Affine Cipher - Pair of plain to cipher text

Question

Consider an affine cipher.

The cryptanalyst observed the following plaintext/ciphertext pairs $(p,c)$: $(8,15)$ and $(5,16)$.

1. Recover the key $(a,b)$ used in the encryption system above.
2. What is the ciphertext corresponding to the plaintext $p=3$?

For #1, I came to this result:

$$\begin{aligned} 8 \cdot a + b &\equiv 15, \\ 5 \cdot a + b &\equiv 16 \\ \implies a &\equiv 17, \\ b &\equiv 9. \end{aligned} \pmod{26} $$

For #2, congruence gave me:

$$\begin{aligned} 3a &\equiv -1, \\ \text{which is}\quad 3a &\equiv 25, \\ \implies 3^{-1} &\equiv 27, \end{aligned} \pmod{26} $$

not $17$. Am I doing something wrong?

Answer 1

After $3a=-1 \pmod{26}$ note that mod $26$ we have $-1 = 25=51=3\cdot 17$, so $a=17$.

Alternatively, note that $3\cdot9=27=1$ so we can multiply both sides by $3^{-1}=9$ and $-9=17\pmod{26}$.

And then $5\cdot17 +b = 9$, and $5\cdot 17=85=7$ and so $b=9$.

Now substitute $p=3$ in the encryption formula $c=17p+9$ for the final question.

Answer 2

It appears that you already know that, if the plaintext character is $p$, then the corresponding ciphertext character will be $(pa + b) \bmod 26$.

You've correctly inserted the given plaintext/ciphertext pairs into this formula to obtain the linear congruences $8a+b \equiv 15$ and $5a+b \equiv 16$, and solved them modulo $26$ to obtain the coefficients $a \equiv 17$ and $b \equiv 9$.

All you need to do now is apply the same formula given above to $p=3$, i.e. to calculate $(3a + b) \bmod 26 = (3\times17+9) \bmod 26$. This will give you the ciphertext corresponding to the plaintext $3$.

(If you were instead given the ciphertext value $3$ and told to decrypt it, then you'd have to solve the linear congruence $17p+9 \equiv 3$ modulo $26$, which would involve calculating the modular inverse of the multiplicative factor $17$ modulo $26$. But since you're only told to encrypt a plaintext value, you don't need to calculate any inverses.)