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+--- Day 25: Combo Breaker ---
+
+You finally reach the check-in desk. Unfortunately, their registration systems are currently
+offline, and they cannot check you in. Noticing the look on your face, they quickly add that tech
+support is already on the way! They even created all the room keys this morning; you can take yours
+now and give them your room deposit once the registration system comes back online.
+
+The room key is a small RFID card. Your room is on the 25th floor and the elevators are also
+temporarily out of service, so it takes what little energy you have left to even climb the stairs
+and navigate the halls. You finally reach the door to your room, swipe your card, and -
+beep - the light turns red.
+
+Examining the card more closely, you discover a phone number for tech support.
+
+"Hello! How can we help you today?" You explain the situation.
+
+"Well, it sounds like the card isn't sending the right command to unlock the door. If you go back to
+the check-in desk, surely someone there can reset it for you." Still catching your breath, you
+describe the status of the elevator and the exact number of stairs you just had to climb.
+
+"I see! Well, your only other option would be to reverse-engineer the cryptographic handshake the
+card does with the door and then inject your own commands into the data stream, but that's
+definitely impossible." You thank them for their time.
+
+Unfortunately for the door, you know a thing or two about cryptographic handshakes.
+
+The handshake used by the card and the door involves an operation that transforms a
+subject number. To transform a subject number, start with the value 1. Then, a number
+of times called the loop size, perform the following steps:
+
+
+ - Set the value to itself multiplied by the subject number.
+ - Set the value to the remainder after dividing the value by 20201227.
+
+
+The card always uses a specific, secret loop size when it transforms a subject number.
+The door always uses a different, secret loop size.
+
+The cryptographic handshake works like this:
+
+
+ - The card transforms the subject number of 7 according to the
+card's secret loop size. The result is called the card's public key.
+ - The door transforms the subject number of 7 according to the
+door's secret loop size. The result is called the door's public key.
+ - The card and door use the wireless RFID signal to transmit the two public keys (your puzzle
+input) to the other device. Now, the card has the door's public key, and
+the door has the card's public key. Because you can eavesdrop on the
+signal, you have both public keys, but neither device's loop size.
+ - The card transforms the subject number of the door's public key
+according to the card's loop size. The result is the encryption key.
+ - The door transforms the subject number of the card's public key
+according to the door's loop size. The result is the same encryption key
+as the card calculated.
+
+
+If you can use the two public keys to determine each device's loop size, you will have enough
+information to calculate the secret encryption key that the card and door use to
+communicate; this would let you send the unlock command directly to the door!
+
+For example, suppose you know that the card's public key is 5764801. With a little trial and error,
+you can work out that the card's loop size must be 8, because transforming the initial
+subject number of 7 with a loop size of 8 produces 5764801.
+
+Then, suppose you know that the door's public key is 17807724. By the same process, you can
+determine that the door's loop size is 11, because transforming the initial subject
+number of 7 with a loop size of 11 produces 17807724.
+
+At this point, you can use either device's loop size with the other device's public key to calculate
+the encryption key. Transforming the subject number of 17807724 (the door's public key)
+with a loop size of 8 (the card's loop size) produces the encryption key, 14897079.
+(Transforming the subject number of 5764801 (the card's public key) with a loop size of 11 (the
+door's loop size) produces the same encryption key: 14897079.)
+
+What encryption key is the handshake trying to establish?
+
+