Efficient and secure distributed signing protocol for mobile devices in wireless networks

1. A system for generating a distributed cryptographic key and a cryptographic signature comprising:

an electronic computing device operating as a key generating center;

a first electronic device;

a second electronic device;

the key generating center, the first electronic device, and the second electronic device in communication with each other to generate a distributed cryptographic key by:

creating a first part of a private cryptographic key;

creating a second part of the private cryptographic key;

creating a public cryptographic key;

transmitting the public cryptographic key and the first part of the private cryptographic key to the first electronic device; and

transmitting the public cryptographic key and the second part of the private cryptographic key to the second electronic device;

the first electronic device to provide a cryptographic signature to the second electronic device by

transmitting from the first electronic device to the second electronic device an indicator that a cryptographic signature for a message will be transmitted;

upon receiving the first transmission, the second electronic device computing a first intermediate value and a second intermediate value using a first random number and a second random number and transmitting to the first electronic device the first intermediate value and the second intermediate value;

upon receiving the second transmission, the first electronic device computing a third intermediate value using a third random number, a fourth random number, and the message, and transmitting to the second electronic device the third intermediate value;

upon receiving the third transmission, the second electronic device computing a fourth intermediate value and a fifth intermediate value and transmitting to the first electronic device the fourth intermediate value and the fifth intermediate value;

upon receiving the fourth transmission, the first electronic device computing the cryptographic signature for the message and transmitting the cryptographic signature to the second electronic device; and

upon receiving the cryptographic signature, the second electronic device verifying the cryptographic signature by computing a sixth intermediate value and comparing the sixth intermediate value with the cryptographic signature.

2. The system of claim 1 , wherein the distributed cryptographic key

is based upon an identity of the first electronic device.

3. The system of claim 2 , wherein generating the distributed cryptographic key comprises using the identity of the first electronic device as an input to a hash function.

4. The system of claim 1 , wherein the Paillier method is used to generate the distributed cryptographic key.

5. The system of claim 1 , wherein the first electronic device, upon receiving the second transmission, computes the third intermediate value using an output of a hash function wherein the message is an input of the hash function.

6. A method of generating a distributed cryptographic key and a cryptographic signature comprising the steps of:

creating, at an electronic computing device operating as a key generation center, a first part of a private cryptographic key;

creating, at the key generation center, a second part of the private cryptographic key;

creating, at the key generation center, a public cryptographic key;

transmitting the public cryptographic key and the first part of the private cryptographic key from the key generation center to a first electronic device;

transmitting the public cryptographic key and the second part of the private cryptographic key from the key generation center to a second electronic device;

transmitting from the first electronic device to the second electronic device an indicator that a cryptographic signature for a message will be transmitted;

upon receiving the first transmission, the second electronic device computing a first intermediate value and a second intermediate value using a first random number and a second random number and transmitting to the first electronic device the first intermediate value and the second intermediate value;

upon receiving the second transmission, the first electronic device computing a third intermediate value using a third random number, a fourth random number, and the message, and transmitting to the second electronic device the third intermediate value;

upon receiving the third transmission, the second electronic device computing a fourth intermediate value and a fifth intermediate value and transmitting to the first electronic device the fourth intermediate value and the fifth intermediate value;

upon receiving the fourth transmission, the first electronic device computing the cryptographic signature for the message and transmitting the cryptographic signature to the second electronic device;

upon receiving the cryptographic signature, the second electronic device verifying the cryptographic signature by computing a sixth intermediate value and comparing the sixth intermediate value with the cryptographic signature.

7. The method of claim 6 , wherein the distributed cryptographic key is based upon an identity of the first electronic device.

8. The method of claim 7 , wherein generating the distributed cryptographic key comprises using the identity of the first electronic device as an input to a hash function.

9. The method of claim 6 , wherein the Paillier method is used to generate the distributed cryptographic key.

10. The method of claim 6 , wherein the first electronic device, upon receiving the second transmission, computes the third intermediate value using an output of a hash function wherein the message is an input of the hash function.