Algorithms for generating parameters for genus 2 hyperelliptic curve cryptography
View Patent ↗An exemplary method includes defining a CM field, representing coefficients of a Frobenius element of a hyperelliptic curve over a prime field as non-linear polynomials that are functions of an integer x and selecting a value for x whereby the product of the Frobenius element and its complex conjugate is a prime number. Such a method may further include determining the order of the Jacobian of the hyperelliptic curve, for example, where the order is an almost prime number. Various other methods, devices, systems, etc., are also disclosed, which may be optionally used for cryptography.
1. A method, implemented at least in part by a computing device, comprising:
defining, by the computing device, a Complex Multiplication (CM) field;
representing coefficients of a Frobenius element of a hyperelliptic curve over a prime field as non-linear polynomials that are functions of an integer x wherein the Frobenius element comprises an element of a ring of integers of the CM field; and
iteratively selecting, by the computing device, a value for x until a product of the Frobenius element and a complex conjugate of the Frobenius element is a prime number.
2. The method of claim 1 further comprising determining an order of a Jacobian of the hyperelliptic curve.
3. The method of claim 2 further comprising determining if the order comprises an almost prime number, the almost prime number being represented as a product of a large prime number and a small cofactor.
4. The method of claim 3 further comprising implementing a discrete logarithm-based cryptography protocol using at least the prime number or the almost prime number.
5. The method of claim 4 wherein the protocol comprises a Diffie-Hellman protocol.
6. The method of claim 2 wherein the selected value for x generates an almost prime number for the order as a value of the non-linear polynomials.
7. The method of claim 1 wherein the non-linear polynomials comprise at least one second order polynomial.
8. The method of claim 1 comprising searching for factorizations of polynomials.
9. The method of claim 1 wherein the selected value for x generates a prime number as a value of the non-linear polynomials.
10. A method, implemented at least in part by a computing device, comprising:
parameterizing coefficients of the Frobenius element of a hyperelliptic curve as non-linear polynomials that include polynomials that depend on a common variable; and
based at least in part on the parameterizing, determining, by the computing device, the common variable that yields a prime number as a product of the Frobenius element and a complex conjugate of the Frobenius element and determining, by the computing device, one or more orders for a Jacobian of the hyperelliptic curve wherein at least one of the orders comprises an almost prime number.
11. The method of claim 10 wherein the common variable comprises integer values.
12. The method of claim 10 wherein the determining one or more orders comprises determining two orders.
13. The method of claim 10 wherein the non-linear polynomials comprise a second order polynomial.
14. A computing device comprising:
one or more processors; and
processor executable instructions to define a Complex Multiplication (CM) field, to represent coefficients of a Frobenius element of a hyperelliptic curve over a prime field as non-linear polynomials that are functions of an integer x wherein the Frobenius element comprises an element of a ring of integers of the CM field and to select a value for x that yields a prime number as a solution for a closed form equation that includes the non-linear polynomials, wherein the closed form equation represents a product of the Frobenius element and a complex conjugate of the Frobenius element as the prime number.
15. The computing device of claim 14 further comprising instructions to determine an order of a Jacobian of the hyperelliptic curve wherein the order comprises an almost prime number.
16. The computing device of claim 15 further comprising a network interface configured to communicate the prime number, the almost prime number or the prime number and the almost prime number for use in a cryptographic protocol.
17. The computing device of claim 16 wherein the cryptographic protocol comprises a discrete logarithm-based protocol.
18. The computing device of claim 16 wherein the cryptographic protocol comprises a protocol selected from a group consisting of an ElGammal protocol, a Diffie-Hellman protocol and a digital signature protocol.
19. The computing device of claim 14 further comprising instructions to protect a software product using, directly or indirectly, the Jacobian of a hyperelliptic curve determined by the selected value of the integer x.