The most Bitcoin users are very careful about keeping their anonymity. In this work we investigate the use of disposable addresses, a very common method adopted by users to protect their identity and propose a method to recognize these addresses, We applied it on the rst 180,000 blocks of the Bitcoin Blockchain. Results highlighted that a large part of Bitcoin transactions involves disposable addresses. Further, they showed that many of these transactions form chains whose length is characterized by a power-law distribution.
In 2008, Satoshi Nakamoto invented a payment system, called "Bitcoin cash
system", which allows electronic payments without the need for centralized
system (eg. credit institutions) [
The Blockchain is a shared database containing a continuously growing list of blocks of Bitcoin transactions. It does not store information about the users' identity, and hence identifying the users through the only analysis of the Blockchain is a hard work. Bitcoin system guarantees the pseudo-anonymity (pseudonymity). Users can exchange bitcoins through their pseudonyms called addresses. A Bitcoin address is an alphanumeric identi er that represents the sender (input section) or the destination (output section) of the Bitcoin transaction. The balance of each address is computed as the value of the unspent transactions output (UTXO).
In order to preserve and reinforce the anonymity of the Bitcoin users, many
strategies have been proposed. Some of these strategies improve the privacy and
anonymity including mixing protocols (eg. CoinShu e, CoinJoin and CoinParty
[
We modeled the Blockchain as a Petri net, a bipartite oriented graph N , de ned as N = (P ; T; P re; P ost), where P is the set of the places (addresses), T is the set of the transitions (transactions), P re is the Pre-incidence matrix and P ost is the Post-incidence matrix. The element ij in the Pre matrix de nes how many times the address i is in the input section of the transaction j, instead, in the Post matrix, it de nes how many times the address i is in the output section of the transaction j. After having built of the Petri net N , we focused our attention on the chains of disposable addresses, and hence on the transactions having only one address, a, in the input section and only two addresses, b and c, in the output section. In more detail, the address a in the input section is used by a user u1 to send bitcoins to one of the addresses in the output section, b, belonging to a user u2. The other address, c, in the output section is created by the user u1 to collect the change. We created the set of potentially disposable addresses Ad, starting from the set A of the addresses and from the set of the transactions in the Blockchain.
Let d be the set of transaction d such that: d = f d : jIN ( d)j = 1; jOU T ( d)j = 2; IN ( d) 2 Ad; 9 2 OU T ( d) : 2 Ad; 8 d 2 dg:
In order to build a chain, for each d we need to know the previous transactions dp = P REV ( d). Using P re and P ost matrices, it is very easy to look for these previous transactions. We call ds d the set of transaction ds that could be considered the starting point of a chain because it does not have a previous transaction inside d . We denote with ds the address in input to a transaction ds. Finally, we call N EXT ( d) the transaction d0 which has, in the input section, the disposable address that is contained in the output section of the transaction d. To nd the chains c of disposable addresses, we de ned and implemented the following algorithm: 1. Let C = ; be a set of empty chains, c, 2. for each ds 2 ds: (a) take a empty chain, c, (b) insert ds in c 3. for each c 2 C
A Petri net-based model for investigating disposable addresses (a) take the last element inserted in c, d, (b) while 9 d0 = N EXT ( d)
i. insert d0 in c,
The algorithm returns a set C of chains c. Each chain c contains the transactions ordered by execution order. 3
We processed the rst 180,000 blocks, analysing the Json les downloaded from the website blockchain.info. We built a Petri net model composed of 3,730,480 places (or addresses), 3,142,019 transitions (or transactions), and of the connections between nodes, expressed by Pre and Post matrices.
Description
Value Potential disposable address d 2,897,577
Involved transaction d 1,350,010
Number of chains c 122,155
The computation of the chains is a three steps process. The rst step is to identify the potential disposable addresses. The second step is to recognize transactions in which are involved disposable addresses. The third step is to build chains. The result of the performed analysis are illustrated in Tab 1 e 2. The analysis allow us to compute the dimensions of sets of potential disposable addresses and the transactions, which are involved in the computation, and the number of found chains ( see Tab. 1). We found that over one third of addresses inside the Blockchain are actually disposable addresses. The chains length is highly variable. The found longest chain contains 3,658 transactions and involves an equal number of disposable addresses. The rst ve chains ordered by length, the number of blocks where each chain appears (calculated as the di erence between the ending block number and starting block number) and the rate of transaction execution per block (calculated as the ratio between the length of the chain and the number of blocks) are summarized in Tab. 3. It is interesting to note that some long chains were executed in the time of few tens of blocks (in temporal terms, in few hours). We computed the Complementary Cumulative Distribution Function (CCDF) of the lengths. The graph of the distribution, in Log-Log scale, is showed in Fig. 1. This distribution follows a Power-law in the tail (starting from a length about 20). 4
This work focuses on the problem of the anonymity in the Bitcoin system, and in particular, on the massive use of disposable addresses to protect the user'
Statistics
Value
Min Max Mean 2.00 3,658.00 11.05
Median 3.00
Variance 1230.629 identity. We modeled Bitcoin Blockchain as A Petri net-based model, and analyzed the connections between place (bitcoin addresses) and transitions (bitcoin transactions) thanks to the Petri net formalismm, speci cally thanks to Pre and Post matrices Results show that over one-third of the addresses registered in the Blockchain are actually disposable addresses. The lengths of the chains are very inhomogeneous and their statistical distributions follow partially a power low. Further, we found that the chains are executed quickly, without waiting for con rmation. In future works we will analyse a higher number of blocks and more complex techniques of anonymization.