Miscellaneous generic functions

A collection of functions implementing generic algorithms in arbitrary groups, including additive and multiplicative groups.

In all cases the group operation is specified by a parameter ‘operation’, which is a string either one of the set of multiplication_names or addition_names specified below, or ‘other’. In the latter case, the caller must provide an identity, inverse() and op() functions.

multiplication_names = ( 'multiplication', 'times', 'product', '*')
addition_names       = ( 'addition', 'plus', 'sum', '+')

Also included are a generic function for computing multiples (or powers), and an iterator for general multiples and powers.

EXAMPLES:

Some examples in the multiplicative group of a finite field:

  • Discrete logs:

    sage: K = GF(3^6,'b')
    sage: b = K.gen()
    sage: a = b^210
    sage: discrete_log(a, b, K.order()-1)
    210
    
  • Linear relation finder:

    sage: F.<a>=GF(3^6,'a')
    sage: a.multiplicative_order().factor()
    2^3 * 7 * 13
    sage: b=a^7
    sage: c=a^13
    sage: linear_relation(b,c,'*')
    (13, 7)
    sage: b^13==c^7
    True
    
  • Orders of elements:

    sage: k.<a> = GF(5^5)
    sage: b = a^4
    sage: order_from_multiple(b,5^5-1,operation='*')
    781
    sage: order_from_bounds(b,(5^4,5^5),operation='*')
    781
    

Some examples in the group of points of an elliptic curve over a finite field:

  • Discrete logs:

    sage: F=GF(37^2,'a')
    sage: E=EllipticCurve(F,[1,1])
    sage: F.<a>=GF(37^2,'a')
    sage: E=EllipticCurve(F,[1,1])
    sage: P=E(25*a + 16 , 15*a + 7 )
    sage: P.order()
    672
    sage: Q=39*P; Q
    (36*a + 32 : 5*a + 12 : 1)
    sage: discrete_log(Q,P,P.order(),operation='+')
    39
    
  • Linear relation finder:

    sage: F.<a>=GF(3^6,'a')
    sage: E=EllipticCurve([a^5 + 2*a^3 + 2*a^2 + 2*a, a^4 + a^3 + 2*a + 1])
    sage: P=E(a^5 + a^4 + a^3 + a^2 + a + 2 , 0)
    sage: Q=E(2*a^3 + 2*a^2 + 2*a , a^3 + 2*a^2 + 1)
    sage: linear_relation(P,Q,'+')
    (1, 2)
    sage: P == 2*Q
    True
    
  • Orders of elements:

    sage: k.<a> = GF(5^5)
    sage: E = EllipticCurve(k,[2,4])
    sage: P = E(3*a^4 + 3*a , 2*a + 1 )
    sage: M = E.cardinality(); M
    3227
    sage: plist = M.prime_factors()
    sage: order_from_multiple(P, M, plist, operation='+')
    3227
    sage: Q = E(0,2)
    sage: order_from_multiple(Q, M, plist, operation='+')
    7
    sage: order_from_bounds(Q, Hasse_bounds(5^5), operation='+')
    7
    
sage.groups.generic.bsgs(a, b, bounds, operation='*', identity=None, inverse=None, op=None)

Totally generic discrete baby-step giant-step function.

Solves \(na=b\) (or \(a^n=b\)) with \(lb\le n\le ub\) where bounds==(lb,ub), raising an error if no such \(n\) exists.

\(a\) and \(b\) must be elements of some group with given identity, inverse of x given by inverse(x), and group operation on x, y by op(x,y).

If operation is ‘*’ or ‘+’ then the other arguments are provided automatically; otherwise they must be provided by the caller.

INPUT:

  • a - group element
  • b - group element
  • bounds - a 2-tuple of integers (lower,upper) with 0<=lower<=upper
  • operation - string: ‘*’, ‘+’, ‘other’
  • identity - the identity element of the group
  • inverse() - function of 1 argument x returning inverse of x
  • op() - function of 2 arguments x, y returning x*y in group

OUTPUT:

An integer \(n\) such that \(a^n = b\) (or \(na = b\)). If no such \(n\) exists, this function raises a ValueError exception.

NOTE: This is a generalization of discrete logarithm. One situation where this version is useful is to find the order of an element in a group where we only have bounds on the group order (see the elliptic curve example below).

ALGORITHM: Baby step giant step. Time and space are soft \(O(\sqrt{n})\) where \(n\) is the difference between upper and lower bounds.

EXAMPLES:

sage: b = Mod(2,37);  a = b^20
sage: bsgs(b, a, (0,36))
20

sage: p=next_prime(10^20)
sage: a=Mod(2,p); b=a^(10^25)
sage: bsgs(a, b, (10^25-10^6,10^25+10^6)) == 10^25
True

sage: K = GF(3^6,'b')
sage: a = K.gen()
sage: b = a^210
sage: bsgs(a, b, (0,K.order()-1))
210

sage: K.<z>=CyclotomicField(230)
sage: w=z^500
sage: bsgs(z,w,(0,229))
40

An additive example in an elliptic curve group:

sage: F.<a> = GF(37^5)
sage: E = EllipticCurve(F, [1,1])
sage: P = E.lift_x(a); P
(a : 28*a^4 + 15*a^3 + 14*a^2 + 7 : 1)  # 32-bit
(a : 9*a^4 + 22*a^3 + 23*a^2 + 30 : 1)  # 64-bit

This will return a multiple of the order of P:

sage: bsgs(P,P.parent()(0),Hasse_bounds(F.order()),operation='+')
69327408

AUTHOR:

  • John Cremona (2008-03-15)
sage.groups.generic.discrete_log(a, base, ord=None, bounds=None, operation='*', identity=None, inverse=None, op=None)

Totally generic discrete log function.

INPUT:

  • a - group element
  • base - group element (the base)
  • ord - integer (multiple of order of base, or None)
  • bounds - a priori bounds on the log
  • operation - string: ‘*’, ‘+’, ‘other’
  • identity - the group’s identity
  • inverse() - function of 1 argument x returning inverse of x
  • op() - function of 2 arguments x, y returning x*y in group

a and base must be elements of some group with identity given by identity, inverse of x by inverse(x), and group operation on x, y by op(x,y).

If operation is ‘*’ or ‘+’ then the other arguments are provided automatically; otherwise they must be provided by the caller.

OUTPUT: Returns an integer \(n\) such that \(b^n = a\) (or \(nb = a\)), assuming that ord is a multiple of the order of the base \(b\). If ord is not specified, an attempt is made to compute it.

If no such \(n\) exists, this function raises a ValueError exception.

Warning

If x has a log method, it is likely to be vastly faster than using this function. E.g., if x is an integer modulo \(n\), use its log method instead!

ALGORITHM: Pohlig-Hellman and Baby step giant step.

EXAMPLES:

sage: b = Mod(2,37);  a = b^20
sage: discrete_log(a, b)
20
sage: b = Mod(2,997);  a = b^20
sage: discrete_log(a, b)
20

sage: K = GF(3^6,'b')
sage: b = K.gen()
sage: a = b^210
sage: discrete_log(a, b, K.order()-1)
210

sage: b = Mod(1,37);  x = Mod(2,37)
sage: discrete_log(x, b)
Traceback (most recent call last):
...
ValueError: No discrete log of 2 found to base 1
sage: b = Mod(1,997);  x = Mod(2,997)
sage: discrete_log(x, b)
Traceback (most recent call last):
...
ValueError: No discrete log of 2 found to base 1

See trac\#2356:
sage: F.<w> = GF(121)
sage: v = w^120
sage: v.log(w)
0

sage: K.<z>=CyclotomicField(230)
sage: w=z^50
sage: discrete_log(w,z)
50

An example where the order is infinite: note that we must give an upper bound here:

sage: K.<a> = QuadraticField(23)
sage: eps = 5*a-24        # a fundamental unit
sage: eps.multiplicative_order()
+Infinity
sage: eta = eps^100
sage: discrete_log(eta,eps,bounds=(0,1000))
100

In this case we cannot detect negative powers:

sage: eta = eps^(-3)
sage: discrete_log(eta,eps,bounds=(0,100))
Traceback (most recent call last):
...
ValueError: No discrete log of -11515*a - 55224 found to base 5*a - 24

But we can invert the base (and negate the result) instead:

sage: - discrete_log(eta^-1,eps,bounds=(0,100))
-3

An additive example: elliptic curve DLOG:

sage: F=GF(37^2,'a')
sage: E=EllipticCurve(F,[1,1])
sage: F.<a>=GF(37^2,'a')
sage: E=EllipticCurve(F,[1,1])
sage: P=E(25*a + 16 , 15*a + 7 )
sage: P.order()
672
sage: Q=39*P; Q
(36*a + 32 : 5*a + 12 : 1)
sage: discrete_log(Q,P,P.order(),operation='+')
39

An example of big smooth group:

sage: F.<a>=GF(2^63)
sage: g=F.gen()
sage: u=g**123456789
sage: discrete_log(u,g)
123456789

AUTHORS:

  • William Stein and David Joyner (2005-01-05)
  • John Cremona (2008-02-29) rewrite using dict() and make generic
sage.groups.generic.discrete_log_generic(a, base, ord=None, bounds=None, operation='*', identity=None, inverse=None, op=None)

Alias for discrete_log.

sage.groups.generic.discrete_log_lambda(a, base, bounds, operation='*', hash_function=<built-in function hash>)

Pollard Lambda algorithm for computing discrete logarithms. It uses only a logarithmic amount of memory. It’s useful if you have bounds on the logarithm. If you are computing logarithms in a whole finite group, you should use Pollard Rho algorithm.

INPUT:

  • a - a group element
  • base - a group element
  • bounds - a couple (lb,ub) representing the range where we look for a logarithm
  • operation - string: ‘+’, ‘*’ or ‘other’
  • hash_function – having an efficient hash function is critical for this algorithm

OUTPUT: Returns an integer \(n\) such that \(a=base^n\) (or \(a=n*base\))

ALGORITHM: Pollard Lambda, if bounds are (lb,ub) it has time complexity
O(sqrt(ub-lb)) and space complexity O(log(ub-lb))

EXEMPLES:

sage: F.<a> = GF(2^63)
sage: discrete_log_lambda(a^1234567, a, (1200000,1250000))
1234567

sage: F.<a> = GF(37^5)
sage: E = EllipticCurve(F, [1,1])
sage: P = E.lift_x(a); P
(a : 9*a^4 + 22*a^3 + 23*a^2 + 30 : 1)  # 32-bit
(a : 28*a^4 + 15*a^3 + 14*a^2 + 7 : 1)  # 64-bit

This will return a multiple of the order of P:

sage: discrete_log_lambda(P.parent()(0), P, Hasse_bounds(F.order()), operation='+')
69327408

sage: K.<a> = GF(89**5)
sage: hs = lambda x: hash(x) + 15
sage: discrete_log_lambda(a**(89**3 - 3), a, (89**2, 89**4), operation = '*', hash_function = hs)  # long time (10s on sage.math, 2011)
704966

AUTHOR:

– Yann Laigle-Chapuy (2009-01-25)
sage.groups.generic.discrete_log_rho(a, base, ord=None, operation='*', hash_function=<built-in function hash>)

Pollard Rho algorithm for computing discrete logarithm in cyclic group of prime order. If the group order is very small it falls back to the baby step giant step algorithm.

INPUT:

  • a – a group element
  • base – a group element
  • ord – the order of base or None, in this case we try to compute it
  • operation – a string (default: '*') denoting whether we are in an additive group or a multiplicative one
  • hash_function – having an efficient hash function is critical for this algorithm (see examples)

OUTPUT: an integer \(n\) such that \(a = base^n\) (or \(a = n*base\))

ALGORITHM: Pollard rho for discrete logarithm, adapted from the article of Edlyn Teske, ‘A space efficient algorithm for group structure computation’.

EXAMPLES:

sage: F.<a> = GF(2^13)
sage: g = F.gen()
sage: discrete_log_rho(g^1234, g)
1234

sage: F.<a> = GF(37^5)
sage: E = EllipticCurve(F, [1,1])
sage: G = (3*31*2^4)*E.lift_x(a)
sage: discrete_log_rho(12345*G, G, ord=46591, operation='+')
12345

It also works with matrices:

sage: A = matrix(GF(50021),[[10577,23999,28893],[14601,41019,30188],[3081,736,27092]])
sage: discrete_log_rho(A^1234567, A)
1234567

Beware, the order must be prime:

sage: I = IntegerModRing(171980)
sage: discrete_log_rho(I(2), I(3))
Traceback (most recent call last):
...
ValueError: for Pollard rho algorithm the order of the group must be prime

If it fails to find a suitable logarithm, it raises a ValueError:

sage: I = IntegerModRing(171980)
sage: discrete_log_rho(I(31002),I(15501))
Traceback (most recent call last):
...
ValueError: Pollard rho algorithm failed to find a logarithm

The main limitation on the hash function is that we don’t want to have \(hash(x*y) = hash(x) + hash(y)\):

sage: I = IntegerModRing(next_prime(2^23))
sage: def test():
....:     try:
....:          discrete_log_rho(I(123456),I(1),operation='+')
....:     except Exception:
....:          print "FAILURE"
sage: test()  # random failure
FAILURE

If this happens, we can provide a better hash function:

sage: discrete_log_rho(I(123456),I(1),operation='+', hash_function=lambda x: hash(x*x))
123456

AUTHOR:

  • Yann Laigle-Chapuy (2009-09-05)
sage.groups.generic.linear_relation(P, Q, operation='+', identity=None, inverse=None, op=None)

Function which solves the equation a*P=m*Q or P^a=Q^m.

Additive version: returns \((a,m)\) with minimal \(m>0\) such that \(aP=mQ\). Special case: if \(\left<P\right>\) and \(\left<Q\right>\) intersect only in \(\{0\}\) then \((a,m)=(0,n)\) where \(n\) is Q.additive_order().

Multiplicative version: returns \((a,m)\) with minimal \(m>0\) such that \(P^a=Q^m\). Special case: if \(\left<P\right>\) and \(\left<Q\right>\) intersect only in \(\{1\}\) then \((a,m)=(0,n)\) where \(n\) is Q.multiplicative_order().

ALGORITHM:

Uses the generic bsgs() function, and so works in general finite abelian groups.

EXAMPLES:

An additive example (in an elliptic curve group):

sage: F.<a>=GF(3^6,'a')
sage: E=EllipticCurve([a^5 + 2*a^3 + 2*a^2 + 2*a, a^4 + a^3 + 2*a + 1])
sage: P=E(a^5 + a^4 + a^3 + a^2 + a + 2 , 0)
sage: Q=E(2*a^3 + 2*a^2 + 2*a , a^3 + 2*a^2 + 1)
sage: linear_relation(P,Q,'+')
(1, 2)
sage: P == 2*Q
True

A multiplicative example (in a finite field’s multiplicative group):

sage: F.<a>=GF(3^6,'a')
sage: a.multiplicative_order().factor()
2^3 * 7 * 13
sage: b=a^7
sage: c=a^13
sage: linear_relation(b,c,'*')
(13, 7)
sage: b^13==c^7
True
sage.groups.generic.merge_points(P1, P2, operation='+', identity=None, inverse=None, op=None, check=True)

Returns a group element whose order is the lcm of the given elements.

INPUT:

  • P1 – a pair \((g_1,n_1)\) where \(g_1\) is a group element of order \(n_1\)

  • P2 – a pair \((g_2,n_2)\) where \(g_2\) is a group element of order \(n_2\)

  • operation – string: ‘+’ (default ) or ‘*’ or other. If other, the following must be supplied:

    • identity: the identity element for the group;

    • inverse(): a function of one argument giving the inverse of a group element;

    • op(): a function of 2 arguments defining the group

      binary operation.

OUTPUT:

A pair \((g_3,n_3)\) where \(g_3\) has order \(n_3=\hbox{lcm}(n_1,n_2)\).

EXAMPLES:

sage: F.<a>=GF(3^6,'a')
sage: b = a^7
sage: c = a^13
sage: ob = (3^6-1)//7
sage: oc = (3^6-1)//13
sage: merge_points((b,ob),(c,oc),operation='*')
(a^4 + 2*a^3 + 2*a^2, 728)
sage: d,od = merge_points((b,ob),(c,oc),operation='*')
sage: od == d.multiplicative_order()
True
sage: od == lcm(ob,oc)
True

sage: E=EllipticCurve([a^5 + 2*a^3 + 2*a^2 + 2*a, a^4 + a^3 + 2*a + 1])
sage: P=E(2*a^5 + 2*a^4 + a^3 + 2 , a^4 + a^3 + a^2 + 2*a + 2)
sage: P.order()
7
sage: Q=E(2*a^5 + 2*a^4 + 1 , a^5 + 2*a^3 + 2*a + 2 )
sage: Q.order()
4
sage: R,m = merge_points((P,7),(Q,4), operation='+')
sage: R.order() == m
True
sage: m == lcm(7,4)
True
sage.groups.generic.multiple(a, n, operation='*', identity=None, inverse=None, op=None)

Returns either \(na\) or \(a^n\), where \(n\) is any integer and \(a\) is a Python object on which a group operation such as addition or multiplication is defined. Uses the standard binary algorithm.

INPUT: See the documentation for discrete_logarithm().

EXAMPLES:

sage: multiple(2,5)
32
sage: multiple(RealField()('2.5'),4)
39.0625000000000
sage: multiple(2,-3)
1/8
sage: multiple(2,100,'+') == 100*2
True
sage: multiple(2,100) == 2**100
True
sage: multiple(2,-100,) == 2**-100
True
sage: R.<x>=ZZ[]
sage: multiple(x,100)
x^100
sage: multiple(x,100,'+')
100*x
sage: multiple(x,-10)
1/x^10

Idempotence is detected, making the following fast:

sage: multiple(1,10^1000)
1

sage: E=EllipticCurve('389a1')
sage: P=E(-1,1)
sage: multiple(P,10,'+')
(645656132358737542773209599489/22817025904944891235367494656 : 525532176124281192881231818644174845702936831/3446581505217248068297884384990762467229696 : 1)
sage: multiple(P,-10,'+')
(645656132358737542773209599489/22817025904944891235367494656 : -528978757629498440949529703029165608170166527/3446581505217248068297884384990762467229696 : 1)
class sage.groups.generic.multiples(P, n, P0=None, indexed=False, operation='+', op=None)

Return an iterator which runs through P0+i*P for i in range(n).

P and P0 must be Sage objects in some group; if the operation is multiplication then the returned values are instead P0*P**i.

EXAMPLES:

sage: list(multiples(1,10))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: list(multiples(1,10,100))
[100, 101, 102, 103, 104, 105, 106, 107, 108, 109]

sage: E=EllipticCurve('389a1')
sage: P=E(-1,1)
sage: for Q in multiples(P,5): print Q, Q.height()/P.height()
(0 : 1 : 0) 0.000000000000000
(-1 : 1 : 1) 1.00000000000000
(10/9 : -35/27 : 1) 4.00000000000000
(26/361 : -5720/6859 : 1) 9.00000000000000
(47503/16641 : 9862190/2146689 : 1) 16.0000000000000

sage: R.<x>=ZZ[]
sage: list(multiples(x,5))
[0, x, 2*x, 3*x, 4*x]
sage: list(multiples(x,5,operation='*'))
[1, x, x^2, x^3, x^4]
sage: list(multiples(x,5,indexed=True))
[(0, 0), (1, x), (2, 2*x), (3, 3*x), (4, 4*x)]
sage: list(multiples(x,5,indexed=True,operation='*'))
[(0, 1), (1, x), (2, x^2), (3, x^3), (4, x^4)]
sage: for i,y in multiples(x,5,indexed=True): print "%s  times %s = %s"%(i,x,y)
0  times x = 0
1  times x = x
2  times x = 2*x
3  times x = 3*x
4  times x = 4*x

sage: for i,n in multiples(3,5,indexed=True,operation='*'):  print "3 to the power %s = %s"%(i,n)
3 to the power 0 = 1
3 to the power 1 = 3
3 to the power 2 = 9
3 to the power 3 = 27
3 to the power 4 = 81
next()

Returns the next item in this multiples iterator.

sage.groups.generic.order_from_bounds(P, bounds, d=None, operation='+', identity=None, inverse=None, op=None)

Generic function to find order of a group element, given only upper and lower bounds for a multiple of the order (e.g. bounds on the order of the group of which it is an element)

INPUT:

  • P - a Sage object which is a group element
  • bounds - a 2-tuple (lb,ub) such that m*P=0 (or P**m=1) for some m with lb<=m<=ub.
  • d - (optional) a positive integer; only m which are multiples of this will be considered.
  • operation - string: ‘+’ (default ) or ‘*’ or other. If other, the following must be supplied:
    • identity: the identity element for the group;
    • inverse(): a function of one argument giving the inverse of a group element;
    • op(): a function of 2 arguments defining the group binary operation.

Note

Typically lb and ub will be bounds on the group order, and from previous calculation we know that the group order is divisible by d.

EXAMPLES:

sage: k.<a> = GF(5^5)
sage: b = a^4
sage: order_from_bounds(b,(5^4,5^5),operation='*')
781
sage: E = EllipticCurve(k,[2,4])
sage: P = E(3*a^4 + 3*a , 2*a + 1 )
sage: bounds = Hasse_bounds(5^5)
sage: Q = E(0,2)
sage: order_from_bounds(Q, bounds, operation='+')
7
sage: order_from_bounds(P, bounds, 7, operation='+')
3227

sage: K.<z>=CyclotomicField(230)
sage: w=z^50
sage: order_from_bounds(w,(200,250),operation='*')
23
sage.groups.generic.order_from_multiple(P, m, plist=None, factorization=None, check=True, operation='+')

Generic function to find order of a group element given a multiple of its order.

INPUT:

  • P - a Sage object which is a group element;
  • m - a Sage integer which is a multiple of the order of P, i.e. we require that m*P=0 (or P**m=1);
  • check - a Boolean (default:True), indicating whether we check if m really is a multiple of the order;
  • factorization - the factorization of m, or None in which case this function will need to factor m;
  • plist - a list of the prime factors of m, or None - kept for compatibility only, prefer the use of factorization;
  • operation - string: ‘+’ (default) or ‘*’.

Note

It is more efficient for the caller to factor m and cache the factors for subsequent calls.

EXAMPLES:

sage: k.<a> = GF(5^5)
sage: b = a^4
sage: order_from_multiple(b,5^5-1,operation='*')
781
sage: E = EllipticCurve(k,[2,4])
sage: P = E(3*a^4 + 3*a , 2*a + 1 )
sage: M = E.cardinality(); M
3227
sage: F = M.factor()
sage: order_from_multiple(P, M, factorization=F, operation='+')
3227
sage: Q = E(0,2)
sage: order_from_multiple(Q, M, factorization=F, operation='+')
7

sage: K.<z>=CyclotomicField(230)
sage: w=z^50
sage: order_from_multiple(w,230,operation='*')
23

sage: F=GF(2^1279,'a')
sage: n=F.cardinality()-1 # Mersenne prime
sage: order_from_multiple(F.random_element(),n,factorization=[(n,1)],operation='*')==n
True

sage: K.<a> = GF(3^60)
sage: order_from_multiple(a, 3^60-1, operation='*', check=False)
42391158275216203514294433200

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