I've seen confusion about prefix searches for Bitcoin puzzles in
this thread, and I've decided to provide a third, more detailed explanation of their function, explaining why, for "lucky hunters" the best option right now is prefix searches, as they offer a significant advantage when luck is involved.
It was already concluded in
this thread that in an exhaustive search comparing sequential searches versus prefix searches, there are no notable statistical advantages. However, this often confuses "treasure hunters" since most of them don't have access to GPU or CPU farms, making this comparison useless in home use. They need to change their mindset from needing to find the key at all costs when trying their luck, since exhaustively searching the entire space is pointless in both cases. With the limited resources of home computing, it's theoretically impossible to scan the entire range.
This is where probabilistic search strategy comes in to increase your chances of success, since the prefix strategy is faster and consumes fewer resources, statistically increasing your probability of success in this "lottery".
If your hardware is limited to a home PC or a couple of GPUs, the goal cannot be "brute force" or "sequential search". Your goal must be probabilistic efficiency.
The point is not the feasibility of finding the solution, but rather that between trying sequentially versus using prefixes, prefixes are statistically superior and are
THE ONLY RATIONAL OPTION.
Focusing on a limited space like 71 bits, it would take home hardware many years to cover the entire range. "Sequential" searching (checking the entire range) is therefore a guaranteed path to failure due to lack of time.
The puzzle finder must maximize the number of "lucky hits" per unit of time and computation.
The Optimal Prefix-Block RelationshipThe essence of the prefix search strategy lies not in magic numbers, but in maintaining an optimal probabilistic relationship between two key variables:
The Critical Variables:•
Prefix Length (L): Number of hexadecimal characters of the target hash used as a filter
•
Block Size (N): Number of private keys we search for in each segment
The Key Equation:16^L ≈ NWhere:•
16^L represents the total space of possible prefixes (16 hexadecimal characters per position)
•
N is the number of keys in each block
Why Is This Relationship Optimal?• If
16^L ≫ N: Too many possible prefixes, low probability of finding matches → you lose efficiency
• If
16^L ≪ N: Too many prefix matches, high risk of missing the target block → you lose accuracy
The Ideal Probability:When
16^L ≈ N, the probability that at least one key in the block has the target prefix is approximately:
P(match) ≈ 1 - (1 - 1/16^L)^N ≈ 63.2%
This is the optimal range where the method reaches its maximum efficiency.
test scriptimport hashlib
import random
import time
import math
import statistics
import scipy.stats as stats
import statsmodels.stats.power as smp
from math import ceil
# Configuration
TOTAL_SIZE = 100_000
RANGE_SIZE = 4_096
PREFIX_LENGTH = 3
SIMULATIONS = 100000
SECP256K1_ORDER = int("0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16)
print(f"""
=== Configuration ===
Total numbers: {TOTAL_SIZE:,}
Block size: {RANGE_SIZE:,}
Total blocks needed: {ceil(TOTAL_SIZE/RANGE_SIZE)}
Prefix: {PREFIX_LENGTH} characters (16^{PREFIX_LENGTH} = {16**PREFIX_LENGTH:,} combinations)
Simulations: {SIMULATIONS}
secp256k1 order: {SECP256K1_ORDER}
""")
def generate_h160(data):
h = hashlib.new('ripemd160', str(data).encode('utf-8'))
return h.hexdigest()
def shuffled_blck(total_blocks):
blocks = list(range(total_blocks))
random.shuffle(blocks)
return blocks
def sequential_search(dataset, block_size, target_hash, block_order):
checks = 0
for block_idx in block_order:
start = block_idx * block_size
end = min(start + block_size, len(dataset))
for i in range(start, end):
checks += 1
if generate_h160(dataset[i]) == target_hash:
return {"checks": checks, "found": True, "index": i}
return {"checks": checks, "found": False}
def prefix_search(dataset, block_size, prefix_len, target_hash, block_order):
prefix = target_hash[:prefix_len]
checks = 0
for block_idx in block_order:
start = block_idx * block_size
end = min(start + block_size, len(dataset))
found_prefix = False
for i in range(start, end):
checks += 1
current_hash = generate_h160(dataset[i])
if current_hash == target_hash:
return {"checks": checks, "found": True, "index": i}
if not found_prefix and current_hash.startswith(prefix):
found_prefix = True
break
return {"checks": checks, "found": False}
def comp_cohens_d(list1, list2):
if len(list1) < 2 or len(list2) < 2:
return float('nan')
n1, n2 = len(list1), len(list2)
m1, m2 = statistics.mean(list1), statistics.mean(list2)
s1, s2 = statistics.stdev(list1), statistics.stdev(list2)
pooled_std = math.sqrt(((n1-1)*s1**2 + (n2-1)*s2**2) / (n1+n2-2))
if pooled_std == 0:
return float('nan')
return (m1 - m2) / pooled_std
def coeff_variation(data):
if not data or statistics.mean(data) == 0:
return float('nan')
return (statistics.stdev(data) / statistics.mean(data)) * 100
def longest_streak(outcomes, letter):
max_streak = current = 0
for o in outcomes:
current = current + 1 if o == letter else 0
max_streak = max(max_streak, current)
return max_streak
def ascii_bar(label, value, max_value, bar_length=50):
bar_count = int((value / max_value) * bar_length) if max_value > 0 else 0
return f"{label:12}: {'#' * bar_count} ({value})"
def conf_interval(data, confidence=0.95):
if len(data) < 2:
return (0, 0)
try:
return stats.t.interval(
confidence=confidence,
df=len(data)-1,
loc=statistics.mean(data),
scale=stats.sem(data)
)
except:
return (statistics.mean(data), statistics.mean(data))
def statistical_analysis(seq_checks, pre_checks):
analysis = {}
analysis['seq_mean'] = statistics.mean(seq_checks) if seq_checks else 0
analysis['pre_mean'] = statistics.mean(pre_checks) if pre_checks else 0
analysis['seq_ci'] = conf_interval(seq_checks)
analysis['pre_ci'] = conf_interval(pre_checks)
if len(seq_checks) > 1 and len(pre_checks) > 1:
analysis['t_test'] = stats.ttest_ind(seq_checks, pre_checks, equal_var=False)
analysis['mann_whitney'] = stats.mannwhitneyu(seq_checks, pre_checks)
analysis['cohen_d'] = comp_cohens_d(seq_checks, pre_checks)
effect_size = abs(analysis['cohen_d'])
if effect_size > 0:
analysis['power'] = smp.tt_ind_solve_power(
effect_size=effect_size,
nobs1=len(seq_checks),
alpha=0.05,
ratio=len(pre_checks)/len(seq_checks)
)
else:
analysis['power'] = 0
else:
analysis['t_test'] = None
analysis['mann_whitney'] = None
analysis['cohen_d'] = 0
analysis['power'] = 0
return analysis
def compare_methods():
results = {
"sequential": {"wins": 0, "checks": [], "times": []},
"prefix": {"wins": 0, "checks": [], "times": []},
"ties": 0,
"both_failed": 0
}
outcome_history = []
total_blocks = ceil(TOTAL_SIZE / RANGE_SIZE)
valid_cases = 0
for _ in range(SIMULATIONS):
max_offset = SECP256K1_ORDER - TOTAL_SIZE - 1
offset = random.randint(0, max_offset)
dataset = [offset + i for i in range(TOTAL_SIZE)]
target_num = random.choice(dataset)
target_hash = generate_h160(target_num)
block_order = shuffled_blck(total_blocks)
start = time.perf_counter()
seq_res = sequential_search(dataset, RANGE_SIZE, target_hash, block_order)
seq_time = time.perf_counter() - start
start = time.perf_counter()
pre_res = prefix_search(dataset, RANGE_SIZE, PREFIX_LENGTH, target_hash, block_order)
pre_time = time.perf_counter() - start
if seq_res["found"] and pre_res["found"]:
valid_cases += 1
results["sequential"]["checks"].append(seq_res["checks"])
results["prefix"]["checks"].append(pre_res["checks"])
results["sequential"]["times"].append(seq_time)
results["prefix"]["times"].append(pre_time)
if seq_res["checks"] < pre_res["checks"]:
results["sequential"]["wins"] += 1
outcome_history.append("S")
elif pre_res["checks"] < seq_res["checks"]:
results["prefix"]["wins"] += 1
outcome_history.append("P")
else:
results["ties"] += 1
outcome_history.append("T")
elif not seq_res["found"] and not pre_res["found"]:
results["both_failed"] += 1
else:
continue
def get_stats(data):
if not data:
return {"mean": 0, "min": 0, "max": 0, "median": 0, "stdev": 0}
return {
"mean": statistics.mean(data),
"min": min(data),
"max": max(data),
"median": statistics.median(data),
"stdev": statistics.stdev(data) if len(data) > 1 else 0
}
seq_stats = get_stats(results["sequential"]["checks"])
pre_stats = get_stats(results["prefix"]["checks"])
seq_time_stats = get_stats(results["sequential"]["times"])
pre_time_stats = get_stats(results["prefix"]["times"])
total_comparisons = results["sequential"]["wins"] + results["prefix"]["wins"] + results["ties"]
seq_win_rate = results["sequential"]["wins"] / total_comparisons if total_comparisons > 0 else 0
pre_win_rate = results["prefix"]["wins"] / total_comparisons if total_comparisons > 0 else 0
cv_seq = coeff_variation(results["sequential"]["checks"])
cv_pre = coeff_variation(results["prefix"]["checks"])
stats_analysis = statistical_analysis(
seq_checks=results["sequential"]["checks"],
pre_checks=results["prefix"]["checks"]
)
print(f"""
=== FINAL ANALYSIS ===
Valid cases (both found target): {valid_cases}/{SIMULATIONS}
[Performance Metrics]
| Sequential | Prefix
---------------+---------------------+--------------------
Checks (mean) | {seq_stats['mean']:>12,.1f} ± {seq_stats['stdev']:,.1f} | {pre_stats['mean']:>12,.1f} ± {pre_stats['stdev']:,.1f}
Time (mean ms) | {seq_time_stats['mean']*1000:>12.2f} ± {seq_time_stats['stdev']*1000:.2f} | {pre_time_stats['mean']*1000:>12.2f} ± {pre_time_stats['stdev']*1000:.2f}
Min checks | {seq_stats['min']:>12,} | {pre_stats['min']:>12,}
Max checks | {seq_stats['max']:>12,} | {pre_stats['max']:>12,}
Coef. Variation| {cv_seq:>11.1f}% | {cv_pre:>11.1f}%
[Comparison Results]
Sequential wins: {results['sequential']['wins']} ({seq_win_rate:.1%})
Prefix wins: {results['prefix']['wins']} ({pre_win_rate:.1%})
Ties: {results['ties']}
Both failed: {results['both_failed']}
=== STATISTICAL ANALYSIS ===
[Confidence Intervals]
Checks Sequential: {seq_stats['mean']:.1f} ({stats_analysis['seq_ci'][0]:.1f} - {stats_analysis['seq_ci'][1]:.1f})
Checks Prefix: {pre_stats['mean']:.1f} ({stats_analysis['pre_ci'][0]:.1f} - {stats_analysis['pre_ci'][1]:.1f})
[Statistical Tests]
Welch's t-test: {'t = %.3f, p = %.4f' % (stats_analysis['t_test'].statistic, stats_analysis['t_test'].pvalue) if stats_analysis['t_test'] else 'N/A'}
Mann-Whitney U: {'U = %.1f, p = %.4f' % (stats_analysis['mann_whitney'].statistic, stats_analysis['mann_whitney'].pvalue) if stats_analysis['mann_whitney'] else 'N/A'}
Effect Size (Cohen's d): {stats_analysis['cohen_d']:.3f}
[Power Analysis]
Statistical Power: {stats_analysis['power']:.1%}
""")
if outcome_history:
non_tie_outcomes = [o for o in outcome_history if o != "T"]
streak_analysis = f"""
=== STREAK ANALYSIS ===
Longest Sequential streak: {longest_streak(outcome_history, 'S')}
Longest Prefix streak: {longest_streak(outcome_history, 'P')}
Expected max streak: {math.log(len(non_tie_outcomes), 2):.1f} (for {len(non_tie_outcomes)} trials)
"""
print(streak_analysis)
max_wins = max(results["sequential"]["wins"], results["prefix"]["wins"], results["ties"])
print("=== WIN DISTRIBUTION ===")
print(ascii_bar("Sequential", results["sequential"]["wins"], max_wins))
print(ascii_bar("Prefix", results["prefix"]["wins"], max_wins))
print(ascii_bar("Ties", results["ties"], max_wins))
if __name__ == '__main__':
compare_methods()
=== FINAL ANALYSIS ===
Valid cases (both found target): 63761/100000
[Performance Metrics]
| Sequential | Prefix
---------------+---------------------+--------------------
Checks (mean) | 49,583.6 ± 28,856.7 | 32,119.8 ± 19,013.3
Time (mean ms) | 118.70 ± 70.17 | 79.24 ± 47.81
Min checks | 1 | 1
Max checks | 99,997 | 84,519
Coef. Variation| 58.2% | 59.2%
[Comparison Results]
Sequential wins: 0 (0.0%)
Prefix wins: 59696 (93.6%)
Ties: 4065
Both failed: 0
=== STATISTICAL ANALYSIS ===
[Confidence Intervals]
Checks Sequential: 49583.6 (49359.6 - 49807.6)
Checks Prefix: 32119.8 (31972.2 - 32267.4)
[Statistical Tests]
Welch's t-test: t = 127.607, p = 0.0000
Mann-Whitney U: U = 2742363164.5, p = 0.0000
Effect Size (Cohen's d): 0.715
[Power Analysis]
Statistical Power: 100.0%
=== STREAK ANALYSIS ===
Longest Sequential streak: 0
Longest Prefix streak: 148
Expected max streak: 15.9 (for 59696 trials)
=== WIN DISTRIBUTION ===
Sequential : (0)
Prefix : ################################################## (59696)
Ties : ### (4065)
The analysis of 100,000 simulations is conclusive: Prefix is 35.3% more efficient than Sequential for each attempt.
Prefix Search sacrifices reliability for speed.
Loss Minimization: Most of the computation time in Sequential Search is wasted checking thousands of numbers in an incorrect block. Prefix Search avoids this by using a probability of 1 in 4,096 (for 3 characters) as a skip mechanism.
Advantage Accumulation: By quickly skipping incorrect blocks, the method accumulates a massive savings in checks (approximately 17,500 per successful attempt). This maximizes the number of "lucky" attempts your hardware can make.
Result: Total Dominance: The effort reduction is so significant that Prefix Search wins 93.6% of the time in a direct efficiency comparison.
If you don't own a GPU farm, you have no other rational option.
Sequential search is a failed brute-force strategy that exhausts your resources on useless ranges.
Prefix search is an optimized luck strategy that gives you a 93.6% probability of making the most efficient use of your computation time.
You must accept the risk (approximately 36% failure per partial attempt due to the "jump") in exchange for the possibility of success (approximately 64%) with minimal effort. This is the only way to tackle the vastness of the keyspace with limited resources.
You sacrifice: 100% theoretical coverage (impossible to achieve anyway)
You gain: 35% greater efficiency, 50% more attempts, 93% higher probability of success per resource invested.
In a search where time is the scarcest resource, sacrificing a theoretical guarantee for a practical advantage of 35% is not an option... it is an obligation.
In short, prefixes influence probabilities by speeding up the search process, allowing for more attempts per unit of time. They don't change the probability of an individual attempt, but they do increase the overall probability of success by increasing the frequency of attempts. It's the rational strategy for those who rely on luck!
