How to calclulate multiplicative inverse of e mod $\phi(n)$?

Question

In this wikipedia article about RSA, At step 5, How are they calclulating value of $d$? Can anybody give me a step-by-step explanation?

Compute $d$, the modular multiplicative inverse of $e \pmod{\phi(n)}$ yielding $d=2753$.

Answer 1

The most efficient way would be the Euclidean algorithm.

http://en.wikipedia.org/wiki/Euclidean_algorithm

You use the Euclidean algorithm to find $d, a$ such that $d \cdot e + a \cdot \phi(n) = 1$. Taking this mod $\phi(n)$ gives $d \cdot e \equiv 1$.

For your specific case where $\phi(n) = 3120$ and $e = 17$, you start with $3120 / 17$, which is 183 with a remainder of 9. Write this as:

$$3120 = 17 \cdot 183 + 9$$ Now, repeat this process with 17 and 9. $$17 = 9 \cdot 1 + 8$$ Now, repeat this process with 9 and 8. $$9 = 8 \cdot 1 + 1$$ Now that you have obtained 1, "solve" for 1 using back substitution, with an end goal of getting 1 as a linear combination of 3120 and 17. First, the three above equations become

$$\begin{align*} 1 &= 9 - 8 \\ 8 &= 17 - 9 \\ 9 &= 3120 - 17 \cdot 183 \end{align*}$$

So, doing back substitution gives

$$1 = 9 - 8 = 9 - (17 - 9) = 9 \cdot 2 - 17 = [3120 - 17 \cdot 183] \cdot 2 - 17 = 3120 \cdot 2 - 17 \cdot 367$$

As I said above, taking everything mod 3120 gives $$17 \cdot -367 \equiv 1$$ and since $-367 + 3120 = 2753$, we see that $-367 \equiv 2753$ mod 3120. Thus, $-367$, or $2753$ is the multiplicative inverse of 17 mod 3120.

Answer 2

You use the Extended Euclidean algorithm.

You know $\phi(n)$ (because you know $n=pq$ with $p$ and $q$ primes, so that $\phi(n)=(p-1)(q-1)$). So it is an easy matter to apply the Extended Euclidean Algorithm to $e$ and to $(p-1)(q-1)$ to obtain integers $x$ and $y$ such that $$1 = xe + y\phi(n).$$ (If $\gcd(e,\phi(n))\gt 1$, you are supposed to pick a different $e$).

The number $d$ you are looking for is the coefficient $x$ in this expression.

In the specific example given: $p=61$, $q=53$, $n=61\times 53=3233$, $\phi(n) = 60\times 52 = 3120$, and $e=17$.

Divide $\phi(n)$ by $e$ with remainder: $$3120 = 183\times {\color{magenta}{17}} + {\color{blue}9}. \tag{1}$$ Now divide ${\color{magenta}{17}}$ by the remainder ${\color{blue}9}$: $${\color{magenta}{17}} = 1\times {\color{blue}9} + {\color{red}8}.\tag{2}$$ Now divide ${\color{blue}9}$ by the remainder ${\color{red}8}$: $${\color{blue}9} = 1\times{\color{red}8} + {\color{green}1}.\tag{3}$$ This will be the last step in the Euclidean algorithm, since when you divide ${\color{red}8}$ by $\color{green}1$, you will obtain a remainder of $0$; the gcd is the last nonzero remainder.

Now backtrack to write ${\color{green}1}$ as a linear combination of $3120$ and $17$: $$\begin{align*} {\color{green}1} &= {\color{blue}9} - 1\times{\color{red}8} &\text{(by (3))}\\ &= {\color{blue}9} - 1\times({\color{magenta}{17}} - 1\times{\color{blue}9}) &\text{(by (2))}\\ &= {\color{blue}9} - {\color{magenta}{17}} + {\color{blue}9}\\ &= 2\times{\color{blue}9} - 1\times{\color{magenta}{17}}\\ &= 2\times(3120 - 183\times{\color{magenta}{17}}) - 1\times{\color{magenta}{17}}&\text{(by (1))}\\ &= 2\times 3120 - 366\times{\color{magenta}{17}} - {\color{magenta}{17}}\\ &= 2\times 3120 - 367\times{\color{magenta}{17}}. \end{align*}$$ So $d\equiv -367 \equiv 3120-367 \equiv 2753\pmod{3120}$.

Answer 3

I spent a lot of time trying to figure out the algebra from the above answer. I personally like parenthesis. They tend to make math more clear not less clear so i am rewriting the above answer for people that have the same preference. I hope this helps some people, and great job to the person above me. It helped a lot.

Now backtrack to write 1 as a linear combination of 3120 and 17:

 1=9−1×8,
 1=9−1×(17−1×9),
 1=9+(-1)(17-9),
 1=9−17+9,
 1=2×9−17,
 1=(2x9)-(1x17),
 1=2×[3120−(183×17)]−1×17,
 1=(2)(3120)-(2)(183x17)-1x17,
 1=(2x3120)+(-2)(183x17)-1x17,
 1=(2x3120)+(-366x17)-(1x17),
 1=(2x3120)+[17(-366-1)],
 1=(2x3120)+[17x-367],
 1=(2x3120)+[-367x17],
 1=(2x3120)+(-1x367x17),
 1=(2x3120)-(1x367x17),
 1=(2x3120)-(367x17),

I know many of these parenthesis are not necessary, but I believe they make the math easier to follow. It is just my preference, and I typed this up to help the next person who has the same preference. Anyway, thanks for the above answers. They helped me a great deal.