In recent years, cryptocurrency wallets on Android devices have become increasingly popular, which has led to increased attention from hackers. One of the key security issues is the generation of private keys using the function SecureRandom(), which is supposed to ensure randomness and uniqueness of keys. However, due to implementation errors and vulnerabilities in the function itself, collisions of private keys can occur, making them vulnerable to attacks. In this article, we will look at the mathematical aspects of the errors in SecureRandom()and their impact on the security of Bitcoin wallets on Android.

Theoretical basis

Generating private keys

In cryptography, private keys are generated using cryptographically secure pseudo-random number generators (CSPRNGs). The SecureRandom()Android feature is designed to provide such randomness. However, if the generator is not truly random, or if there is a bug in its implementation, it can result in keys being repeated or unpredictable.

Collisions of private keys

A private key collision occurs when two or more users have the same private key. This can happen due to errors in key generation, such as using predictable seeds or insufficient entropy. In the case of Bitcoin wallets, key collisions can allow attackers to access someone else’s funds.

Mathematical aspects of errors in SecureRandom()

Implementation errors

Implementation errors SecureRandom()may include:

1. Insufficient entropy : If the generator does not receive enough random information, it may produce predictable sequences of numbers.
2. Predictable Seeds : If the generator’s seed values ​​are predictable, attackers can recreate the key sequence.
3. Library vulnerabilities : Using outdated or vulnerable libraries may lead to attacks on the generator.

Mathematically, if $$ P $$ is the probability of a key collision, then it increases as the entropy of the generator decreases. For a cryptographically secure generator, $$ P $$ must be extremely small.

Mathematical model

Suppose that the generator produces keys of length $$ n $$ bits. The probability of a collision between two keys can be estimated by the formula:

$$
P = 1 — \frac{(2^n — 1)}{2^n}
$$

However, if the generator is not perfectly random, this probability can be significantly higher.

Impact on Bitcoin Wallet Security

Android Vulnerabilities

Android devices are particularly vulnerable due to issues with operating system and app updates, which can lead to outdated and vulnerable versions being used SecureRandom()[2]. Additionally, many wallet apps do not use secure encryption protocols and algorithms [3].

Consequences of the attacks

If attackers gain access to private keys due to collisions or generation errors, they can completely control the wallet and transfer funds without the owner’s knowledge. This can lead to significant financial losses.

Conclusion

Implementation errors SecureRandom()and private key collisions pose a serious security threat to Bitcoin wallets on Android. To prevent such attacks, it is necessary to use cryptographically secure pseudo-random number generators and regularly update the software. In addition, wallet app developers should pay special attention to security and use proven cryptographic methods for key generation.

Recommendations

1. Using secure generators : Developers should use only proven and secure pseudo-random number generators.
2. Regular updates : Regular updates to your operating system and applications can help patch vulnerabilities.
3. Cryptographic Validation : Wallet applications must undergo regular cryptographic audits to check for vulnerabilities.

Implementing these measures can significantly improve the security of cryptocurrency wallets on Android devices.

Citations:
[1] https://hashtelegraph.com/crocodilus-obnaruzhen-opasnyj-trojan-dlja-android-ohotjashhijsja-na-kriptovaljutu/
[2] https://www.forbes.ru/tehnologii/354403-mobilnyy-kriptoseyf-kak-zashchitit-bitkoiny-ot-hakerov
[3] https://bosfera.ru/press-release/mobilnye-bitkoin-koshelki-proverili-na-uyazvimosti?amp
[4] https://rg.ru/2024/05/05/sledite-za-koshelkom.html
[5] https://www.rbc.ru/crypto/news/5f0c803a9a79477907996d2b
[6] https://www.kaspersky.ru/blog/vulnerability-in-hot-cryptowallets-from-2011-2015/36592/
[7] https://ru.tradingview.com/news/forklog:3031939c867b8:0/
[8] https://2bitcoins.ru/hakery-nezametno-kradut-kriptu/