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ernst.py
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ernst.py
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import logging
from sage.all import RR
from sage.all import ZZ
from sage.all import gcd
from shared import small_roots
def integer_trivariate_1(f, m, t, W, X, Y, Z, check_bounds=True, roots_method="groebner"):
"""
Computes small integer roots of a trivariate polynomial.
More information: Ernst M. et al., "Partial Key Exposure Attacks on RSA Up to Full Size Exponents" (Section 4.1.1)
:param f: the polynomial
:param m: the parameter m
:param t: the parameter t
:param W: the parameter W
:param X: an approximate bound on the x roots
:param Y: an approximate bound on the y roots
:param Z: an approximate bound on the z roots
:param check_bounds: perform bounds check (default: True)
:param roots_method: the method to use to find roots (default: "groebner")
:return: a generator generating small roots (tuples of x, y, and z roots) of the polynomial
"""
pr = f.parent()
x, y, z = pr.gens()
tau = t / m
if check_bounds and RR(X) ** (1 + 3 * tau) * RR(Y) ** (2 + 3 * tau) * RR(Z) ** (1 + 3 * tau + 3 * tau ** 2) > RR(W) ** (1 + 3 * tau):
logging.debug(f"Bound check failed for {m = }, {t = }")
return
R = int(f.constant_coefficient())
assert R != 0
while gcd(R, X) != 1:
X += 1
while gcd(R, Y) != 1:
Y += 1
while gcd(R, Z) != 1:
Z += 1
while gcd(R, W) != 1:
W += 1
n = (X * Y) ** m * Z ** (m + t) * W
assert gcd(R, n) == 1
f_ = (pow(R, -1, n) * f % n).change_ring(ZZ)
logging.debug("Generating shifts...")
shifts = []
for i in range(m + 1):
for j in range(m - i + 1):
for k in range(j + 1):
g = x ** i * y ** j * z ** k * f_ * X ** (m - i) * Y ** (m - j) * Z ** (m + t - k)
shifts.append(g)
for k in range(j + 1, j + t + 1):
h = x ** i * y ** j * z ** k * f_ * X ** (m - i) * Y ** (m - j) * Z ** (m + t - k)
shifts.append(h)
for i in range(m + 2):
j = m + 1 - i
for k in range(j + 1):
g_ = n * x ** i * y ** j * z ** k
shifts.append(g_)
for k in range(j + 1, j + t + 1):
h_ = n * x ** i * y ** j * z ** k
shifts.append(h_)
L, monomials = small_roots.create_lattice(pr, shifts, [X, Y, Z])
L = small_roots.reduce_lattice(L)
polynomials = small_roots.reconstruct_polynomials(L, f, n, monomials, [X, Y, Z])
for roots in small_roots.find_roots(pr, [f] + polynomials, method=roots_method):
yield roots[x], roots[y], roots[z]
def integer_trivariate_2(f, m, t, W, X, Y, Z, check_bounds=True, roots_method="groebner"):
"""
Computes small integer roots of a trivariate polynomial.
More information: Ernst M. et al., "Partial Key Exposure Attacks on RSA Up to Full Size Exponents" (Section 4.1.2)
:param f: the polynomial
:param m: the parameter m
:param t: the parameter t
:param W: the parameter W
:param X: an approximate bound on the x roots
:param Y: an approximate bound on the y roots
:param Z: an approximate bound on the z roots
:param check_bounds: perform bounds check (default: True)
:param roots_method: the method to use to find roots (default: "groebner")
:return: a generator generating small roots (tuples of x, y, and z roots) of the polynomial
"""
pr = f.parent()
x, y, z = pr.gens()
tau = t / m
if check_bounds and RR(X) ** (2 + 3 * tau) * RR(Y) ** (3 + 6 * tau + 3 * tau ** 2) * RR(Z) ** (3 + 3 * tau) > RR(W) ** (2 + 3 * tau):
logging.debug(f"Bound check failed for {m = }, {t = }")
return
R = int(f.constant_coefficient())
assert R != 0
while gcd(R, X) != 1:
X += 1
while gcd(R, Y) != 1:
Y += 1
while gcd(R, Z) != 1:
Z += 1
while gcd(R, W) != 1:
W += 1
n = X ** m * Y ** (m + t) * Z ** m * W
assert gcd(R, n) == 1
f_ = (pow(R, -1, n) * f % n).change_ring(ZZ)
logging.debug("Generating shifts...")
shifts = []
for i in range(m + 1):
for j in range(m - i + 1):
for k in range(m - i + 1):
g = x ** i * y ** j * z ** k * f_ * X ** (m - i) * Y ** (m + t - j) * Z ** (m - k)
shifts.append(g)
for j in range(m - i + 1, m - i + t + 1):
for k in range(m - i + 1):
h = x ** i * y ** j * z ** k * f_ * X ** (m - i) * Y ** (m + t - j) * Z ** (m - k)
shifts.append(h)
for i in range(m + 2):
for j in range(m + t + 2 - i):
k = m + 1 - i
g_ = n * x ** i * y ** j * z ** k
shifts.append(g_)
for i in range(m + 1):
j = m + t + 1 - i
for k in range(m - i + 1):
h_ = n * x ** i * y ** j * z ** k
shifts.append(h_)
L, monomials = small_roots.create_lattice(pr, shifts, [X, Y, Z])
L = small_roots.reduce_lattice(L)
polynomials = small_roots.reconstruct_polynomials(L, f, n, monomials, [X, Y, Z])
for roots in small_roots.find_roots(pr, [f] + polynomials, method=roots_method):
yield roots[x], roots[y], roots[z]