Find the result of this dual key cipher

I came up with a neat little cipher that I'm calling the dual key cipher. The basic premise is that there are two cipher keys that, when used together can decode the cipher text against a given alphabet. Here is a basic example.

Key 1: CAT
Key 2: DOG
Alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ
Cipher Text: ISA


The result would be:

HEN

Can you solve for this one? Bonus points for showing your work.

Puzzle 1
Key 1: FISH
Key 2: DUCK
Alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ
Cipher Text: ZUBG


Additional (off-topic) questions: Does anyone know if this type of cipher exists and if it has an official name? I've developed an algorithm that does this at a massive scale and wanted to know if anyone finds it interesting. Here is a somewhat more difficult version using my algorithm.

Puzzle 2
Key 1: Chocolate
Key 2: Strawberry
Alphabet: abcdefghijklmnopqrstuvwxyz ABCDEFGHIJKLMNOPQRSTUVWXYZ
Cipher Text: lmqgtbeyVXPERdpbX


Hint: The space between "z A" is part of this "alphabet".

The cipher is decoded by

rotating each character in the ciphertext through the "alphabet" by the index in the "alphabet" of the corresponding character in the first key minus the index of the corresponding character in the second key. If the end of one of the keys is reached, it wraps around.

The answer to the first puzzle is:

"BIRD":

  Z   U   B   G
+ F   I   S   H
- D   U   C   K
+2  -12 +16 -3
B   I   R   D

The answer to the second puzzle is:

"Vanilla Ice Cream":

  l   m   q   g   t   b   e   y   V   X   P   E   R   d   p   b   X
+ C   h   o   c   o   l   a   t   e   C   h   o   c   o   l   a   t
- S   t   r   a   w   b   e   r   r   y   S   t   r   a   w   b   e
-16 -12 -3  +2  -8  +10 -4  +2  -13 +5  -38 -5  -15 +14 -11 -1  +15
V   a   n   i   l   l   a       I   c   e       C   r   e   a   m

• Bingo! And I really should call "alphabet", character set, I suppose. – tremor Apr 6 at 14:30

The results for both puzzles are:

BIRD

and an incomplete one:

vanilla ...

The cipher is a variant of:

But:

Instead of using 1 key, it uses 2 keys. The result can be found by original Vigenere Cipher by this step:
- Let Key 3 be equal to deciphering Key 2 with Passphrase of Key 1
- The result will be equal to deciphering the Cipher Text with Passphrase of Key 3

A note for the alphabets:

Instead of using simple order from A to Z, we can change the order with specified Alphabet. But deciphering this needs a computer program or some works with hands.. I'm sorry I'm too lazy to solve the Puzzle 2 because of this.. ><

• Worth an upvote for being close. – tremor Apr 6 at 5:00