The limits of Bitcoin derivations: Can all seeds produce all public keys?
Bitcoin’s unique cryptography is based on the BIP 32 derivation route, which allows users to create multiple public keys from a single seed. However, although this concept may seem that it offers unlimited possibilities to generate new keys, a more detailed examination reveals that not all seeds can produce all possible combinations of derivations.
What are BIP 32 referrals?
In BIP 32 protocol of Bitcoin, a « seed » is a key used to obtain multiple public keys of. These derivation routes are created using the following rules:
- Each derivation route consists of a set of two parameters:
M, which is the main key (a sheet node) andn, which is the number of times the root key must be concatenated.
- The first parameter,
m, can be a single sheet node (for example, 0) or an empty chain ('').
- The second parameter,
n, specifies how many times the main key must be concatenated.
By combining these two parameters in several ways, users can create multiple derivations that finally produce different public keys. For example:
| m | n | PERFORMATION ROUTE |
| — | — | — |
| 0 | 2 | « M = 0, n = 2 » |
| 0 | 3 | « M = 0, n = 3 » |
| 1 | 2 | « M = », n = 2″ |
| … | … | … |
Theoretical limits of derivations
When examining the possible combinations of BIP 32 derivation routes, it becomes clear that not all seeds can produce all possible combinations. The main reason for this limitation is that each seed is associated with a specific set of public keys.
In Bitcoin, the private key of a user (that is, its seed) corresponds to a unique public key (p). To create multiple public keys of the same seed, users must obtain different roots from the same main key. However, since each derivation route requires two parameters (m yn), there are only possible combinations.
For example, consider a user with a seed that produces two different public keys:
- P1 (root) | Root hash | PERIORATION ROUTE
| — | — | — |
| A | ABCDEFG | « M = 0, n = 2 » |
| H | xyzdefgh | « M = », n = 2″ |
As you can see, there are only two possible derivation routes for each seed (since m andn they can take values from 0 to 1). This is because each derivation route requires a specific combination of the main key (m) and the number of concatenations (n). No matter how many seeds it has, not all combinations of derivations will produce all possible public key.
Conclusion
While it may seem that the BIP 32 derivation system of Bitcoin allows unlimited possibilities to generate new keys, reality is more nuanced. The theoretical limits of the derivations mean that not all seeds can produce all possible combinations of roots and derivatives, resulting in a finite set of public keys associated with each seed.
In practice, users can still create multiple public keys different from a single seed using several techniques, such as using different values for `o n`. However, the inherent limitations of BIP 32 derivations mean that not all seeds will produce all possible combinations of derivation roads, which ultimately limits the number of public keys available.
