# Modular Arithmetic problem

Having confusion. Plaintext = 25, Key=15 and modulus =26. Cipher=(15 + 25) mod 26=14. Now , plaintext = (15-14) mod 26=1

Here I am not getting the plaintext correctly. How to do it?

• Why 15-14? 14-15. – Meysam Ghahramani Jul 7 '18 at 19:58

Let CT be the ciphertext, PT the plaintext, K the key, n the modulus.

Since CT=(PT+K) mod n, PT=(CT-K) mod n. In your case, PT=(14-15) mod 26= -1 mod 26=25, as expected. Remember that for x<0, x mod n= (n+x) mod n (not n-x, as -x would be positive) by definition.

What you did was PT=(K-CT) mod n, which is not true, because this would imply PT=(K-(PT+K)) mod n= -PT mod n, which in general is not true.

For example, 10 mod 26=10, while -10 mod 26=(26-10) mod 26=16, and clearly 10 is not 16.

In fact, this is only true for PT=n/2(and its multiples) because only in that case PT=(n-PT) mod n.

• Okay. Understood the solution. So PT=(CT-K) mod N is giving the correct answer. Thank you ☺ – Avisek Manna Jul 7 '18 at 18:36
• One more thing, if I use CT=(PT +K1+K2) mod N, then PT=(K2-K1-CT)mod N. Is it correct? – Avisek Manna Jul 7 '18 at 18:44
• @avisekmanna No, it's PT=(CT-K1-K2) mod n. First, CT, then the key or keys. Indeed, if you consider K=K1+K2 as a single key, applying the formula I wrote in my answer you have PT=(CT-K) mod n, which is (CT-(K1+K2)) mod n or, equivalently, (CT-K1-K2) mod n. – A. Darwin Jul 7 '18 at 18:48