Let's map the alphabet and some symbols onto the numbers 1 to 31. And let us use the number 9 as
encryption key as follows: C = 9* P mod 31 (P a plaintext number and C a ciphertext number, both in the range 131, where we interpret 31 as o).
Accordingly we can encrypt the word "LOVE" as follows:
Replace the letters with their numeric value: LOVE = 12,15,22,5
Encrypt each plaintext letter P into its corresponding ciphertext letter C
So 12 becomes 12*9 mod 31 = 15,
15 becomes 15*9 mod 31 = 11
22 becomes 22*9 mod 31 = 12
5 becomes 5*9 mod 31 = 14
Replace the numeric values with their letters: 15,11,12,14 becomes: OKLN
so the plaintext "LOVE" becomes the ciphertext "OKLN"
If we try to use the same key for decryption we get a plaintext: KFOB which is different from the plaintext word "LOVE". So a symmetric key won't work!
However, if we use a different key K=7 (not 9) and apply to the ciphertext then we get:
15 becomes 15*7 mod 31 = 12
11 becomes 11*7 mod 31 = 15
12 becomes 12*7 mod 31 = 22
14 becomes 14*7 mod 31 = 5
which translates back to the original plaintext "LOVE".
This illustrates a simple asymmetrical cipher that does the job.
The math behind the illustration:
Let e be the encryption key; d, the decryption key. A letter x is encrypted to y=ex mod 31, then decrypted back to x = dy = dex mod 31. So we must have de = 1 mod 31. for e=9 and d=7, we have 7*9=63 = 31*2 + 1 = 1 mod31.

