Quick Python Double-Check and Visualization of Bitcoin Whitepaper
source: Bitcoin, p.7
import math
import warnings
warnings.filterwarnings('ignore')
def AttackerSuccessProbability(q, z):
p = 1.0 - q
lambda_ = z * (q/p)
sum_ = 1.0
k = 0
poisson = math.exp(-lambda_)
while k <= z:
sum_ -= ((lambda_**k * poisson)/math.factorial(k)) * (1 - (q/p)**(z-k))
k += 1
return sum_
AttackerSuccessProbability(.1, 1)
0.20458727394278242
running some results
source: Bitcoin, p.8
q = 0.1
print "q=",q
for z in range(11):
print "z=%d" % z, "\t", "P=%.7f" % AttackerSuccessProbability(q, z)
print ""
q = 0.3
print "q=",q
for z in range(0,51,5):
print "z=%d" % z, "\t", "P=%.7f" % AttackerSuccessProbability(q, z)
q= 0.1
z=0 P=1.0000000
z=1 P=0.2045873
z=2 P=0.0509779
z=3 P=0.0131722
z=4 P=0.0034552
z=5 P=0.0009137
z=6 P=0.0002428
z=7 P=0.0000647
z=8 P=0.0000173
z=9 P=0.0000046
z=10 P=0.0000012
q= 0.3
z=0 P=1.0000000
z=5 P=0.1773523
z=10 P=0.0416605
z=15 P=0.0101008
z=20 P=0.0024804
z=25 P=0.0006132
z=30 P=0.0001522
z=35 P=0.0000379
z=40 P=0.0000095
z=45 P=0.0000024
z=50 P=0.0000006
solving for P < 0.1%
source: Bitcoin, p.8
import numpy as np
def P_lessthanpoint001(q_beg=0.1, q_end=0.5, q_step=.05, verbose=False):
if verbose==True:
print "P < 0.001"
q_s = []
z_s = []
for q in np.arange(q_beg, q_end, q_step):
P = 1.0
z = 0
while P > 0.001:
P = AttackerSuccessProbability(q, z)
z += 1
if verbose==True:
print "q=%.2f \t z=%d" % (q, z-1)
else:
q_s.append(q)
z_s.append(z-1)
if verbose==False:
return zip(q_s, z_s)
P_lessthanpoint001()
[(0.10000000000000001, 5),
(0.15000000000000002, 8),
(0.20000000000000004, 11),
(0.25000000000000006, 15),
(0.30000000000000004, 24),
(0.35000000000000009, 41),
(0.40000000000000013, 89),
(0.45000000000000007, 148)]
attacker_probs = np.array(P_lessthanpoint001(.01, .52, .01))
attacker_probs[-10:]
array([[ 0.42, 137. ],
[ 0.43, 151. ],
[ 0.44, 150. ],
[ 0.45, 148. ],
[ 0.46, 147. ],
[ 0.47, 146. ],
[ 0.48, 145. ],
[ 0.49, 144. ],
[ 0.5 , 144. ],
[ 0.51, 143. ]])
graphing the above
import matplotlib.pyplot as plt
import matplotlib
matplotlib.style.use('ggplot')
%matplotlib inline
plt.figure(figsize=(10,6))
plt.plot(attacker_probs[:,0], attacker_probs[:,1], '-')
plt.xlim(0.,.45)
plt.title("Solving for P less than 0.1%...")
plt.xlabel("P the attacker finds the next block")
plt.ylabel("z blocks behind at which attacker's P < 0.001")
plt.show()
