Let $\phi=1103481364240094378710324387523817552607307673089944349316644000826050949$ and $N=1050467212358431669174928621845479661$. I want to generate an integer $e=N^{\alpha}$ of $59$ bits and an integer $d=N^{\delta}$ satisfying $ed\equiv 1\bmod \phi$ 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 $$\alpha+\delta > \log_N(\phi)\approx 2.000.$$
Asked: 2024-06-23 01:40:07 +0100
Seen: 155 times
Last updated: Jun 23
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.
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