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scalar Multiplication

asked 2017-12-15 06:31:11 +0200

anonymous user

Anonymous

i have implemented point addition and point doubling algorithm over elliptic curve. i want to use same algorithm for scalar multiplication i am unable to generate code for the same please guide. Both the sage math codes are given here.

point Addition

import time p =71 A=41 B=18 print"\n p=",p print"\n A=",A print"\n B=",B

F = GF(p) E = EllipticCurve( F, [A,B] );

S. = PolynomialRing( F ) K. = GF( p^2);#K.modulus#, modulus=W^2+W+1 ) print "\n Modulus of K is =", K.modulus()

R.<z> = PolynomialRing( K, sparse=True )

x= (65a + 42)z^71 + (6a + 30)z

y= (6a + 65)z^71 + (65a + 6)z

f=(22a + 9)z^213 + (23a + 52)z^143 + 68z^142 + (48a + 27)z^73 + 6z^72 + (33a + 53)z^71 + (49a + 53)z^3 + 68z^2 + (38a + 48)*z + 53

print "\n Elliptic Curve fE(z) = ", f

P = E.random_point() Q = E.random_point()

print "\n point1 =", P.xy() print "\n point2 =", Q.xy()

start1 = time.time() H = P + Q; end1 = time.time() Hx, Hy = H.xy() print "\n H = P + Q has the components:\nHx = %s\nHy = %s" % ( Hx, Hy ) print "\n Time for this operation (existing approach): %s" % ( end1 - start1 )

if P[0] != Q[0] : start = time.time()

g = ( + ( Q[0] - P[0] ) * y
      - ( Q[1] - P[1] ) * x
      - ( P[1]*Q[0] - Q[1]*P[0] ) )
print "\n Line equation =", g

f1 = ( f % g ).monic()
print "\n f1 = gcd (fE,fL) = ", f1

f2 = ( z - P[0] - P[1]*a ) * ( z - Q[0] - Q[1]*a )
print "\n point equation =", f2

f3 = f1 // f2
print "\n Division of two polynomials f1 & f2 =", f3

c = f3.constant_coefficient()
M0, M1 = S(c).coefficients()

G = E.point( ( -M0, M1 ) )
end = time.time()

Gx, Gy = G.xy()
print "\n G = P + Q has the components:\nGx = %s\nGy = %s" % ( Gx, Gy )
print "\n Time for this operation (new approach): %s" % ( end - start )

point Doubling

import time p =71 A=41 B=18 print"\n p=",p print"\n A=",A print"\n B=",B

F = GF(p) E = EllipticCurve( F, [A,B] );

S. = PolynomialRing( F ) K. = GF( p^2);#K.modulus#, modulus=W^2+W+1 ) print "\n Modulus of K is =", K.modulus()

R.<z> = PolynomialRing( K, sparse=True )

x= (65a + 42)z^71 + (6a + 30)z

y= (6a + 65)z^71 + (65a + 6)z

f=(22a + 9)z^213 + (23a + 52)z^143 + 68z^142 + (48a + 27)z^73 + 6z^72 + (33a + 53)z^71 + (49a + 53)z^3 + 68z^2 + (38a + 48)*z + 53

print "\n Elliptic Curve fE(z) = ", f

P = E(63,32)#.random_point()

Q = E.random_point()

print "\n point1 =", P.xy()

print "\n point2 =", Q.xy()

start1 = time.time() H = 2*P; end1 = time.time() Hx, Hy = H.xy() print "\n H = P + Q has the components:\nHx = %s\nHy = %s" % ( Hx, Hy ) print "\n Time for this operation (existing approach): %s" % ( end1 - start1 )

start = time.time()

g = ( ( 2 * P[1] * y ) - ( ( 3 * P[0]^2) + A ) * x -(-P[0]^3 + ( A * P[0] ) + 2 * B) ) print "\n Line equation =", g

f1 = ( f % g ).monic() print "\n f1 = gcd (fE,fL) = ", f1

f2 = ( z - P[0] - P[1]a )( z - P[0] - P[1]*a ) print "\n point equation =", f2

f3 = f1 // f2 print "\n Division of two polynomials f1 & f2 =", f3

c = f3.constant_coefficient() f4=(-c) M0, M1 = S(f4).coefficients();M0,M1

G = E.point( ( M0, -M1 ) ) end = time.time()

Gx, Gy = G.xy() print "\n G = P + Q has the components:\nGx = %s\nGy = %s" % ( Gx, Gy ) print "\n Time for this operation (new approach): %s" % ( end - start )

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answered 2017-12-15 13:07:14 +0200

dan_fulea gravatar image

The code had to be arranged (lots of work, please insert correctly the code, so that it can be pasted, and so that it works).

  • The implemented algorithms did not came as a function, instead, there was code doing the job for some, but not all cases. - Also, there was no description of the algorithms, so there is also a small shape of mathematics and understanding what the code does involved in between.
  • if there is a simpler (structural) way to post the code, please do it.
  • The following is not optimal, it was partially typed in train, then bug-fixed (not improved) till it worked. Please rearrange it for the own needs.
  • Using globals is not a good solution, please rewrite for the own needs. (Depending on needs, either speed or solid style are relevant.)

Here is the code, that did the job for me.

# # # ==============================
# GLOBALS 
p = 71
F = GF(p)

A = F(41)
B = F(18)

E = EllipticCurve( F, [A, B] )
ZERO = E(0)

S.<W> = PolynomialRing( F )
K.<a> = GF( p^2, modulus=W^2+W+1 )

R.<z> = PolynomialRing( K, sparse=True )

x = (a + 2  )/3 * (z^p-a*z)
y = (1 + 2*a)/3 * (z^p-  z)
f = y^2 - ( x^3 + A*x + B )

# The following routines are not optimal, please improve
def get_coefficients( pol, padto=2 ):
    """To be sure that we always have two coefficients,
    also for polynomials (of degree one) 
    with the one or the other vanishing coefficient.
    """
    coefs = pol.coefficients( sparse=False )
    if len(coefs) < padto:
        return coefs + ( ( padto-len(coefs) ) * [ F(0), ] )
    return coefs


def add( P, Q ):
    """ad-hoc addition of two points on the same curve
    """
    # print "Adding", P, Q
    if P == ZERO:
        return Q
    if Q == ZERO:
        return P

    if P[0] == Q[0]:
        if P[1] == Q[1]:
            return double(P)
        elif P[1] == - Q[1]:
            return ZERO
        else:
            return "*** INTERNAL ERROR"

    g = ( + ( Q[0] - P[0] ) * y
          - ( Q[1] - P[1] ) * x
          - ( P[1]*Q[0] - Q[1]*P[0] ) )

    f1 = ( f % g ).monic()
    f2 = ( z - P[0] - P[1]*a ) * ( z - Q[0] - Q[1]*a )
    f3 = f1 // f2
    c  = f3.constant_coefficient()
    M0, M1 = get_coefficients( S(c) )

    return E.point( ( -M0, M1 ) )


def double( P ):
    """ad-hoc doubling map
    """
    # print "doubling", P
    if P == ZERO or P[1] == 0:
        return ZERO

    g = ( 2 * P[1] * y
          - ( 3 * P[0]^2 + A ) * x
          - ( -P[0]^3 + A * P[0] + 2*B ) )

    f1 = ( f % g ).monic()
    f2 = ( z - P[0] - P[1]*a ) * ( z - P[0] - P[1]*a )
    f3 = f1 // f2
    c  = f3.constant_coefficient()
    M0, M1 = get_coefficients( S(-c) )

    return E.point( ( M0, -M1 ) )

def mult( P, k ):
    """ad-hoc multiplication P -> k*P on the elliptic curve E
    """
    # if k not in ZZ:
    #     return "*** ERROR"
    # print "computing %s * %s" % ( k, str(P) )
    if k == 0:
        return P.parent(0)
    if k<0:
        return -mult(P, -k)
    rest = k%2
    khalf = k//2
    if khalf:
        Q = double( mult(P, khalf) )
    else:
        Q = P.parent(0)
    if rest:
        return add(P, Q)
    return Q

# # # ================================================
print "TEST SUITE:"
bad = []
k   = 21378964192843
print "k =", k
count = 0
for P in E.points():
    count += 1
    Q1, Q2 = mult(P, k), k*P
    print ( "%2d Point P = %13s | mult(P,k) = %13s | k*P = %13s | %s"
            % ( count, P, Q1, Q2, bool(Q1==Q2) ) )

    if Q1 != Q2:
        bad.append(P)

if bad:
    print "ERRORS:", bad
else:
    print "OK"

Results for the tests:

TEST SUITE:
k = 21378964192843
 1 Point P =   (0 : 1 : 0) | mult(P,k) =   (0 : 1 : 0) | k*P =   (0 : 1 : 0) | True
 2 Point P =  (0 : 35 : 1) | mult(P,k) = (43 : 63 : 1) | k*P = (43 : 63 : 1) | True
 3 Point P =  (0 : 36 : 1) | mult(P,k) =  (43 : 8 : 1) | k*P =  (43 : 8 : 1) | True
 4 Point P =  (1 : 29 : 1) | mult(P,k) =  (0 : 35 : 1) | k*P =  (0 : 35 : 1) | True
 5 Point P =  (1 : 42 : 1) | mult(P,k) =  (0 : 36 : 1) | k*P =  (0 : 36 : 1) | True
 6 Point P =  (2 : 26 : 1) | mult(P,k) = (27 : 17 : 1) | k*P = (27 : 17 : 1) | True
 7 Point P =  (2 : 45 : 1) | mult(P,k) = (27 : 54 : 1) | k*P = (27 : 54 : 1) | True
 8 Point P =   (5 : 8 : 1) | mult(P,k) = (68 : 62 : 1) | k*P = (68 : 62 : 1) | True
 9 Point P =  (5 : 63 : 1) | mult(P,k) =  (68 : 9 : 1) | k*P =  (68 : 9 : 1) | True
10 Point P =  (6 : 14 : 1) | mult(P,k) =   (5 : 8 : 1) | k*P =   (5 : 8 : 1) | True
11 Point P =  (6 : 57 : 1) | mult(P,k) =  (5 : 63 : 1) | k*P =  (5 : 63 : 1) | True
12 Point P =   (7 : 3 : 1) | mult(P,k) = (45 : 64 : 1) | k*P = (45 : 64 : 1) | True
13 Point P =  (7 : 68 : 1) | mult(P,k) =  (45 : 7 : 1) | k*P =  (45 : 7 : 1) | True
14 Point P =  (8 : 19 : 1) | mult(P,k) = (50 : 64 : 1) | k*P = (50 : 64 : 1) | True
15 Point P =  (8 : 52 : 1) | mult(P,k) =  (50 : 7 : 1) | k*P =  (50 : 7 : 1) | True
16 Point P = (10 : 24 : 1) | mult(P,k) = (24 : 49 : 1) | k*P = (24 : 49 : 1) | True
17 Point P = (10 : 47 : 1) | mult(P,k) = (24 : 22 : 1) | k*P = (24 : 22 : 1) | True
18 Point P =  (11 : 5 : 1) | mult(P,k) =  (1 : 42 : 1) | k*P =  (1 : 42 : 1) | True
19 Point P = (11 : 66 : 1) | mult(P,k) =  (1 : 29 : 1) | k*P =  (1 : 29 : 1) | True
20 Point P = (12 : 26 : 1) | mult(P,k) = (38 : 43 : 1) | k*P = (38 : 43 : 1) | True
21 Point P = (12 : 45 : 1) | mult(P,k) = (38 : 28 : 1) | k*P = (38 : 28 : 1) | True
22 Point P = (13 : 11 : 1) | mult(P,k) = (15 : 48 : 1) | k*P = (15 : 48 : 1) | True
23 Point P = (13 : 60 : 1) | mult(P,k) = (15 : 23 : 1) | k*P = (15 : 23 : 1) | True
24 Point P = (15 : 23 : 1) | mult(P,k) =  (56 : 2 : 1) | k*P =  (56 : 2 : 1) | True
25 Point P = (15 : 48 : 1) | mult(P,k) = (56 : 69 : 1) | k*P = (56 : 69 : 1) | True
26 Point P = (17 : 27 : 1) | mult(P,k) = (12 : 26 : 1) | k*P = (12 : 26 : 1) | True
27 Point P = (17 : 44 : 1) | mult(P,k) = (12 : 45 : 1) | k*P = (12 : 45 : 1) | True
28 Point P = (21 : 22 : 1) | mult(P,k) = (51 : 59 : 1) | k*P = (51 : 59 : 1) | True
29 Point P = (21 : 49 : 1) | mult(P,k) = (51 : 12 : 1) | k*P = (51 : 12 : 1) | True
30 Point P =  (23 : 8 : 1) | mult(P,k) = (26 : 49 : 1) | k*P = (26 : 49 : 1) | True
31 Point P = (23 : 63 : 1) | mult(P,k) = (26 : 22 : 1) | k*P = (26 : 22 : 1) | True
32 Point P = (24 : 22 : 1) | mult(P,k) = (10 : 47 : 1) | k*P = (10 : 47 : 1) | True
33 Point P = (24 : 49 : 1) | mult(P,k) = (10 : 24 : 1) | k*P = (10 : 24 : 1) | True
34 Point P = (25 : 14 : 1) | mult(P,k) = (48 : 16 : 1) | k*P = (48 : 16 : 1) | True
35 Point P = (25 : 57 : 1) | mult(P,k) = (48 : 55 : 1) | k*P = (48 : 55 : 1) | True
36 Point P = (26 : 22 : 1) | mult(P,k) = (13 : 60 : 1) | k*P = (13 : 60 : 1) | True
37 Point P = (26 : 49 : 1) | mult(P,k) = (13 : 11 : 1) | k*P = (13 : 11 : 1) | True
38 Point P = (27 : 17 : 1) | mult(P,k) = (36 : 15 : 1) | k*P = (36 : 15 : 1) | True
39 Point P = (27 : 54 : 1) | mult(P,k) = (36 : 56 : 1) | k*P = (36 : 56 : 1) | True
40 Point P = (28 : 16 : 1) | mult(P,k) =  (47 : 7 : 1) | k*P =  (47 : 7 : 1) | True
41 Point P = (28 : 55 : 1) | mult(P,k) = (47 : 64 : 1) | k*P = (47 : 64 : 1) | True
42 Point P =  (29 : 6 : 1) | mult(P,k) = (67 : 28 : 1) | k*P = (67 : 28 : 1) | True
43 Point P = (29 : 65 : 1) | mult(P,k) = (67 : 43 : 1) | k*P = (67 : 43 : 1) | True
44 Point P = (32 : 35 : 1) | mult(P,k) = (17 : 27 : 1) | k*P = (17 : 27 : 1) | True
45 Point P = (32 : 36 : 1) | mult(P,k) = (17 : 44 : 1) | k*P = (17 : 44 : 1) | True
46 Point P = (35 : 33 : 1) | mult(P,k) = (29 : 65 : 1) | k*P = (29 : 65 : 1) | True
47 Point P = (35 : 38 : 1) | mult(P,k) =  (29 : 6 : 1) | k*P =  (29 : 6 : 1) | True
48 Point P = (36 : 15 : 1) | mult(P,k) = (37 : 28 : 1) | k*P = (37 : 28 : 1) | True
49 Point P = (36 : 56 : 1) | mult(P,k) = (37 : 43 : 1) | k*P = (37 : 43 : 1) | True
50 Point P = (37 : 28 : 1) | mult(P,k) = (40 : 57 : 1) | k*P = (40 : 57 : 1) | True
51 Point P = (37 : 43 : 1) | mult(P,k) = (40 : 14 : 1) | k*P = (40 : 14 : 1) | True
52 Point P = (38 : 28 : 1) | mult(P,k) = (39 : 36 : 1) | k*P = (39 : 36 : 1) | True
53 Point P = (38 : 43 : 1) | mult(P,k) = (39 : 35 : 1) | k*P = (39 : 35 : 1) | True
54 Point P = (39 : 35 : 1) | mult(P,k) =  (23 : 8 : 1) | k*P =  (23 : 8 : 1) | True
55 Point P = (39 : 36 : 1) | mult(P,k) = (23 : 63 : 1) | k*P = (23 : 63 : 1) | True
56 Point P = (40 : 14 : 1) | mult(P,k) = (63 : 32 : 1) | k*P = (63 : 32 : 1) | True
57 Point P = (40 : 57 : 1) | mult(P,k) = (63 : 39 : 1) | k*P = (63 : 39 : 1) | True
58 Point P =  (42 : 0 : 1) | mult(P,k) =  (42 : 0 : 1) | k*P =  (42 : 0 : 1) | True
59 Point P =  (43 : 8 : 1) | mult(P,k) = (52 : 30 : 1) | k*P = (52 : 30 : 1) | True
60 Point P = (43 : 63 : 1) | mult(P,k) = (52 : 41 : 1) | k*P = (52 : 41 : 1) | True
61 Point P =  (45 : 7 : 1) | mult(P,k) = (35 : 33 : 1) | k*P = (35 : 33 : 1) | True
62 Point P = (45 : 64 : 1) | mult(P,k) = (35 : 38 : 1) | k*P = (35 : 38 : 1) | True
63 Point P =  (47 : 7 : 1) | mult(P,k) =  (8 : 19 : 1) | k*P =  (8 : 19 : 1) | True
64 Point P = (47 : 64 : 1) | mult(P,k) =  (8 : 52 : 1) | k*P =  (8 : 52 : 1) | True
65 Point P = (48 : 16 : 1) | mult(P,k) = (58 : 46 : 1) | k*P = (58 : 46 : 1) | True
66 Point P = (48 : 55 : 1) | mult(P,k) = (58 : 25 : 1) | k*P = (58 : 25 : 1) | True
67 Point P =  (50 : 7 : 1) | mult(P,k) = (57 : 26 : 1) | k*P = (57 : 26 : 1) | True
68 Point P = (50 : 64 : 1) | mult(P,k) = (57 : 45 : 1) | k*P = (57 : 45 : 1) | True
69 Point P = (51 : 12 : 1) | mult(P,k) = (25 : 57 : 1) | k*P = (25 : 57 : 1) | True
70 Point P = (51 : 59 : 1) | mult(P,k) = (25 : 14 : 1) | k*P = (25 : 14 : 1) | True
71 Point P = (52 : 30 : 1) | mult(P,k) = (28 : 55 : 1) | k*P = (28 : 55 : 1) | True
72 Point P = (52 : 41 : 1) | mult(P,k) = (28 : 16 : 1) | k*P = (28 : 16 : 1) | True
73 Point P =  (56 : 2 : 1) | mult(P,k) = (32 : 36 : 1) | k*P = (32 : 36 : 1) | True
74 Point P = (56 : 69 : 1) | mult(P,k) = (32 : 35 : 1) | k*P = (32 : 35 : 1) | True
75 Point P = (57 : 26 : 1) | mult(P,k) =  (11 : 5 : 1) | k*P =  (11 : 5 : 1) | True
76 Point P = (57 : 45 : 1) | mult(P,k) = (11 : 66 : 1) | k*P = (11 : 66 : 1) | True
77 Point P = (58 : 25 : 1) | mult(P,k) = (21 : 49 : 1) | k*P = (21 : 49 : 1) | True
78 Point P = (58 : 46 : 1) | mult(P,k) = (21 : 22 : 1) | k*P = (21 : 22 : 1) | True
79 Point P = (63 : 32 : 1) | mult(P,k) = (64 : 13 : 1) | k*P = (64 : 13 : 1) | True
80 Point P = (63 : 39 : 1) | mult(P,k) = (64 : 58 : 1) | k*P = (64 : 58 : 1) | True
81 Point P = (64 : 13 : 1) | mult(P,k) =  (6 : 14 : 1) | k*P =  (6 : 14 : 1) | True
82 Point P = (64 : 58 : 1) | mult(P,k) =  (6 : 57 : 1) | k*P =  (6 : 57 : 1) | True
83 Point P = (66 : 16 : 1) | mult(P,k) = (66 : 55 : 1) | k*P = (66 : 55 : 1) | True
84 Point P = (66 : 55 : 1) | mult(P,k) = (66 : 16 : 1) | k*P = (66 : 16 : 1) | True
85 Point P = (67 : 28 : 1) | mult(P,k) =   (7 : 3 : 1) | k*P =   (7 : 3 : 1) | True
86 Point P = (67 : 43 : 1) | mult(P,k) =  (7 : 68 : 1) | k*P =  (7 : 68 : 1) | True
87 Point P =  (68 : 9 : 1) | mult(P,k) =  (2 : 26 : 1) | k*P =  (2 : 26 : 1) | True
88 Point P = (68 : 62 : 1) | mult(P,k) =  (2 : 45 : 1) | k*P =  (2 : 45 : 1) | True
OK
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Comments

The above code is based on my new approach or existing approach of addition and doubling. the function mult(P,K) is based on new approach.??

santoshi gravatar imagesantoshi ( 2017-12-15 18:04:39 +0200 )edit
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Asked: 2017-12-15 06:31:11 +0200

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Last updated: Dec 15 '17

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