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Paper: Approximate Divisor Multiples  Factoring with Only a Third of the Secret CRTExponents
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Conference:  EUROCRYPT 2022 
Abstract:  We address Partial Key Exposure attacks on CRTRSA on secret exponents $d_p, d_q$ with small public exponent $e$. For constant $e$ it is known that the knowledge of half of the bits of one of $d_p, d_q$ suffices to factor the RSA modulus $N$ by Coppersmith's famous {\em factoring with a hint} result. We extend this setting to nonconstant $e$. Somewhat surprisingly, our attack shows that RSA with $e$ of size $N^{\frac 1 {12}}$ is most vulnerable to Partial Key Exposure, since in this case only a third of the bits of both $d_p, d_q$ suffices to factor $N$ in polynomial time, knowing either most significant bits (MSB) or least significant bits (LSB). Let $ed_p = 1 + k(p1)$ and $ed_q = 1 + \ell(q1)$. On the technical side, we find the factorization of $N$ in a novel twostep approach. In a first step we recover $k$ and $\ell$ in polynomial time, in the MSB case completely elementary and in the LSB case using Coppersmith's latticebased method. We then obtain the prime factorization of $N$ by computing the root of a univariate polynomial modulo $kp$ for our known $k$. This can be seen as an extension of HowgraveGraham's {\em approximate divisor} algorithm to the case of {\em approximate divisor multiples} for some known multiple $k$ of an unknown divisor $p$ of $N$. The point of {\em approximate divisor multiples} is that the unknown that is recoverable in polynomial time grows linearly with the size of the multiple $k$. Our resulting Partial Key Exposure attack with known MSBs is completely rigorous, whereas in the LSB case we rely on a standard Coppersmithtype heuristic. We experimentally verify our heuristic, thereby showing that in practice we reach our asymptotic bounds already using small lattice dimensions. Thus, our attack is highly practical. 
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BibTeX
@inproceedings{eurocrypt202231860, title={Approximate Divisor Multiples  Factoring with Only a Third of the Secret CRTExponents}, publisher={SpringerVerlag}, author={Alexander May and Julian Nowakowski and Santanu Sarkar}, year=2022 }