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Capital controls are most common in developing or unstable economies with unsound money. The regulations help maintain a steady balance of imports and exports and also prevent runs on domestic banks. However, capital controls can be detrimental to residents who have restrictions on their financial transactions. Assets such as Bitcoin, which are subject to less government regulation, are commonly used to bypass capital controls.

For example, if you create a signature for the text “Hello”, this signature is invalid for the text “Hey”. This trait allows digital signatures to be publicized without risk of fraud.

There are three parts to a digital signature: the data being signed, the public key of the signer, and the signature itself. The data can be anything digital, including text, an image, an audio file, and more. The public key is a pseudonymous form of identity, informing the public that the owner of the corresponding private key has signed this message. Lastly the signature is mathematical proof that the owner of the public key and the corresponding private key signed the exact data provided.

The magic here is that the private key is required to create the signature, but not required to verify it. Anyone can verify that the signature is valid using only the public key. Furthermore, since the signature is generated using the hash of the data it is signing, verifiers can be sure that the data provided is unaltered since it was signed.

Most systems that require passwords ask users to include uppercase letters, numbers, and special characters in order to decrease the efficacy of brute force attacks. This works because the greater the number of possible passwords, the harder it is to brute force.

The difficulty of a brute force can be calculated simply by taking the number of valid values—usually just one—and dividing it by the number of possible values. For example, a Bitcoin private key is usually 256 bits long. In order to brute force a specific Bitcoin private key, an attacker would have to guess each of the 256 bits correctly, and since each bit has two possible values (1 or 0), the attacker must guess from a range of 2^256 (roughly 10^77) possible values. Comparatively, a credit card number of 16 digits plus a 3 digit security code has a range of 10^19 possibilities. Brute forcing a social security number is even easier, since it is only 9 digits long. This is what makes Bitcoin vastly more secure than traditional financial systems.

In the context of cryptography, a problem or calculation is considered “hard” if the best possible method of solving it is a brute force attack. This is because a brute force attack is usually only used when no algorithm or other method can solve the problem.