Algorithms for generating parameters for genus 2 hyperelliptic curve cryptography
AssigneeMicrosoft Corporation
InventorsKristin E. Lauter, Ning Shang
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Abstract
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.
Claims20
A method, implemented at least in part by a computing device, comprising:
- 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 wherein the Frobenius element comprises an element of a ring of integers of the CM field;
- and
- selecting a value for x whereby the product of the Frobenius element and its complex conjugate is a prime number.
A method, implemented at least in part by a computing device, comprising:
- parameterizing the coefficients of the Frobenius element of a hyperelliptic curve as non-linear polynomials;
- and
- based at least in part on the parameterizing, determining a prime number as the product of the Frobenius element and its complex conjugate and determining one or more orders for a Jacobian of the hyperelliptic curve wherein at least one of the orders comprises an almost prime number, the almost prime number a product of a large prime number and a small cofactor.
A computing device comprising:
- a processor;
- and
- processor executable instructions to define a 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 whereby the product of the Frobenius element and its complex conjugate is a prime number.
Dependencies are inferred from the claim text.