## 2 Public Keys and Economically Active Entities  
We first consider the number of public keys and try to estimate the number of active entities within the Bitcoin system. The number of _all public keys_ (or addresses) is the number of public keys present in any transaction, whereas the notion of _used public keys_ corresponds to those public keys that were used as input to any transaction at least once. Such used public keys belong to an actor who has control over these bitcoins. Therefore, we regard this as an economically active entity. Addresses used together as input for a single transaction belong to the same entity, because, in order to use an address as an input, one must be in possession of the corresponding private key for that address (this observation was already made by [PERSON] and [PERSON] [5]). It is of course possible that some entity has never used two or more addresses together, but is still in possession of both private keys. In such a case both addresses would be perceived as belonging to two different entities by an adversary (and our experimental analysis) due to lack of data. Thus, the number of entities reported here serves as an upper bound. As defined by the Bitcoin protocol, the balance associated with an address cannot be divided into smaller amounts. Nevertheless it is possible to use the same input address again as output address; this way only a fraction of the balance can be transferred to another address, whereas the remainder of a balance can be transferred back to the originator. This has, of course, negative implications on privacy: it allows to link different transactions, as an attacker can more accurately estimate the number of active entities (if there was no linkability, the new transaction would look as if it originated from a new entity). To counter this, the Satoshi client creates a new address for the balance in the background; via this mechanism it remains unclear which address is used by the recipient and which address is used for the remaining balance. Figure 1 shows the number of public keys, used public keys, and entities over time. We can see a huge increase in the number of entities in April 2010. The reason for the increased interest in Bitcoin is most likely the fact that around this time (late April 2010) public trading of bitcoins began at an exchange rate of 0.003 USD per one bitcoin (in batches of 1000) [9]. More publications on Slashdot [10] (for \"reaching dollar parity\"), Forbes [11] and Gawker [12] have created further interest in Bitcoin, which can be seen by a fast-increasing number of entities. After all, a hype was created and found its peak in  
June/July 2011, where the exchange rate reached about 30 USD. After the latter article, the exchange rate dropped below 2 USD. Despite this bubble, Bitcoin was still popular enough that a sustainable user base exists and still new users are recruited. The ratio of public keys (used or unused) to the number of entities seems to be stationary at a factor of about two, which means on average two public keys can be assigned to one entity solely based on the Bitcoin block chain. In Figure 1 we can also observe a notable gap between the number of used and all public keys. Addresses that were only used to receive funds are probably generation transactions, which do not have any inputs. If an address appeared as an input at least once, it is considered as used. As addresses are used, they get assigned to entities. Figure 2 explores this in more detail: in particular, the ratio of public keys per entity decreases over time, but saturates to a stationary regime since Bitcoin's popularity. We note that this does not necessarily reflect carelessness of the users, but rather could be attributed to an increased awareness of anonymity issues and thus avoidance of entity mergings. Entities and Their Activity PeriodsWe now investigate the size of entities--where the size of an entity is the number of public addresses that can be associated with the respective entity.