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SarkoxedaF avatar Crypto

Finding small roots of a univariate polynomial modulo N. Don Coppersmith

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SarkoxedaF

I'm currently trying to understand the Coppersmith's method of finding small integer roots of polynomials modulo some integer. I am reading the original paper Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities by Don Coppersmith. Specifically, in section 6 he claims: enter image description here

I really can't get the part, where he says that we can compute $c_g$ values. We are given the reduced lattice basis, containing the $\vec{s}$, from which we can find the $\vec{r}$ by simply solving the equation $\vec{s} = \vec{r} M$. So why would we want to have the polynomial instead? On the other hand, if I am missing something and we really need this polynomial, how could we get it from this reduced basis?

Also I've tried to find another explanations of Coppersmith's method. But they all use different Matrices to span the lattice. I would be very grateful if someone could clarify this point for me.

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