Communication system

A device effective to decrypt an encrypted message, the device comprising:

• a memory, wherein the memory is effective to include
• a first function,
• a list of submonoid generators,
• a second function, and
• a first monoid element,
• a first module in communication with the memory, the first module effective to receive a monoid expression in the encrypted message, the first module effective to insert the submonoid generators into the monoid expression to produce a second monoid element;
• a second module in communication with the memory and the first module, the second module effective to
• receive a third monoid element in the encrypted message,
• receive the second function,
• receive the first monoid element,
• receive the second monoid element, and
• apply the second function to the third monoid element, to the inverse of the second monoid element, and to the inverse of the first monoid element to produce a fourth monoid element;
• a third module in communication with the memory and the second module, the third module effective to apply the first function to the fourth monoid element to produce the message.

A system effective to communicate a message, the system comprising:

• a first device in communication with a second device over a network;
• wherein the first device includes
• a first memory, wherein the first memory is effective to include
• a first function,
• a list of submonoid generators,
• a second function, and
• a first monoid element,
• a first module in communication with the first memory, the first module effective to receive a message and apply the first function to the message to produce a second monoid element;
• a second module in communication with the memory, the second module effective to receive and insert the submonoid generators into at least one monoid expression to produce a third monoid element;
• and
• a third module in communication with the memory, in communication with the first module, and in communication with the second module, the third module effective to apply the second function to the first monoid element, the second monoid element, the monoid expression, and the third monoid element to produce an encrypted message including a fourth monoid element and the monoid expression;
• wherein the second device includes
• a second memory, wherein the second memory is effective to include
• the first function,
• the list of submonoid generators,
• the second function, and
• the first monoid element,
• a fourth module in communication with the second memory, the fourth module effective to receive the monoid expression in the encrypted message, the fourth module effective to insert the submonoid generators into the monoid expression to re-produce the third monoid element;
• a fifth module in communication with the second memory and the fourth module, the fifth module effective to
• receive the fourth monoid element in the encrypted message,
• receive the second function,
• receive the first monoid element,
• receive the third monoid element, and
• apply the second function to the fourth monoid element, the inverse of the first monoid element and the inverse of the third monoid element to produce the second monoid element;
• and
• a sixth module in communication with the second memory and the fifth module, the sixth module effective to apply the first function to the second monoid element to re-produce the message.