The following answer does not implement the two functions "add" and "double" on an elliptic curve, instead, it uses the already existing functionality of sage. (It would be too much to implement also these basic functions depending on the form of the elliptic curve and the base field characteristic.)
Code:
def left_to_right_double_and_add(n, G):
"""n is a positive integer number,
G is a rational point of an elliptic curve E,
defined over some finite field F.
This routine computes
n*G
in an ad-hoc manner.
We use the standard add, and doubling on E,
and do not implemented them here.
"""
digits = ZZ(n).digits(base=2)
E = G.curve()
# F = E.base_field()
P = G # this corresponds to 1*G, init
nG = E(0) # and we add further contributions
for digit in digits:
if digit:
nG = nG.__add__(P) # or simply nG += P
P = P.__mul__(2) # or simply P *= 2
# print "\tdigit=%s P=%s nG=%s" % (digit, P, nG)
return nG
# quick test on the given curve with the given poing:
E = EllipticCurve( GF(37), [7, 25] )
G = E.point( [33, 9] )
for n in [1..9]:
nG = left_to_right_double_and_add(n, G)
print ( "sage %s*G = %13s | ad-hoc %s*G = %13s | Are they equal? %s"
% ( n, n*G, n, nG, n*G == nG ) )
This gives in the test:
sage 1*G = (33 : 9 : 1) | ad-hoc 1*G = (33 : 9 : 1) | Are they equal? True
sage 2*G = (9 : 15 : 1) | ad-hoc 2*G = (9 : 15 : 1) | Are they equal? True
sage 3*G = (2 : 11 : 1) | ad-hoc 3*G = (2 : 11 : 1) | Are they equal? True
sage 4*G = (35 : 15 : 1) | ad-hoc 4*G = (35 : 15 : 1) | Are they equal? True
sage 5*G = (15 : 8 : 1) | ad-hoc 5*G = (15 : 8 : 1) | Are they equal? True
sage 6*G = (30 : 22 : 1) | ad-hoc 6*G = (30 : 22 : 1) | Are they equal? True
sage 7*G = (1 : 25 : 1) | ad-hoc 7*G = (1 : 25 : 1) | Are they equal? True
sage 8*G = (31 : 27 : 1) | ad-hoc 8*G = (31 : 27 : 1) | Are they equal? True
sage 9*G = (17 : 32 : 1) | ad-hoc 9*G = (17 : 32 : 1) | Are they equal? True
the above code gives me following error when run it on sagemath cloud Error in lines 13-16 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 4 salvus.execute_with_code_decorators(*_salvus_parsing.dec_args[Integer(3)]) ^ SyntaxError: invalid syntax
Without the $[n \bmod order]G$ check this code will fail as see on Stackoverflow : order of a point is not regular in Hessian Curve implementation
Asked: 2018-03-29 07:31:23 +0100
Seen: 2,379 times
Last updated: Mar 30 '18
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In a dialog with the sage interpreter, by "coincidence", the following data ould be digested:
(I was trying to rather guess the framework.)
What is/are in this context the "left to right double and add scalar multiplication"?
(As seen, sage simply computes the multiplication
21*G.)tx for your reply but i want the compution using above algorithm. also by defining function point_mul(k,p) where k is scalar and p is the point.
Which algorithm should be used? (There is no description of an algorithm, no reference for a specific one.) Why not simply use the already implemented sage algorithm?
Note that
21*Gis computed above by using the class method__mul__for the class of the instanceG.For instance:
left to right double and add scalar multiplication algorithm for above parameter KP=k_0p+k_12p+K_24P+.........+K_l2^l-1*P I/P point P which is on elliptic curve , K=(k_l-1K_l-2.....K_0)_2 (that is binary representation of scalar) output KP uses P and Q
Q= infinity
for i=l-1 to 0 do Q=2Q if k_i=1 then Q=Q+P
return Q