Abstract: A system and method for generating a secret key to facilitate secure communications between users. A first and second and a function between the two monoids are selected, the function being a monoid homomorphism. A group and a group action of the group on the first monoid is selected. Each user is assigned a submonoid of the first monoid so that these submonoids satisfy a special symmetry property determined by the function, a structure of the first and second monoids, and the action of the group. A multiplication of an element in the second monoid and an element in the first monoid is obtained by combining the group action and the monoid homomorphism. First and second users choose private keys which are sequences of elements in their respective submonoids. A first result is obtained by multiplying an identity element by the first element of the sequence in a respective submonoid.

Abstract: A system effective to communicate a message between two devices. A first device may include a plaintext to monoid element module effective to receive a plaintext message and apply a first function to the plaintext message to produce a first monoid element. A monoid element evaluator module may be effective to receive and insert submonoid generators into a monoid expression to produce a second monoid element in response. An encryption device module may be effective to apply a second function to the first monoid element, the second monoid element, the monoid expression, and a third monoid element to produce an encrypted plaintext message. Decryption may be performed on the encrypted plaintext message knowing the private key which includes the first function, the second function, the third monoid element and the submonoid generators list.

Abstract: A first module divides a string into a number of blocks. A second module associates the blocks with monoid elements in a list of first monoid elements to produce second monoid elements. A third module applies a first function to an initial monoid element and a first of the second monoid elements producing a first calculated monoid element and evaluates an action of the initial monoid element on the first function producing a second function. A fourth module applies the second function to the first calculated monoid element and to a second of the second monoid elements producing a second calculated monoid element and evaluates the action of the first calculated monoid element on the first function producing a third function.

Abstract: A first module divides a string into blocks. A second module associates the blocks with monoid elements in a list of first monoid elements to produce second monoid elements. A third module applies a first function to an initial monoid element and a first of the second monoid elements producing a first calculated monoid element and evaluates an action of the initial monoid element on the first function producing a second function. A fourth module applies the second function to the first calculated monoid element and to a second of the second monoid elements producing a second calculated monoid element and evaluates the action of the first calculated monoid element on the first function producing a third function.

Abstract: A first module divides a string into a number of blocks. A second module associates the blocks with monoid elements in a list of first monoid elements to produce second monoid elements. A third module applies a first function to an initial monoid element and a first of the second monoid elements producing a first calculated monoid element and evaluates an action of the initial monoid element on the first function producing a second function. A fourth module applies the second function to the first calculated monoid element and to a second of the second monoid elements producing a second calculated monoid element and evaluates the action of the first calculated monoid element on the first function producing a third function.

Abstract: A system and method for generating a secret key to facilitate secure communications between users. A first and second and a function between the two monoids are selected, the function being a monoid homomorphism. A group and a group action of the group on the first monoid is selected. Each user is assigned a submonoid of the first monoid so that these submonoids satisfy a special symmetry property determined by the function, a structure of the first and second monoids, and the action of the group. A multiplication of an element in the second monoid and an element in the first monoid is obtained by combining the group action and the monoid homomorphism. First and second users choose private keys which are sequences of elements in their respective submonoids. A first result is obtained by multiplying an identity element by the first element of the sequence in a respective submonoid.

Abstract: A system and method for generating a secret key to facilitate secure communications between users. A first and second and a function between the two monoids are selected, the function being a monoid homomorphism. A group and a group action of the group on the first monoid is selected. Each user is assigned a submonoid of the first monoid so that these submonoids satisfy a special symmetry property determined by the function, a structure of the first and second monoids, and the action of the group. A multiplication of an element in the second monoid and an element in the first monoid is obtained by combining the group action and the monoid homomorphism. First and second users choose private keys which are sequences of elements in their respective submonoids. A first result is obtained by multiplying an identity element by the first element of the sequence in a respective submonoid.

Abstract: A system and method for generating a secret key to facilitate secure communications between users. A first and second and a function between the two monoids are selected, the function being a monoid homomorphism. A group and a group action of the group on the first monoid is selected. Each user is assigned a submonoid of the first monoid so that these submonoids satisfy a special symmetry property determined by the function, a structure of the first and second monoids, and the action of the group. A multiplication of an element in the second monoid and an element in the first monoid is obtained by combining the group action and the monoid homomorphism. First and second users choose private keys which are sequences of elements in their respective submonoids. A first result is obtained by multiplying an identity element by the first element of the sequence in a respective submonoid.

Abstract: A system and method for generating a secret key to facilitate secure communications between users. A first and second and a function between the two monoids are selected, the function being a monoid homomorphism. A group and a group action of the group on the first monoid is selected. Each user is assigned a submonoid of the first monoid so that these submonoids satisfy a special symmetry property determined by the function, a structure of the first and second monoids, and the action of the group. A multiplication of an element in the second monoid and an element in the first monoid is obtained by combining the group action and the monoid homomorphism. First and second users choose private keys which are sequences of elements in their respective submonoids. A first result is obtained by multiplying an identity element by the first element of the sequence in a respective submonoid.

Abstract: The present invention is a method and apparatus for providing cryptographically secure algebraic key establishment protocols that use monoids and groups possessing certain algorithmic properties. Special fast algorithms associated with certain monoids and groups are used to optimize both key agreement and key transport protocols. The cryptographic security of the algorithms is based on the difficulty of solving the conjugacy problem in groups and other known hard algebraic problems. Braid groups and their associated algorithms are the basis for highly rapid key agreement and key transport protocols which employ modest computational resources.