28ff420dd0
Co-authored-by: Jorge Chavez-Saab <jorgechavezsaab@gmail.com> Co-authored-by: Maria Corte-Real Santos <36373796+mariascrs@users.noreply.github.com> Co-authored-by: Luca De Feo <github@defeo.lu> Co-authored-by: Jonathan Komada Eriksen <jonathan.eriksen97@gmail.com> Co-authored-by: Basil Hess <bhe@zurich.ibm.com> Co-authored-by: Antonin Leroux <18654258+tonioecto@users.noreply.github.com> Co-authored-by: Patrick Longa <plonga@microsoft.com> Co-authored-by: Lorenz Panny <lorenz@yx7.cc> Co-authored-by: Francisco Rodríguez-Henríquez <francisco.rodriguez@tii.ae> Co-authored-by: Sina Schaeffler <108983332+syndrakon@users.noreply.github.com> Co-authored-by: Benjamin Wesolowski <19474926+Calodeon@users.noreply.github.com>
116 lines
3.9 KiB
Python
Executable File
116 lines
3.9 KiB
Python
Executable File
#!/usr/bin/env sage
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proof.all(False) # faster
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from sage.misc.banner import require_version
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if not require_version(9, 8, print_message=True):
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exit('')
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################################################################
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from parameters import p
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num = 7 #TODO how many extra maximal orders to precompute?
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################################################################
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# Underlying theory:
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# - Ibukiyama, On maximal orders of division quaternion algebras with certain optimal embeddings
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# - https://ia.cr/2023/106 Lemma 10
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from sage.algebras.quatalg.quaternion_algebra import basis_for_quaternion_lattice
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bfql = lambda els: basis_for_quaternion_lattice(els, reverse=True)
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Quat.<i,j,k> = QuaternionAlgebra(-1, -p)
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assert Quat.discriminant() == p # ramifies correctly
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orders = []
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q = 1
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while len(orders) < num:
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q = next_prime(q)
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if q == 2:
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continue
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Quat2.<ii,jj,kk> = QuaternionAlgebra(-q, -p)
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if Quat2.discriminant() != p: # ramifies incorrectly
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continue
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x, y = QuadraticForm(QQ, 2, [1,0,p]).solve(q)
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gamma = x + j*y
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assert gamma.reduced_norm() == q
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ims = [Quat(1), i*gamma, j, k*gamma]
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assert ims[1]^2 == -q
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assert ims[2]^2 == -p
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assert ims[1]*ims[2] == ims[3]
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assert ims[2]*ims[1] == -ims[3]
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# (1,ii,jj,kk)->ims is an isomorphism Quat2->Quat
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r = min(map(ZZ, Mod(-p, 4*q).sqrt(all=True)))
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if q % 4 == 3:
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bas2 = [
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Quat2(1),
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(1 + ii) / 2,
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jj * (1 + ii) / 2,
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(r + jj) * ii / q,
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]
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else:
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bas2 = [
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Quat2(1),
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ii,
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(1 + jj) / 2,
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(r + jj) * ii / 2 / q,
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]
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O2 = Quat2.quaternion_order(bas2)
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assert O2.discriminant() == p # is maximal
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bas = [sum(c*im for c,im in zip(el,ims)) for el in bas2]
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bas = bfql(bas)
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O = Quat.quaternion_order(bas)
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assert O.discriminant() == p # is maximal
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assert j in O # p-extremal
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mat = matrix(map(list, bas))
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# print(f'{q = }\nsqrt(-q) = {ims[1]}\n {(chr(10)+" ").join(map(str,bas))}', file=sys.stderr)
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assert mat[0] == vector((1,0,0,0))
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orders.append((q, ims[1], mat))
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################################################################
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gram = matrix(ZZ, [
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[((gi+gj).reduced_norm() - gi.reduced_norm() - gj.reduced_norm()) / 2
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for gi in Quat.basis()] for gj in Quat.basis()])
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O0mat = matrix([list(g) for g in [Quat(1), i, (i+j)/2, (1+k)/2]])
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################################################################
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from cformat import Ibz, Object, ObjectFormatter
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algobj = [Ibz(p), [[Ibz(v) for v in vs] for vs in gram]]
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O0ord = [Ibz(O0mat.denominator()), [[Ibz(v*O0mat.denominator()) for v in vs] for vs in O0mat.transpose()]]
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O0obj = [O0ord, [Ibz(1), [Ibz(c) for c in (0,1,0,0)]], [Ibz(1), [Ibz(c) for c in (0,0,1,0)]], 1]
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objs = [[[Ibz(mat.denominator()), [[Ibz(v*mat.denominator()) for v in vs] for vs in mat.transpose()]], [Ibz(mat.denominator()), [Ibz(c*mat.denominator()) for c in ii]], [Ibz(1), [Ibz(c) for c in (0,0,1,0)]], q] for q,ii,mat in orders]
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objs = ObjectFormatter([
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Object('quat_alg_t', 'QUATALG_PINFTY', algobj),
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Object('quat_order_t', 'MAXORD_O0', O0ord),
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Object('quat_p_extremal_maximal_order_t', 'STANDARD_EXTREMAL_ORDER', O0obj),
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Object('quat_p_extremal_maximal_order_t[]', 'ALTERNATE_EXTREMAL_ORDERS', objs),
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])
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with open('include/quaternion_data.h','w') as hfile:
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with open('quaternion_data.c','w') as cfile:
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print(f'#include <intbig.h>', file=hfile)
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print(f'#include <quaternion.h>', file=hfile)
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print(f'#include <stddef.h>', file=cfile)
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print(f'#include <stdint.h>', file=cfile)
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print(f'#include <quaternion_data.h>', file=cfile)
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print(f'#define NUM_ALTERNATE_EXTREMAL_ORDERS {len(orders)}', file=hfile)
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objs.header(file=hfile)
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objs.implementation(file=cfile)
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