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RSA and a higher degree diophantine equation
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Abstract: | Let $N=pq$ be an RSA modulus where $p$, $q$ are large primes of the same bitsize. We study the class of the public exponents $e$ for which there exist an integer $m$ with $1\leq m\leq {\log{N}\over \log{32}}$ and small integers $u$, $X$, $Y$ and $Z$ satisfying $$(e+u)Y^m-\psi(N)X^m=Z,$$ where $\psi(N)=(p+1)(q-1)$. First we show that these exponents are of improper use in RSA cryptosystems. Next we show that their number is at least $O\left(mN^{{1\over 2}+{\a\over m}-\a-\e}\right)$ where $\a$ is defined by $N^{1-\a}=\psi(N)$. |
BibTeX
@misc{eprint-2006-21586, title={RSA and a higher degree diophantine equation}, booktitle={IACR Eprint archive}, keywords={public-key cryptography / RSA cryptosystem, Continued fractions, Coppersmith's algorithm}, url={http://eprint.iacr.org/2006/093}, note={ nitaj@math.unicaen.fr 13216 received 9 Mar 2006}, author={Abderrahmane Nitaj}, year=2006 }