https://github.com/simonmasson/efficient-glv-curves-generation

Generation of efficient four-dimensional GLV curves with high security (256-bit)
https://github.com/simonmasson/efficient-glv-curves-generation

cryptography curves elliptic glv sage

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Generation of efficient four-dimensional GLV curves with high security (256-bit)

• Host: GitHub
• URL: https://github.com/simonmasson/efficient-glv-curves-generation
• Owner: simonmasson
• License: mit
• Created: 2018-02-26T12:30:43.000Z (over 8 years ago)
• Default Branch: master
• Last Pushed: 2018-03-27T16:28:10.000Z (over 8 years ago)
• Last Synced: 2025-03-30T11:45:31.615Z (over 1 year ago)
• Topics: cryptography, curves, elliptic, glv, sage
• Language: Python
• Size: 185 KB
• Stars: 0
• Watchers: 0
• Forks: 1
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
# Efficient GLV curves generation

Generation of efficient four-dimensional GLV curves with high security (256-bit)

## How to use ?

`sage : load('generating_curves.sage')`


NB : if you want to use other primes, just change the line 84 of `generating_curves.sage` with `primes = [my, new, primes]`.

## Files description

 - auxiliary_functions.sage
 

 `separate_square_factors(r)` returns s and D such that r = s^2*D with s rational and D square-free.


 `roots_d2(P)` returns the roots of a degree 2 polynomial.


 `get_hasegawa_j_inv(d)` returns the j-invariant of a Hasegawa Q-curve of degree d.


 `get_hasegawa_reduction_coefficients(p, d, s, D)` returns A and B in GF(p^2) defining the Hasegawa

 curve y^2 = x^3 + A*x + B parametrized by s, d and Delta.

 - CM_Q-curves.sage


 `contruct_Discs_D0_f` return the list of possible discriminants for a CM Q-curve with deg(H_D) = 1 or 2.


 `construct_CM_j_roots` returns the possible j-invariants for a CM Q-curve with deg(H_D) = 1 or 2.

 

 - Hase_CM_curves.sage


  `get_Qfamily_equation_from_j(d, j0)` returns the equation satisfied by (s, Delta) to get j(d, s, Delta) = j0.


  `compute_Hase_CM_curves.sage` returns the list of (d, s, Delta) for which a CM Q-curve of degree d arises.

 

 - generating_primes.sage
 

 Computes a list of 256-bit primes with efficient finite field arithmetic, and stores it in a file `primes.sage`.

 

 - generating_curves.sage


 `get_possible_traces(p, Delta)` returns the list of the twisted curves traces for a given quadratic finite field F_{p^2} and a given endomorphisms algebra Q(√Δ).


 `get_curve_order_and_twist_orders(P, Q1, traces)` (Q1 = [p]P) return [a, L] where a is the order of the curve, and L is the list of the twisted curves orders.


 `write_curve(cpt, file, p, A, B, order, N, twist_orders, E, d, s, D)` write a new curve in file, for which  there is an efficient arithmetic, and for which we can apply the four-dimensional GLV method. The list is stored in `GLV4_curves.sage`. 

- primes.sage


File created by `generating_primes.sage`.

- GLV4_curves.sage


File created by `generating_curves.sage`. Each element of the list represents a curve, given in the form : `[p, A, B, order, N, [twists], Delta, d, s, D]` where the curve is given by E_{d, s, D} : y^2 = x^3 + A*x + B with A,B in F_{p^2} = F_p(sqrt(Delta)), #E = order, N|#E is prime, and [twists] are the orders of the twists.