Diophantine system for digital signatures
Digital signatures using the Diophantine system of equations are implemented. A digital signature is an authentication mechanism that enables the creator of a message to attach a code that acts as a signature. A digital signature scheme typically includes three algorithms: a key generation algorithm, a signing algorithm, and a signature verifying algorithm. The key generation algorithm selects a private key uniformly at random from a set of possible private keys. The key generation algorithm outputs the private key and a corresponding public key. The signing algorithm produces a signature given a message and a private key. The signature verifying algorithm either accepts or rejects a message's claim to authenticity based at least in part on the message, the public key, and the signature.
1. A method programmed in a non-transitory memory of a device of generating a digital signature for data, comprising:
generating, with a processor a hash of the data;
generating the digital signature, with the processor, based at least in part on the hash, random data comprising a first set of non-singular matrices, and a private key comprising a first set of singular matrices and a second set of non-singular matrices, wherein the digital signature corresponds to a set of matrices and to a relation which corresponds to a Diophantine second order equation; and
communicating from the device, over a network, the digital signature and the data to a remote device.
2. The method of claim 1 wherein the hash comprises a second set of singular matrices.
3. The method of claim 1 further comprising verifying the digital signature based at least in part on a public key comprising singular matrices.
4. The method of claim 1 further comprising registering a signer user with an authentication server by generating an authentication transaction value and sending the authentication transaction value to the authentication server.
5. The method of claim 4 , further comprising:
receiving a server digital signature and a hash of the authentication transaction value from the authentication server; and
verifying the server digital signature and the hash of the authentication transaction value.
6. The method of claim 4 , further comprising authenticating a device of the signer user by:
performing a database lookup for a stored name associated with the device of the signer user;
performing an XOR operation on a received secret token using a secret key associated with the stored name to decrypt the received secret token; and
comparing the decrypted token with a stored token.
7. An apparatus for generating a digital signature for data, comprising:
a non-transitory memory configured for storing an application and a processor for processing the application, the application configured for:
generating, with the processor, a hash of the data; and
generating the digital signature, with the processor, based at least in part on the hash, random data comprising a first set of non-singular matrices, and a private key comprising a first set of singular matrices and a second set of non-singular matrices, wherein the digital signature corresponds to a set of matrices and to a relation which corresponds to a Diophantine second order equation; and
communicating from the apparatus, over a network, the digital signature and the data to a remote device.
8. The apparatus of claim 7 wherein the hash comprises a second set of singular matrices.
9. The apparatus of claim 7 wherein the application is further for verifying the digital signature based at least in part on a public key comprising singular matrices.
10. The apparatus of claim 7 wherein the application is further for registering a signer user with an authentication server by generating an authentication transaction value and sending the authentication transaction value to the authentication server.
11. The apparatus of claim 10 , wherein the application is further for:
receiving a server digital signature and a hash of the authentication transaction value from the authentication server; and
verifying the server digital signature and the hash of the authentication transaction value.
12. The apparatus of claim 10 , wherein the application is further for authenticating a device of the signer user by:
performing a database lookup for a stored name associated with the device of the signer user;
performing an XOR operation on a received secret token using a secret key associated with the stored name to decrypt the received secret token; and
comparing the decrypted token with a stored token.
13. A method programmed in a non-transitory memory of a device of generating a digital signature for data, comprising:
generating, with a processor, a hash of the data, wherein the hash comprises a first set of singular matrices;
generating the digital signature, with the processor, based at least in part on the hash, random data comprising a first set of matrices, and a private key comprising a second set of singular matrices and a second set of matrices, wherein the digital signature corresponds to a third set of matrices in a relation corresponding to a Diophantine second order equation; and
communicating from the device, over a network, the digital signature and the data to a remote device.
14. The method of claim 13 further comprising verifying the digital signature based at least in part on a public key comprising singular matrices.
15. The method of claim 13 further comprising registering a signer user with an authentication server by generating an authentication transaction value and sending the authentication transaction value to the authentication server.
16. The method of claim 15 , further comprising:
receiving a server digital signature and a hash of the authentication transaction value from the authentication server; and
verifying the server digital signature and the hash of the authentication transaction value.
17. The method of claim 15 , further comprising authenticating the device of the signer user by:
performing a database lookup for a stored name associated with the device of the signer user;
performing an XOR operation on a received secret token using a secret key associated with the stored name to decrypt the received secret token; and
comparing the decrypted token with a stored token.