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The next Python program computes all of the doable complete lengths as precisely as computationally possible:
from fractions import Fraction
P = 2**256 - 2**32 - 977
N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
def lentable(desk, low, excessive):
# For each worth v in [low, high], increment desk[l], the place l is the size
# of the encoding of v. Returns the variety of values in vary.
complete = 0
whereas low <= excessive:
lowlen = 1 + (low.bit_length() // 8)
maxval = 2**(8*lowlen - 1) - 1
highnow = min(excessive, maxval)
whereas len(desk) <= lowlen:
desk.append(0)
desk[lowlen] += highnow - low + 1
complete += highnow - low + 1
low = highnow + 1
return complete
def analyze(low_r, low_s):
# Desk of possibilities of S size
slen = []
stotal = 0
if low_s:
stotal = lentable(slen, 1, N // 2)
# Excessive s (vary n/2 + 1 .. N-1) is simply mapped to 1 .. N/2 once more.
else:
stotal = lentable(slen, 1, N-1)
# Desk of possibilities of R size
rlen = []
rtotal = 0
if low_r:
rtotal = lentable(rlen, 0, 2**255-1)
# Excessive r trigger a retry till one in 0..2**255-1 is hit.
else:
rtotal = lentable(rlen, 0, P-1)
# Desk of signature lengths
dlen = [0 for _ in range(len(slen) + len(rlen) + 5)]
for sl in vary(len(slen)):
for rl in vary(len(rlen)):
dlen[sl + rl + 6] += Fraction(slen[sl] * rlen[rl], rtotal * stotal)
return dlen
for low_r in [0, 1]:
for low_s in [0, 1]:
print("low_r=%i low_s=%i" % (low_r, low_s))
for dl, freq in enumerate(analyze(low_r, low_s)):
print("* siglen=%i: %.15g" % (dl, freq))
print()
Its output comprises:
low_r=0 low_s=0
- siglen=64: 6.17568333711321e-15
- siglen=65: 1.3553741462502e-12
- siglen=66: 2.8922197969905e-10
- siglen=67: 5.92558535572607e-08
- siglen=68: 1.13845453597605e-05
- siglen=69: 0.00194549560546875
- siglen=70: 0.249996185302734
- siglen=71: 0.498046875
- siglen=72: 0.25
low_r=0 low_s=1
- siglen=64: 1.23444548853768e-14
- siglen=65: 2.7089788745549e-12
- siglen=66: 5.77990988404053e-10
- siglen=67: 1.18395746540045e-07
- siglen=68: 2.27394048124552e-05
- siglen=69: 0.00388339161872864
- siglen=70: 0.498046875
- siglen=71: 0.498046875
low_r=1 low_s=0
- siglen=64: 1.23444548853768e-14
- siglen=65: 2.7089788745549e-12
- siglen=66: 5.77990988404053e-10
- siglen=67: 1.18395746540045e-07
- siglen=68: 2.27394048124552e-05
- siglen=69: 0.00388339161872864
- siglen=70: 0.498046875
- siglen=71: 0.498046875
low_r=1 low_s=1
- siglen=64: 2.46750861930545e-14
- siglen=65: 5.41441891321881e-12
- siglen=66: 1.15507603482001e-09
- siglen=67: 2.36559571931139e-07
- siglen=68: 4.54194378107786e-05
- siglen=69: 0.00775158405303955
- siglen=70: 0.992202758789062
The computation is finished with actual fractions, however printed as floating level. Change the “%.15g” to “%s” if you need fractions within the output.
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