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Being new to cryptology, I'm trying to understand how I would complete RSA encryption by hand. I can only follow the formula so far before becoming very confused.

I want to encrypt the value "123"

First, I am to select 2 primes. I choose: $$p = 101\\ q = 103$$

Next, I compute: $$n = p\cdot q = 10403$$.

After that, I compute: $$\varphi(n) = (p-1)\cdot(q-1) = 10200$$

Now, I want to choose a public exponent, and I choose 3 for this.

I believe the formula to use is: $$d = e^{−1}\bmod\varphi(n)$$

I don't understand how to plug this formula in, nor do I know how I would encrypt "123" using this formula. Also, I don't know how I would find the decryption exponent either.

Any help would be much appreciated!

fgrieu
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Bobby Lefto
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1 Answers1

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Fgrieu essentially gave the answer in comment, I will try to elaborate a bit in answer form.

You can use extended euclidean algorithm to find d from e, but note the e you selected will not work. Because e is not co-prime with $\varphi(n)$ You need to select another one. For efficiency we usually like to select a small e with few set bits, usually of the form $2^k+1$ since 3 doesn't work you can try others. https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm Here is an online calculator: https://planetcalc.com/3298/

It will not only give you the gcd (which needs to be 1) it will also give you a,b so that: $a*e + b*\varphi(n) = 1$ which essentially means $a*e = 1 mod \varphi(n)$ which is what we wanted.

You then encrypt by calculating $c = m^e\space mod(n)$ and decrypt using $m = c^d\space mod(n)$ Both done via https://en.wikipedia.org/wiki/Modular_exponentiation

Meir Maor
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