Vulnerabilities in the libgrand library may allow lost Bitcoin private keys to be recovered using cryptanalytic techniques that exploit weaknesses in key generation or storage. Here’s a simple explanation of how this works:

• Bitcoin keys are based on complex cryptography. The private key is a very large random number (256 bits).
• If a library, such as libgrand, has vulnerabilities in key generation (such as insufficiently random numbers or bugs in the code), an attacker can exploit these weaknesses.
• Using special cryptanalysis algorithms and attacks such as Lattice Attacks, cryptanalysts try to calculate or get closer to the lost private key.
• Lattice Attack allows you to find weaknesses in transactions and keys if the original data is incomplete or has patterns, making it easier to recover the key.
• After successfully finding the private key in HEX format, the user gains full control over the corresponding Bitcoin address and can spend the coins stored there.

Thus, recovery occurs not directly, but through the use of vulnerabilities in the library, which make key generation predictable or allow the key to be found through complex mathematical methods.

In a normal situation, key recovery occurs through a seed phrase (12-24 words), which generates all private keys of the wallet, but if the seed phrase is lost, library vulnerabilities may become the only chance for recovery.

In general, libgrand as a library may in certain cases have vulnerabilities that make private keys susceptible to attacks using cryptanalytic methods, and this makes it technically possible to restore access to lost keys.

If you need more technical and detailed information about specific libgrand vulnerabilities, I can help find and explain these details. Currently, such attacks require deep knowledge of cryptography and powerful computing resources. This information is collected from cryptanalytic sources and research on recovering bitcoin keys with vulnerabilities in software libraries.

Specific vulnerabilities in the libgcrypt library (part of the GnuPG project, often confused with libgrand) are known and well documented. The main one is the vulnerability CVE-2017-7526, which allows the recovery of the contents of the RSA private key via a side-channel attack, specifically using the FLUSH+RELOAD technique.

How it works in simple terms:

• Libgcrypt uses a sliding window exponentiation algorithm for operations on RSA keys.
• This operation leaves traces in the processor cache, which is shared between processes.
• An attacker, having the ability to run his code on the same system, can, through analysis of cache access time delays, calculate what data ended up in the cache.
• This allows the contents of the private RSA key to be recovered bit by bit.

This vulnerability could theoretically be exploited for 2048-bit keys as well, but with much lower probability and with a high computational effort.

It is important to note that libgrand is a different name, and the common vulnerability in libgcrypt, not libgrand, is known as described above. If you are referring specifically to libgcrypt, then that is the vulnerability most known for recovering private keys.

There are also general categories of vulnerabilities that can be found in crypto libraries:

• Weak random number generation (insufficient entropy when creating keys)
• Side-channel leaks (e.g. CPU cache or execution time)
• Buffer overflows and memory errors

If a vulnerability in libgrand is similar to the attacks mentioned above, it is most likely a case of similar side-channel attack methods, or security breaches in key generation or storage.

Briefly:

• The most famous vulnerability used to recover private keys in a similar library libgcrypt is a side-channel attack using the FLUSH+RELOAD technique on RSA operations.
• In the case of libgrand, similar vulnerabilities are possible if the library generates keys with insufficient randomness or leaks through cache and memory.

The CVE-2017-7526 vulnerability in libgcrypt is related to a side-channel attack that uses the processor cache. The features of this vulnerability are as follows:

• The vulnerability exists in an implementation of the RSA algorithm that uses a left-to-right sliding window to perform exponentiation calculations during encryption.
• During these calculations, the data is temporarily stored in the processor cache.
• An attacker with the ability to run arbitrary code on the same system where the private RSA key is used can observe the cache access time.
• Using the FLUSH+RELOAD technique, the attacker gradually reconstructs what data got into the cache by studying changes in access time.
• Using this information, the 1024-bit private RSA key can be completely recovered.
• Theoretically, the method is also applicable to 2048-bit keys, but this will require more computation and the success rate will be about 13%.
• The attack requires the attacker to run the code on the same hardware as the key, such as a machine with multi-user access or a virtual machine near the target.
• The researchers note that in real-world situations it is easier to gain access to the key through other vulnerabilities, but this attack demonstrates the vulnerability of the cache memory architecture and cryptographic implementations.

In summary, CVE-2017-7526 is a vulnerability where a private RSA key can be fully recovered by analyzing indirect information about the state of the processor cache during cryptographic operations in libgcrypt before version 1.7.8.

Patches to address this vulnerability have been released and are recommended for installation to improve security.

Thus, the vulnerability is related to the peculiarities of the processor cache and errors in the implementation of cryptographic operations, which allows one to obtain a private RSA key through an attack on the cache.