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4

A couple things: This article is two years old, so take its predictions with a grain of salt. In the two years that have elapsed, the predicted advances have not materialized, and there is little indication they will soon. The core of those arguments was Joux's 2013 result on the discrete logarithm problem in finite fields of small characteristic. Those ...

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No, in the end the private exponent $d$ is just a number within $0..N$ where $N$ is the modulus. It depends on $N$ what the chance is that the first bit is one, but in more likely to be valued $0$ than $1$ (given that it is well distributed, you would expect it to be $0$ around $\frac23$ of the time). If you generate enough private keys you'll even see ...

1

$d$ must indeed be an integer. To calculate $d$ you need to calculate $d=e^{-1}\bmod{\phi(n)}$ which is called the modular multiplicative inverse of $e\bmod{\phi(n)}$. For $d$ be computable you need to ensure that $$\gcd(e,\phi(n))=\gcd(e,(p-1)(q-1))=1$$ holds, which isn't the case with your sample parameters as $\gcd(3,60)=3\neq1$. As fgrieu pointed out ...

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