bash中的RSA算法（私钥计算）

Question

Looking for help on how to calculate the RSA secret key in a bash script 'd' when p, q & e are given.

Although I know this query has come up before, all I seem to find is theoretical advice on the calculation of the RSA private key when p, q & e are given.

I am trying to put a calculation in a bash script together and so would really like some explanation on the calculations possible in bash (with expr or bc)

I found a site that basically advised the following; (with 't' being the totient of p*q, or rather the totient of 'n')

d=$(echo "((2 * $t) + 1) / $e" | bc)

But that only seems to work on that specific case and did not provide the correct output when trying on different cases.

Any help on how I could write a bash formula for this calculation ?

Answer 1

From the defition of RSA

|e*d| = 1 (mod fi(p*q))

where fi(n) is the Euler function

So you need to calculate the inversion of e in this modulo. Don't forget to check if GCD(e,fi(p*q))=1 or you may not find an inversion of e in the given modulo.

You can achieve this by using Extended Euclidean algorithm. There is a lot of pseudocodes on the web. Do you need more help?