminates without error. In this case, you can do the exact same computations $\bmod p$ and $\bmod q$: All the polynomials you manipulate have the same degree $\bmod n$, $\bmod p$ and $\bmod q$ (due to the leading coefficients being invertible, thus prime with $n$, that is not divisible by $p$ nor $q$), and the final results have the same degrees $\bmod n$, $\bmod p$ and $\bmod q$. By contrapositive, if the degrees are not the same, there must be some non-invertible leading coefficient at some point.

Finding such a non-invertible element means finding an element with a non-trivial gcd with $n$, whence a factor of $n$.

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  • $\begingroup$ Do you know why they pick h(x) to be monic polynomial? $\endgroup$
    – userg93
    Commented Apr 8 at 21:44
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    $\begingroup$ There are several reasons to do so, one being Lemma 3 on the probabilities of a random monic polynomial to be irreducible, or to have a linear factor. But most importantly I guess is the fact that taking $h$ monic over $ℤ_n$ makes it monic over $ℤ_p$ and $ℤ_q$. In "CASE 2" of their proof, I think that non-monic polynomials would be an issue for the isomorphism they use. The real requirement is probably for $h$ to have an invertible leading coefficient, so taking $h$ monic is a good solution. Also, it would not help in any manner to allow $h$ not to be monic. $\endgroup$
    – Bruno
    Commented Apr 10 at 8:06

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