Cryptographic algorithms and protocols process messages as numbers or elements in a finite space. Encoding (encryption) and the necessary decoding (decryption) operations must transform messages to messages so that the transformation obeys a *closure* property inside a finite space of the messages. However, the usual arithmetic over numbers such as addition, subtraction, multiplication and division which are familiar to us do not have a closure property within a finite space (integers or numbers in an interval). Therefore, cryptographic algorithms which operate in a finite space of messages are in general not constructed only using the familiar arithmetic over numbers. Instead, they in general ...

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