I have to find the deciphering multiplier for a multiplication cipher given, Cq=Pi
so q goes to i / 17 goes to 9
I plugged it into the equation 17a = 9(mod 26) then solved using the inverse of 17 to reduce so 23 (17)a = 23 (9)(mod 26)
this left me with a=207(mod 26) so I reduced 207 to 25 as the deciphering shift. I know the answer is supposed to be 15 so if anyone could point out where I'm going wrong I'd be very grateful.
I am not sure if I understand what you're supposed t find. But here's what I have found.
I have mod(23*17,26)=1 (and not 9), so 23 cannot be the multiplier.
I found it by trial and error, that
mod(25*17,26)=9, (encrypting)
mod(25*9,26)=17, (decrypting).
So 25 is the multiplicative key to encrypting and decryption.
Is that what you're supposed to do?
Yes. The math I did was correct and the deciphering multiplier is 25. The answer in my textbook was wrong and claimed it was 15, making me doubt my answer. Thank you for answering :)
To find the deciphering multiplier for a multiplication cipher, you are correct in using the equation Cq ≡ Pi (mod 26), where Cq is the result after multiplication, Pi is the original letter, and 26 represents the number of letters in the English alphabet.
In your case, you have Cq ≡ i and 17 ≡ 9 (mod 26).
To find the deciphering multiplier (a), you correctly set up the equation 17a ≡ 9 (mod 26). The goal is to solve for a.
Next, you mention using the inverse of 17 to reduce and obtain the equation 23(17)a ≡ 23(9) (mod 26). This step is incorrect.
To find the inverse of 17 modulo 26, you need to use modular multiplicative inverse. An important property of modular inverse is that (a * a^(-1)) ≡ 1 (mod n), where a is the number and a^(-1) is its inverse modulo n.
In this case, you need to find the inverse of 17 modulo 26, which means finding a number x such that (17 * x) ≡ 1 (mod 26). In other words, x is the number that, when multiplied by 17 and reduced modulo 26, gives the remainder 1.
After calculating using a method like extended Euclidean algorithm, you'll find that the inverse of 17 modulo 26 is actually 23, not 9.
Therefore, the correct equation should be: 17a ≡ 23 (mod 26).
Now, you can solve this equation to find the value of a.
To solve it, divide both sides of the equation by 17: a ≡ (23 * 9) (mod 26).
Simplifying the right side: a ≡ 207 (mod 26).
To reduce 207 modulo 26, divide 207 by 26: 207 ÷ 26 = 7 with a remainder of 25.
Therefore, a ≡ 25 (mod 26), which means the deciphering multiplier is 25.
This indicates that to decrypt the message, you need to multiply the letter by 25 (modulo 26).
So, if the original letter is q, to decode it, you would multiply it by 25 (modulo 26) to get the plaintext letter.