Geometry-Based Symmetric Cryptosystem Method

A method of communicating information between users of a communication system, the method comprising the steps of:

• generating a module V over a ring R;
• generating an outer component P of encryption key:
• P=(p1, p2, . . . , pk) where each member pj of the sequence belongs to the set {1, 2, . . . , m};
• generating an inner component Q of encryption key that includes elements v1, v2, . . . , vm of said module V and automorphisms g1, g2, . . . , gm of V;
• generating the encryption key K=(P;
• Q), where P is the outer component and Q is the inner component;
• generating an encryption automorphism Te of the module V based on automorphisms T1, T2, . . . , Tm of the module V and on the outer component P=(p1, p2, . . . , pk) of encryption key:
• Te=Tp1°Tp2° . . . Tpk. That is, Te is an automorphism of the module V obtained as a composition of automorphisms T1, T2, . . . , Tm, which composition is performed in the order prescribed by P;
• generating an encrypted message element E as a function of a message element M in V and of the encryption automorphism Te;
• transmitting the encrypted message element E along with the outer component P from one user to another;
• generating an outer component P′=(pk, pk−1, . . . p1), i.e., the sequence that is reversed of that involved in producing the outer component P of the encryption key;
• generating the decryption key K′=(P′;
• Q′), where P′ is the outer component of the decryption key and Q′ is the inner component of the decryption key which is equal to the inner component Q of the encryption key;
• generating a decryption automorphism Td of the module V based on automorphisms T1, T2, . . . , Tm of the module V and on the outer component P′=(pk, pk−1, . . . p1) of the decryption key:
• Te=Tpk° . . . Tp2°Tp1, where T1, T2, . . . , Tm are the same automorphisms of V which have been used in the construction of the encryption automorphism Te;
• determining the message element M as a function of the encrypted message element E and of the decryption automorphism Td, where the function is the same as that one used in generation of E (that is, the decryption method is symmetric to encryption:
• the decryption proceeds as the encryption, but with replacement of the outer component P with the outer component P′).

A method of communicating information between users of a communication system, the method comprising the steps of:

• means for generating a module V over a ring R;
• means for generating an outer component P of encryption key that includes sequence (p1, p2, . . . , pk) where each member pj of the sequence belongs to the set {1, 2, . . . , m};
• means for generating an inner component Q of encryption key that includes elements v1, v2, . . . , vm of V and automorphisms g1, g2, . . . , gm of V;
• means for generating the encryption key K=(P;
• Q), where P is the outer component and Q is the inner component;
• means for generating an encryption automorphism Te of V based on the encryption key K, where Te includes a composition of certain automorphisms T1, T2, . . . , Tm of the module V which composition is performed in the order prescribed by P;
• means for generating an encrypted message element E as a function of a message element M in V and of the encryption automorphism Te;
• means for transmitting the encrypted message element E along with the outer component P from one user to another;
• means for generating the outer component P′ of the decryption key that includes sequence (pk, pk−1, . . . p1), i.e., the sequence that is reversed of that involved in producing the outer component P of the encryption key;
• means for generating the decryption key K′=(P′;
• Q′), where P′ is the outer component of the decryption key and Q′ is the inner component of the decryption key which is equal to the inner component Q of the encryption key;
• means for generating a decryption automorphism Td of V based on the decryption key K′, where Td includes a composition of the automorphisms T1, T2, . . . , Tm, which composition is performed in the order prescribed by P′, e.g., Td is the inverse automorphism of Te;
• means for determining the message element M as a function of the encrypted message element E and of the decryption automorphism Td, where the function is the same as that one used in generation of E (that is, the decryption method is symmetric to encryption:
• the decryption proceeds as the encryption, but with replacement of the outer component P with the outer component P′).