Elliptic curve point transformations
View Patent ↗In an elliptic curve cryptographic system, point coordinates in a first coordinate system are transformed into a second coordinate system. The transformed coordinates are processed by field operations, which have been modified for operating on the transformed point coordinates. In some implementations, the point coordinates are transformed using a linear transformation matrix having coefficients. The coefficients can be fixed, variable or random. In some implementations, the transformation matrix is invertible.
1. A method performed by a system including a processor and memory, the method comprising:
obtaining a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two;
transforming, using the processor, the point from the first elliptic curve coordinate system to a transformed point in a second elliptic curve coordinate system, including performing a linear matrix transformation on the point coordinates using an n row by n column square linear matrix having n*n coefficients, where the linear matrix is invertible; and
performing one or more modified field operations on the transformed point, wherein the modified field operations are modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.
2. The method of claim 1 , where the second elliptic curve coordinate system has more dimensions than the first elliptic curve coordinate system.
3. The method of claim 1 , where the linear matrix includes at least one coefficient that is a variable or random.
4. The method of claim 1 , where the method is used in an elliptic curve encryption or decryption operation.
5. The method of claim 1 , where the method is used in an elliptic curve digital signature generation or verification operation.
6. A method performed by a system including a processor and memory, the method comprising:
obtaining a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two;
transforming, using the processor, the point from the first elliptic curve coordinate system to a transformed point in a second elliptic curve coordinate system, including performing a linear matrix transformation on the point using an n row by n column square linear matrix having n*n coefficients, where the linear matrix is invertible; and
generating ciphertext or a digital signature using the transformed point in combination with at least one modified field operation, the modified field operation being modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.
7. The method of claim 6 , where
the linear matrix includes at least one coefficient that is a variable or random.
8. A method performed by a system including a processor and memory, the method comprising:
obtaining ciphertext generated using a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two;
obtaining the point from the ciphertext;
transforming, using the processor, the point from the first elliptic curve coordinate system to a transformed point in a second elliptic curve coordinate system, including performing a linear matrix transformation on the point using an n row by n column square linear matrix having n*n coefficients, where the linear matrix is invertible; and
generating a plaintext message using the transformed point in combination with at least one modified field operation, the modified field operations being modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.
9. The method of claim 8 , where
the linear matrix includes at least one coefficient that is a variable or random.
10. A method performed by a system including a processor and memory, the method comprising:
obtaining a digital signature, the digital signature generated using a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two;
obtaining the point from the digital signature;
transforming, using the processor, the point from the first elliptic curve coordinate system to a transformed point in a second elliptic curve coordinate system, including performing a linear matrix transformation on the point using an n row by n column square linear matrix having n*n coefficients, where the linear matrix is invertible; and
authenticating the digital signature using the transformed point in combination with at least one modified field operation, wherein the modified field operation is modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.
11. An apparatus comprising:
a processor and a memory;
an interface operable for obtaining one or more point coordinates a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two; and
an encryption engine coupled to the interface and operable for:
transforming the point from the first elliptic curve coordinate system to a transformed point in a second elliptic curve coordinate system, including performing a linear matrix transformation on the point using an n row by n column square linear matrix having n*n coefficients, where the linear matrix is invertible, and
performing one or more modified field operations on the transformed point coordinate, wherein the modified field operation is modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.
12. The apparatus of claim 11 , where the second elliptic curve coordinate system has more dimensions than the first elliptic curve coordinate system.
13. The apparatus of claim 11 , where the linear matrix includes at least one coefficient that is a variable or random.
14. The apparatus of claim 11 , where the transforming is part of an elliptic curve encryption or decryption operation.
15. The apparatus of claim 11 , where the transforming is part of an elliptic curve digital signature generation or verification operation.
16. An apparatus comprising:
a processor and a memory;
an interface operable for obtaining a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two;
an encryption engine coupled to the interface and operable for:
transforming the point from the first elliptic curve coordinate system to a transformed point in a second elliptic curve coordinate system, including performing a linear matrix transformation on the point using an n row by n column square linear matrix having n*n coefficients, where the linear matrix is invertible, and
generating ciphertext or a digital signature using the transformed point in combination with at least one modified field operation, wherein the modified field operation is modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.
17. The apparatus of claim 16 , where
the linear matrix includes at least one coefficient that is a variable or random.
18. An apparatus comprising:
a processor and a memory;
an interface operable for obtaining ciphertext generated using a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two; and
a decryption engine coupled to the interface and operable for:
obtaining the point from the ciphertext, and transforming the point from the first elliptic curve coordinate system to a transformed point in a second elliptic curve coordinate system, including performing a linear matrix transformation on the point using an n row by n column square linear matrix having n*n coefficients, where the linear matrix is invertible, and
generating a plaintext message using the transformed point in combination with at least one modified field operation, wherein the modified field operation is modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.
19. The apparatus of claim 18 , where
the linear matrix includes at least one coefficient that is a variable or random.
20. An apparatus comprising:
a processor and a memory;
an interface operable for obtaining a digital signature, the digital signature generated using a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two; and
a decryption engine coupled to the interface and operable for:
obtaining the point from the digital signature, and transforming the point from the first elliptic curve coordinate system to a transformed point in a second elliptic curve coordinate system, including performing a linear matrix transformation on the point using an n row by n column square linear matrix having n*n coefficients, where the linear matrix is invertible, and
authenticating the digital signature using the transformed point in combination with at least one modified field operation, wherein the modified field operation is modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.
21. A non-transitory computer-readable storage device having instructions stored thereon, which, when executed by a processor, causes the processor to perform operations comprising:
obtaining input specifying one or more point coordinates a point on an elliptic curve specified by n point coordinates for a first elliptic curve coordinate system, wherein n is greater than or equal to two;
transforming the point from the first elliptic curve coordinate system to a transformed point in second elliptic curve coordinate system, including performing a linear matrix transformation on the at least one point using an n row by n column square linear matrix having n*n coefficients, where the matrix is invertible; and
performing one or more modified field arithmetic operations on the transformed point, wherein the modified field arithmetic operations are modified from one or more initial field operations for the first elliptic curve coordinate system for use in the second elliptic curve coordinate system based on an inverse matrix of the linear matrix.