There is a cipher with a built-in retrospect. Each letter corresponds to its position in the alphabet:

A = 0, B = 1, C = 2, …, X = 23, Y = 24, Z = 25

The sum of two letters is derived from the sum of their corresponding positions modulo 26 (the number of letters in the alphabet). At this position in the alphabet we find the letter that is the sum of the two letters. For example, \[ \text{W} (22) + \text{Y} (24) =  \text{U} (20 = 22 + 24\!\!\!\!\mod{26}) \]

Analogously, the difference of two letters is derived from the difference of their corresponding positions modulo 26. At this position in the alphabet we find the letter that is the difference of the two letters. For example, \[ \text{F} (5) - \text{T} (19) =  \text{M} (12 = 5 - 19\!\!\!\!\mod{26}) \]

Each letter of a message is then encoded as the sum of that letter and the previous letter in the message. No distinction is made between uppercase and lowercase letters when determining the sum, but encoding replaces uppercase letters with uppercase letters and lowercase letters with lowercase letters. The first letter of the message has no previous letter, and just remains unchanged. All characters from the message that are not letters also remain unchanged.

Decoding a ciphertext simply reverses the previous process. Each letter of the ciphertext is replaced by the difference of that letter and the previous letter from the original message (which has already been decoded).

Assignment

• Write a function encode_letter that takes a letter $$c$$ (str) from a message. In addition, the function may also take a second letter $$p$$ (str) that represents the previous letter from the message. If no second argument is passed, then $$c$$ is considered to be the first letter of the message. The function must return the encoded version of $$c$$ (str).

• Write a function decode_letter that takes a letter $$c$$ (str) from a ciphertext. In addition, the function may also take a second letter $$p$$ (str) that represents the previous letter from the original message. If no second argument is passed, then $$c$$ is considered to be the first letter of the ciphertext. The function must return the decoded version of $$c$$ (str).

• Write a function encode that takes a message (str). The function must return the corresponding ciphertext (str).

• Write a function decode that takes a ciphertext (str). The function must return the original message (str).

Example

>>> encode_letter('H')
'H'
>>> encode_letter('e', 'H')
'l'
>>> encode_letter('W', 'y')
'U'

>>> decode_letter('H')
'H'
>>> decode_letter('l', 'H')
'e'
>>> decode_letter('U', 'p')
'F'

>>> encode('Henry Walton Jones Jr.')
'Hlrep Uwlehb Wxbrw Ba.'

>>> decode('Hlrep Uwlehb Wxbrw Ba.')
'Henry Walton Jones Jr.'