def galois_field_monic(f, p):
lc = f[0]
if lc == 1:
return list(f)
a = invert(lc, p)
return [(a * b) % p for b in f]
def bottom_up_scan(ex):
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational:
return ex
elif ex.is_Add:
return Add(*[ bottom_up_scan(g) for g in ex.args ])
elif ex.is_Mul:
return Mul(*[ bottom_up_scan(g) for g in ex.args ])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0 and ex.base.is_Add:
coeff, terms = ex.base.as_coeff_add()
elt, _ = primitive_element(terms, polys=True)
alg = ex.base - coeff
# XXX: turn this into eval()
inverse = invert(elt.gen + coeff, elt).as_expr()
base = inverse.subs(elt.gen, alg).expand()
if ex.exp == -1:
return bottom_up_scan(base)
else:
ex = base**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (
ex.base**ex.exp.p).expand(), Rational(1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
return update_mapping(expr, 1/exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex.root not in mapping:
return update_mapping(ex.root, ex.minpoly)
else:
return symbols[ex.root]
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
def bottom_up_scan(ex):
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational:
return ex
elif ex.is_Add:
return Add(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Mul:
return Mul(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0 and ex.base.is_Add:
coeff, terms = ex.base.as_coeff_add()
elt, _ = primitive_element(terms, polys=True)
alg = ex.base - coeff
# XXX: turn this into eval()
inverse = invert(elt.gen + coeff, elt).as_expr()
base = inverse.subs(elt.gen, alg).expand()
if ex.exp == -1:
return bottom_up_scan(base)
else:
ex = base**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (ex.base**ex.exp.p).expand(), Rational(
1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
return update_mapping(expr, 1 / exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex.root not in mapping:
return update_mapping(ex.root, ex.minpoly)
else:
return symbols[ex.root]
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
def bottom_up_scan(ex):
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational:
return ex
elif ex.is_Add:
return Add(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Mul:
return Mul(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0:
minpoly_base = _minpoly_groebner(ex.base, x, cls)
inverse = invert(x, minpoly_base).as_expr()
base_inv = inverse.subs(x, ex.base).expand()
if ex.exp == -1:
return bottom_up_scan(base_inv)
else:
ex = base_inv**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (ex.base**ex.exp.p).expand(), Rational(
1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
return update_mapping(expr, 1 / exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex.root not in mapping:
return update_mapping(ex.root, ex.minpoly)
else:
return symbols[ex.root]
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
def rsa_private_key(p, q, e):
r"""
The RSA *private key* is the pair `(n,d)`, where `n`
is a product of two primes and `d` is the inverse of
`e` (mod `\phi(n)`).
Examples
========
>>> from sympy.crypto.crypto import rsa_private_key
>>> p, q, e = 3, 5, 7
>>> rsa_private_key(p, q, e)
(15, 7)
"""
n = p*q
phi = totient(n)
if isprime(p) and isprime(q) and gcd(e, phi) == 1:
d = int(invert(e,phi))
return n, d
return False
def rsa_private_key(p, q, e):
r"""
The RSA *private key* is the pair `(n,d)`, where `n`
is a product of two primes and `d` is the inverse of
`e` (mod `\phi(n)`).
Examples
========
>>> from sympy.crypto.crypto import rsa_private_key
>>> p, q, e = 3, 5, 7
>>> rsa_private_key(p, q, e)
(15, 7)
"""
n = p * q
phi = totient(n)
if isprime(p) and isprime(q) and gcd(e, phi) == 1:
d = int(invert(e, phi))
return n, d
return False
def bottom_up_scan(ex):
"""
Transform a given algebraic expression *ex* into a multivariate
polynomial, by introducing fresh variables with defining equations.
Explanation
===========
The critical elements of the algebraic expression *ex* are root
extractions, instances of :py:class:`~.AlgebraicNumber`, and negative
powers.
When we encounter a root extraction or an :py:class:`~.AlgebraicNumber`
we replace this expression with a fresh variable ``a_i``, and record
the defining polynomial for ``a_i``. For example, if ``a_0**(1/3)``
occurs, we will replace it with ``a_1``, and record the new defining
polynomial ``a_1**3 - a_0``.
When we encounter a negative power we transform it into a positive
power by algebraically inverting the base. This means computing the
minimal polynomial in ``x`` for the base, inverting ``x`` modulo this
poly (which generates a new polynomial) and then substituting the
original base expression for ``x`` in this last polynomial.
We return the transformed expression, and we record the defining
equations for new symbols using the ``update_mapping()`` function.
"""
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational:
return ex
elif ex.is_Add:
return Add(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Mul:
return Mul(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0:
minpoly_base = _minpoly_groebner(ex.base, x, cls)
inverse = invert(x, minpoly_base).as_expr()
base_inv = inverse.subs(x, ex.base).expand()
if ex.exp == -1:
return bottom_up_scan(base_inv)
else:
ex = base_inv**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (ex.base**ex.exp.p).expand(), Rational(
1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
if exp.is_Integer:
return expr.expand()
else:
return update_mapping(expr, 1 / exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex not in mapping:
return update_mapping(ex, ex.minpoly_of_element())
else:
return symbols[ex]
raise NotAlgebraic("%s does not seem to be an algebraic number" % ex)