Method for restoring public key based on SM2 signature

1. A computer-implemented method for recovering and verifying a public key, the computer-implemented method comprising:

receiving information encoding parameters of an elliptic curve, a published public key, a hash value of a message, a digital signature, and an identification parameter;

generating a recovered public key based on the parameters of the elliptic curve, the hash value of the message, the digital signature, and the identification parameter, wherein generating the recovered public key comprises:

computing a first coordinate parameter based, at least, on the digital signature and the hash value, comprising:

subtracting the hash value from a first parameter included in the digital signature to generate a subtraction result, and

performing a modulo operation on the subtraction result to obtain the first coordinate parameter,

determining that the first coordinate parameter equals unity,

incrementing the first coordinate parameter by a specific prime number that characterizes an upper limit of a bounded range associated with the first parameter, and

computing modulo prime square roots of an elliptic equation characterized by the parameters of the elliptic curve to determine a second coordinate parameter, wherein the modulo prime square roots are computed for the elliptic equation instantiated by the first coordinate parameter; and

comparing the published public key and the recovered public key to verify the published public key.

2. The computer-implemented method of claim 1 , wherein generating the recovered public key further comprises:

verifying whether a second parameter included in the digital signature is within the bounded range; and

in response to determining that the second parameter is not within the bounded range, returning an error message.

3. The computer-implemented method of claim 1 , further comprising:

in response to determining that a particular bit of the identification parameter is unity, setting the second coordinate parameter as an odd member of the modulo prime square roots; and

in response to determining that the particular bit of the identification parameter is zero, setting the second coordinate parameter as an even member of the modulo prime square roots.

4. The computer-implemented method of claim 1 , further comprising:

configuring a coordinate point that is specified by a pairing of the first coordinate parameter and the second coordinate parameter, wherein the recovered public key is generated based, at least, on the first parameter and a second parameter included by the digital signature, the coordinate point specified by the pairing of the first coordinate parameter and the second coordinate parameter, and the parameters of the elliptic curve.

5. The computer-implemented method of claim 1 , wherein comparing the published public key and the recovered public key to verify the published public key further comprises:

in response to determining that a match exists between the published public key and the recovered public key, determining that the published public key is verified; and

in response to determining that a match does not exist between the published public key and the recovered public key, determining that the published public key is not verified.

6. A non-transitory, computer-readable storage medium storing one or more instructions executable by a computer system to perform operations for recovering and verifying a public key, the operations comprising:

receiving information encoding parameters of an elliptic curve, a published public key, a hash value of a message, a digital signature, and an identification parameter;

generating a recovered public key based on the parameters of the elliptic curve, the hash value of the message, the digital signature, and the identification parameter, wherein generating the recovered public key comprises:

computing a first coordinate parameter based, at least, on the digital signature and the hash value, comprising:

subtracting the hash value from a first parameter included in the digital signature to generate a subtraction result, and

performing a modulo operation on the subtraction result to obtain the first coordinate parameter,

determining that the first coordinate parameter equals unity,

incrementing the first coordinate parameter by a specific prime number that characterizes an upper limit of a bounded range associated with the first parameter, and

computing modulo prime square roots of an elliptic equation characterized by the parameters of the elliptic curve to determine a second coordinate parameter, wherein the modulo prime square roots are computed for the elliptic equation instantiated by the first coordinate parameter; and

comparing the published public key and the recovered public key to verify the published public key.

7. The non-transitory, computer-readable storage medium of claim 6 , wherein generating the recovered public key further comprises:

verifying whether a second parameter included in the digital signature is within the bounded range; and

in response to determining that the second parameter is not within the bounded range, returning an error message.

8. The non-transitory, computer-readable storage medium of claim 6 , wherein the operations further comprise:

in response to determining that a particular bit of the identification parameter is unity, setting the second coordinate parameter as an odd member of the modulo prime square roots; and

in response to determining that the particular bit of the identification parameter is zero, setting the second coordinate parameter as an even member of the modulo prime square roots.

9. The non-transitory, computer-readable storage medium of claim 6 , wherein the operations further comprise:

configuring a coordinate point that is specified by a pairing of the first coordinate parameter and the second coordinate parameter, wherein the recovered public key is generated based, at least, on the first parameter and a second parameter included by the digital signature, the coordinate point specified by the pairing of the first coordinate parameter and the second coordinate parameter, and the parameters of the elliptic curve.

10. The non-transitory, computer-readable storage medium of claim 6 , wherein comparing the published public key and the recovered public key to verify the published public key further comprises:

in response to determining that a match exists between the published public key and the recovered public key, determining that the published public key is verified; and

in response to determining that a match does not exist between the published public key and the recovered public key, determining that the published public key is not verified.

11. A computer-implemented system, comprising:

one or more computers; and

one or more computer memory devices interoperably coupled with the one or more computers and having tangible, non-transitory, machine-readable media storing one or more instructions that, when executed by the one or more computers, perform one or more operations for recovering and verifying a public key, the operations comprising:

receiving information encoding parameters of an elliptic curve, a published public key, a hash value of a message, a digital signature, and an identification parameter;

generating a recovered public key based on the parameters of the elliptic curve, the hash value of the message, the digital signature, and the identification parameter, wherein generating the recovered public key comprises:

computing a first coordinate parameter based, at least, on the digital signature and the hash value, comprising:

subtracting the hash value from a first parameter included in the digital signature to generate a subtraction result, and

performing a modulo operation on the subtraction result to obtain the first coordinate parameter,

determining that the first coordinate parameter equals unity,

incrementing the first coordinate parameter by a specific prime number that characterizes an upper limit of a bounded range associated with the first parameter, and

computing modulo prime square roots of an elliptic equation characterized by the parameters of the elliptic curve to determine a second coordinate parameter, wherein the modulo prime square roots are computed for the elliptic equation instantiated by the first coordinate parameter; and

comparing the published public key and the recovered public key to verify the published public key.

12. The system of claim 11 , wherein generating the recovered public key further comprises:

verifying whether a second parameter included in the digital signature is within the bounded range; and

in response to determining that the second parameter is not within the bounded range, returning an error message.

13. The system of claim 11 , wherein the operations further comprise:

in response to determining that a particular bit of the identification parameter is unity, setting the second coordinate parameter as an odd member of the modulo prime square roots; and

in response to determining that the particular bit of the identification parameter is zero, setting the second coordinate parameter as an even member of the modulo prime square roots.

14. The system of claim 11 , wherein the operations further comprise:

configuring a coordinate point that is specified by a pairing of the first coordinate parameter and the second coordinate parameter, wherein the recovered public key is generated based, at least, on the first parameter and a second parameter included by the digital signature, the coordinate point specified by the pairing of the first coordinate parameter and the second coordinate parameter, and the parameters of the elliptic curve.

15. The system of claim 11 , wherein comparing the published public key and the recovered public key to verify the published public key further comprises:

in response to determining that a match exists between the published public key and the recovered public key, determining that the published public key is verified; and

in response to determining that a match does not exist between the published public key and the recovered public key, determining that the published public key is not verified.