Let ϕ=1103481364240094378710324387523817552607307673089944349316644000826050949 and N=1050467212358431669174928621845479661. I want to generate an integer e=Nα of 59 bits and an integer d=Nδ satisfying ed≡1mod and \delta<2-\sqrt{\alpha}. How to do this rapidly ?
This question has nothing to do with Sage, and furthermore what is asked is impossible. We have \log_N(2^{58}) \leq \alpha <\log_N(2^{59}) and \delta < 2-\sqrt{\log_N(2^{58})}. Hence, \alpha+\delta < 2 + \log_N(2^{59}) - \sqrt{\log_N(2^{58})}\approx 1.797. On the other hand, since ed > \phi, we have \alpha+\delta > \log_N(\phi)\approx 2.000.
thank you so much for this observation, what if we take e of 150 bits ? since the inequalities hold
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Last updated: Jun 23 '24
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Is \alpha a real number? Is \delta a real number? Is the situation relevant in real life? If yes, please give references. And moreover, what has this to do with sagemath? Why not https://puzzling.stackexchange.com ?
Yes alpha and delta are small real numbers