Bitcoin: Using BIP32 derivations, can all seeds theoretically produce all public keys?
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Bitcoin derivative limits: can all seeds produce all public keys?
The exclusive Bitcoin encryption is based on the BIP 32 derivative path, which allows users to create many public keys from a sperm. However, although this concept may seem that it offers unlimited possibilities to generate new keys, a narrower study reveals that not all seeds can create any possible combination of derivatives.
What are bip 32 derivatives?
In the 32 Bitcoin bip protocol, the “wheat” is the key to the key to many public keys. These paths are created using the following rules:
- Each derivative path consists of a set of two parameters: `
m ', which is the primary key (leaf node) and "n', which is the number of times in which the root key should be connected.
- The first parameter,M ‘, can be a node with a single leaf (e.g. 0) or an empty string (‘ ”).
- The second parameter,
n ', determines how many times the main key should be connected.
By combining these two parameters in different ways, users can create many derivatives that eventually produce various public keys. For example:
| M | N | Take the derivative
| --- | --- | --- |
| 0 | 2 | "M = 0, n = 2"
| 0 | 3 | "M = 0, n = 3"
| 1 | 2 | "M = '', n = 2"
| ... | ... | ... |
Theoretical boundaries of derivatives
During the study of possible combinations of derivative paths, BIP 32 becomes evident that not all seeds can produce any possible combination. The main reason for this restriction is that each wheat is associated with a specific series of public keys.
In Bitcoin, the user's private key (i.e. its wheat) corresponds to a single public key (P). To create many public keys from the same seeds, users must derive different roots from the same main key. However, since each lead path requires two parameters ('me' n '), there are only 2^n possible combinations.
Consider, for example, a user with wheat that produces two separate public keys:
- P1 (root) Hash of the root | Derivative
| --- | --- | --- |
| And | ABCDEFG | "M = 0, n = 2"
| H | Xyzdefgh | "M = '', n = 2"
As you can see, there are only two possible derivative paths for each sperm (becausen 'can take values from 0 to 1). This is due to the fact that each derivative path requires a specific upper key combination (m ') and the number of concordations (n'). No matter how many seeds you have, not all derivative combinations will produce every possible public key.
Application
Although it may seem that the BIP 32 Bitcoin Bips system allows unlimited possibilities to generate new keys, the reality is more refined. The theoretical boundaries of derivatives mean that not all seeds can produce any possible combination of roots and derivatives, which translates into a complete series of public keys associated with each wheat.
In practice, users can continue to create many public keys separated by a wheat using various techniques, such as the use of different values for "M `n”. However, the intrinsic limits of the bip 32 derivatives mean that not all seeds will produce any possible combination of derivative paths, ultimately limiting the number of public keys available.
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