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### Why is exponentiation of points on elliptic curve so fast?

I am working on elliptic curves in sagemath. I was trying to collect benchmarks for group operation and exponentiation of points on NIST P-256 elliptic curve. When I tried to perform a group operation on 2 points on the curve, it takes roughly 2 micro seconds. When I tried to perform exponentiation on a point in elliptic curve with a random exponent, it takes only 3 micro seconds. How is this even possible? Since I am exponentiating with a 512 bit value, this should at least take time required for 512 group operations, which is 2ms. I am worried if my code is wrong!

p = 115792089210356248762697446949407573530086143415290314195533631308867097853951
order = 115792089210356248762697446949407573529996955224135760342422259061068512044369
b = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b
F = GF(p)
E = EllipticCurve(F, [-3,b])
runs = 10000
G = E.abelian_group()

F2 = GF(order)
exponent = [F2.random_element() for i in range(runs)]

t1 = time()
for i in range(runs):
e = Integer(exponent[i])*e2[i]
t2 = time()
print  "Time per operation = ", (t2 - t1)/runs , " seconds"

e1 = [G.random_element() for i in range(runs)]
e2 = [G.random_element() for i in range(runs)]
t1 = time()
for i in range(runs):
e = e1[i]+e2[i]
t2 = time()
print  "Time per operation = ", (t2 - t1)/runs , " seconds"


### Why is exponentiation of points on elliptic curve so fast?

I am working on elliptic curves in sagemath. I was trying to collect benchmarks for group operation and exponentiation of points on NIST P-256 elliptic curve. When I tried to perform a group operation on 2 points on the curve, it takes roughly 2 micro seconds. When I tried to perform exponentiation on a point in elliptic curve with a random exponent, it takes only 3 micro seconds. How is this even possible? Since I am exponentiating with a 512 256 bit value, this should at least take time required for 512 256 group operations, which is 2ms. more than 1ms. I am worried if my code is wrong!

p = 115792089210356248762697446949407573530086143415290314195533631308867097853951
order = 115792089210356248762697446949407573529996955224135760342422259061068512044369
b = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b
F = GF(p)
E = EllipticCurve(F, [-3,b])
runs = 10000
G = E.abelian_group()

F2 = GF(order)
exponent = [F2.random_element() for i in range(runs)]

t1 = time()
for i in range(runs):
e = Integer(exponent[i])*e2[i]
t2 = time()
print  "Time per operation = ", (t2 - t1)/runs , " seconds"

e1 = [G.random_element() for i in range(runs)]
e2 = [G.random_element() for i in range(runs)]
t1 = time()
for i in range(runs):
e = e1[i]+e2[i]
t2 = time()
print  "Time per operation = ", (t2 - t1)/runs , " seconds"

 3 retagged FrédéricC 4968 ●3 ●41 ●107

### Why is exponentiation of points on elliptic curve so fast?

I am working on elliptic curves in sagemath. I was trying to collect benchmarks for group operation and exponentiation of points on NIST P-256 elliptic curve. When I tried to perform a group operation on 2 points on the curve, it takes roughly 2 micro seconds. When I tried to perform exponentiation on a point in elliptic curve with a random exponent, it takes only 3 micro seconds. How is this even possible? Since I am exponentiating with a 256 bit value, this should at least take time required for 256 group operations, which is more than 1ms. I am worried if my code is wrong!

p = 115792089210356248762697446949407573530086143415290314195533631308867097853951
order = 115792089210356248762697446949407573529996955224135760342422259061068512044369
b = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b
F = GF(p)
E = EllipticCurve(F, [-3,b])
runs = 10000
G = E.abelian_group()

F2 = GF(order)
exponent = [F2.random_element() for i in range(runs)]

t1 = time()
for i in range(runs):
e = Integer(exponent[i])*e2[i]
t2 = time()
print  "Time per operation = ", (t2 - t1)/runs , " seconds"

e1 = [G.random_element() for i in range(runs)]
e2 = [G.random_element() for i in range(runs)]
t1 = time()
for i in range(runs):
e = e1[i]+e2[i]
t2 = time()
print  "Time per operation = ", (t2 - t1)/runs , " seconds"