Recover the Secret

Recover D

The private exponent is the inverse of e modulo phi.

highlighted = computed this step

d is the inverse of e

The private exponent d is chosen so e times d leaves remainder one modulo phi.

ed mod ϕ=oneed\bmod\phi=\text{one}
Recover d as inverseThe RSA key table is recomputed from the recovered factors.Recover d as inverse - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1

Check the product

Here e=3 and d=27, so their product is 81.

3⋅27=813\cdot27=81
Recover d as inverseThe RSA key table is recomputed from the recovered factors.Recover d as inverse - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1

Modulo phi leaves one

Modulo phi=40, the product leaves remainder 1.

81 mod 40=181\bmod{}40=1
Recover d as inverseThe RSA key table is recomputed from the recovered factors.Recover d as inverse - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1

Summary

The recovered private exponent is d=27.

d=27d=27
Recover d as inverseThe RSA key table is recomputed from the recovered factors.Recover d as inverse - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1