Generating Cryptographic Keys Using Weak Pseudo-Random Number Generators (PRNGs)

Cryptographic key generation using weak pseudorandom number generators (PRNGs) is a critical security vulnerability in blockchain systems. This analysis examines the architectural flaws of the function random_key, the mechanisms for exploiting them, and the implications for cryptocurrency wallets.

Theoretical Foundations of PRNG Vulnerabilities

Cryptographic requirements for PRNG :

Statistical indistinguishability from true randomness 3
Unable to restore previous states of generator 3
Defense against future state prediction 3

In the ECDSA algorithm used in Bitcoin, the random number kplays a key role in the signature formation. Its reuse or predictability allows the private key to be calculated using the formula:
s=k−1(H(m)+rd)mod Ns = k^{-1}(H(m) + rd) \mod Ns=k−1(H(m)+rd)modN
where the leak kmakes the equation solvable with respect to d(the private key) 2 .

Analysis of typical vulnerabilities

Attack Type	Implementation mechanism	Example from practice
Weak entropy seed	Using time as an initial value	Cakewallet (2018-2021)1
Algorithmic bookmarks	Deliberate leakage of PRNG states	The Case of BitcoinJS 6 Library
Restoring the state	Output Sequence Analysis	Windows XP PRNG 3 attack

In the case of the BitcoinJS library, which used Math.random()browsers, predictability was enhanced by:

Limited entropy pool (48 bits instead of 128) 6
Deterministic Algorithms in Chrome 2011-2015 6
Cascading errors in JSBN and SecureRandom 6

Practical implications

Cakewallet Incident (2021):
Generation of 95 million potentially vulnerable addresses over 3 years, with the possibility of brute-force testing on average equipment in a few months 1 .
Randstorm Vulnerability :
21,000 BTC (~$750M as of 2023) discovered in wallets with keys generated via vulnerable PRNG 6 .
kBitcoin Duplication Statistics :
- 447 million signatures analyzed 2
- 0.03% of transactions with repeat k2
- Average Damage Per Incident: $45,000 2

Recommendations for improvement

Hardware entropy sources :
Using TRNG (Thermal Noise RNG) with a minimum speed of 2 bits/sec 5 .
Cryptographic protocols :
Implementation of BKRNG type algorithms with a cyclobyte performance of 277.96 cycles/byte versus 546.72 for CTR_DRBG 5 .
Implementation verification :
- NIST SP 800-22 Testing 5
- Audit of the chain of entropy transformations
- Hashing the PRNG output with SHA-3 3

Experimental data show that switching to deterministic BIP-32 key generation schemes with HMAC-SHA512 reduces the predictability risk by 3 orders of magnitude compared to legacy solutions 4 . Modern implementations such as FIPS 140-3 Level 3 provide protection even when the generator state is compromised through continuous test verification mechanisms 5 .