Abstract: A method for establishing key agreement between two communicating parties using a general linear protocol in finite and infinite dimensional spaces. Two topological linear spaces, in particular Euclidean spaces, and a non-trivial degenerate linear operator are selected. Each party respectively selects a secret element, and exchanges with the other party an image under the transformation of a matrix. Key agreement is therefore mutually established between the two communicating parties having the same cryptographic key. Various illustrative embodiments of the general linear operator are disclosed, including a rectangular matrix, a square matrix, a symmetric matrix, a skew symmetric matrix, an upper triangular square matrix, a lower triangular square matrix, a special type of skew symmetric matrix to generate a modified cross product protocol, a series of matrices to generate a sequential key protocol, and a combination of circulant matrices.

Abstract: Using the present cryptographic key exchange protocol, a first of two cryptographic units selects quantities "A" and "B." That same unit then applies a mathematical function ".PHI..sub.1 " to A and B to compute a third quantity "C.". A and B and the function .PHI..sub.1 posses the property that knowing A or B, and C, and the function .PHI..sub.1, it is mathematically impossible to compute the unknown quantity B or A. The first unit then transmits A together C to the second unit while retaining B. The second unit selects a fourth quantity "D" and then applies a second mathematical function .PHI..sub.2 to A and D to compute a quantity "E" which the second unit transmits to the first unit while retaining D. The quantities A and D and the function .PHI..sub.2 must posses the property that knowing D and E, and the function .PHI..sub.2, it is mathematically impossible to compute the unknown quantity D. Then the second unit applies a third mathematical function ".PSI..sub.2 " to D and C to compute the secure key "K.