Fungible and non-fungible tokens with snapshots in Java

Many blockchain applications exchange tokens, such as bitcoin and ether, or implement them through smart contracts. A trend in blockchain is to apply standards for token interoperability, unchanged, from platform to platform, easing the design challenges with trusted and widely-used specifications. However, the exploitation of the target language semantics can result in technological advantages and more efficient contracts. This paper presents a re-engineering of OpenZeppelin’s implementation of the ERC-20 and ERC-721 standards in Takamaka, a Java framework for programming smart contracts. It describes a sound solution to the issue about the types allowed for the token holders and a novel implementation for making snapshots of tokens, based on tree maps, that is possible in Java, but not in Solidity, more efficient than the literal translation in Java from Solidity, within the Java virtual machine. Moreover, it applies to ERC-721 as well, where a snapshot mechanism was previously missing. The same snapshot mechanism can also be applied beyond the smart contracts for tokens.


Introduction
Blockchains exploit the redundant, concurrent execution of the same transactions on a decentralized network of machines to enforce their execution in accordance with a set of predefined rules. Namely, blockchains make it hard, for a single machine, to disrupt the semantics of the transactions or their ordering: a misbehaving single machine gets immediately put out of consensus and isolated. Bitcoin [3,24] has been the first blockchain's success story. Bitcoin transactions are transfers of cryptocurrency between accounts, with the specific rule that the same inputs cannot be spent twice. Bitcoin's cryptocurrency is called bitcoin itself and has been the first example of a blockchain token.
A few years after Bitcoin, another blockchain, called Ethereum [4,6], introduced the possibility of programming transactions in an actual, imperative programming language, called Solidity, whose code is compiled for the Ethereum virtual machine (EVM). Ethereum's transactions are still paid in terms of its native ether token, but they execute much more than native token transfers. Namely, transactions can also run object constructors and methods of code units called smart contracts, so that the Ethereum blockchain becomes a sort of world computer that persists the same objects in the memory of all the machines in the blockchain's network. The transactions included in the blocks of the blockchain must be the same in every machine of the network and must lead to the same outcome. Machines that do not abide to this rule will be put out of consensus and their future transactions will be rejected by the other machines.
Typically, smart contracts are written using domainspecific languages (DSLs) such as Solidity, that have specific features and restrictions for blockchain. More recently, the trend in smart contract development shifted to Marco Crosara, Luca Olivieri, Fausto Spoto, and Fabio Tagliaferro have contributed equally to this work. the usage of well-known general-purpose programming languages providing useful syntactical features missing in DSLs, along with ready-to-use available developer toolbelts. This has opened the opportunity to re-engineer and optimize the implementation of several existing standards for different blockchains. Java is among these generalpurpose languages, since it enjoys large popularity [7,38] and a modern set of programming tools.
A popular class of Solidity smart contracts implements a dynamic ledger of coin transfers between accounts. These coins are not native tokens, but rather new, derived tokens, implemented in software through a smart contract. 1 Native and derived tokens can be categorized in many ways [14,25,37]. The most popular classification is between fungible and non-fungible tokens. Fungible tokens are interchangeable with each other since they have an identical nominal value, that is not tied to any specific token instance. Both native tokens and traditional (fiat) currencies are fungible tokens. Their main application is in the area of crowdfunding and in initial coin offers to support startups. On the contrary, non-fungible tokens have a value that depends on their specific instance. Hence, in general, they are not interchangeable. Their main application is currently in the art market, where they represent a written declaration of the author's rights concession to the holder, in gaming and, in general, in notarization.
A few standards have emerged for fungible and nonfungible tokens, that should guarantee correctness [28], accessibility, interoperability, management, and security of the smart contracts that run the tokens. Among them, the Ethereum Request for Comment #20 (ERC-20 [39]) and #721 (ERC-721 [13]) are the most popular for fungible and non-fungible tokens, respectively, also outside Ethereum [17,18,22]. They provide developers with a list of rules required for the correct integration of tokens with other smart contracts and with applications external to the blockchain, such as wallets, block explorers, decentralized finance protocols, and games.
The most popular implementations of the ERC-20 standard are in Solidity, by OpenZeppelin [26], a team of programmers in the Ethereum community who deliver useful and secure smart contracts and libraries, and by ConsenSys [8], later deprecated in favor of OpenZeppelin's. As it will be discussed in detail in Sect. 2, OpenZeppelin's implementation of the ledger uses a map from each owner O to an integer stating the number of tokens that O owns. This map is dynamic, hence it allows one to track the process of token creation and transfer. OpenZeppelin extends ERC-20 with snapshots, i.e. immutable views of the state of a token contract, that show its ledger at a specific instant of time. They are useful for investigating the consequences of an attack, for creating forks of the token and for implementing mechanisms based on token balances such as weighted voting. Snapshots are essential also to provide an immutable view of the ledger that can be queried by a client without the risk that it changes during the query, which would result in a race condition.
In the case of ERC-721, the standard implementation is in Solidity, again by OpenZeppelin [27]. That implementation does not provide a snapshot mechanism, despite the usefulness of such feature. The reason is that the already very tricky implementation in Solidity of snapshots for ERC-20 becomes intractable for the more complicated ERC-721 standard.
A controversial issue about ERC-20 and ERC-721 tokens is about who can hold tokens. It is universally accepted that externally owned accounts can hold tokens: they are accounts controlled by humans or external applications, hence their behavior is not fixed. Once they receive a token, the human or the application can decide to keep it or sell it forward. In this case, there is no risk that the token remains stuck. However, the ERC-20 and ERC-721 standards also allow contracts to hold tokens. This is problematic, since contracts are controlled by their code, which is immutable. If a contract receives a token and its code is not prepared to deal with it, the result is that the token gets stuck forever: the contract will never use it and it will never sell it forward either. There is no solution, in Solidity, to this problem, since Solidity represents both externally owned accounts and contracts with the same address type, making them indistinguishable. ERC-721 implementations have tried to limit this problem in a buggy and fragile way, by requiring that contracts receiving ERC-721 tokens must implement an interface IERC721Receiver. At least, this acknowledges that the programmer of the contract was aware that the latter could receive tokens. Unfortunately, Solidity has no instanceof operator to check for implementation of an interface, because address values are unboxed in Ethereum and they carry no dynamic type information [9]. As a consequence, the solution, based on the ERC-165 standard [30], is tricky, fragile, and finally voluntary, so that it can be circumvented very easily if contracts cheat about the interfaces they implement.
Contributions The contributions of this paper are the following: • a detailed analysis of OpenZeppelin's Solidity implementation of ERC-20 and ERC-721; • a re-engineered solution that exploits Java features, not applicable in Solidity, to create cleaner implementations of both standards; • a sound check that contracts holding ERC-721 tokens actually implement IERC721Receiver; • an implementation of a mechanism for efficient snapshots within the Java virtual machine (JVM), for both ERC-20 and ERC-721, that has been actually applied also beyond those two kinds of smart contracts.
The smart contracts that we developed are available in the support library of Takamaka, as non-proprietary, opensource code [34]. They run in the Hotmoka blockchain [16] hence not in Ethereum-like blockchains. In particular, the smart contracts of Hotmoka are written in a subset of Java called Takamaka [35,36], hence not in Solidity. Takamaka uses bytecode instrumentation and code annotations (marks starting with @, such as @View or @FromContract) to implement concepts specific to smart contracts. ERC-20 and ERC-721 tokens were not ported previously to Takamaka, hence this is the first version of ERC-20 and ERC-721 for that platform. Moreover, this is the first implementation of ERC-721 tokens that correctly check holders to implement IERC721Receiver and that provide snapshots, as far as we know.
Paper structure Section 2 presents the ERC-20 standard and its OpenZeppelin implementation. Section 3 presents the ERC-721 standard and its OpenZeppelin implementation. Section 4 shows how to perform a code language migration from Solidity to Takamaka and which heuristics have helped in the translation of OpenZeppelin's implementations. Section 5 shows an implementation of ERC-20 contracts in Takamaka, with snapshots, that mimics as much as possible the Solidity code structure of OpenZeppelin's implementation, discussing its drawbacks. Section 6 shows a more efficient implementation of snapshots, possible in Takamaka but not in Solidity, and that works for both ERC-20 and ERC-721 tokens. Section 7 shows, experimentally, that this new implementation is more efficient than what discussed in Sect. 5, inside the JVM. Section 8 presents related work. Section 9 concludes. This paper is an extended version of [10]. Compared to that previous version, which was limited to ERC-20 tokens only, the current one adds the ERC-721 tokens as well and the solution to the issue related to the kind of holders allowed to hold tokens, that Solidity tries to solve by using the partial and fragile approach of the ERC-165 standard. Moreover, all sections have been expanded and clarified.

ERC-20 and its OpenZeppelin implementation
The ERC-20 standard [39] defines an interface with nine functions and two events, i.e. immutable marks saved in blockchain to attest some logical turning points. Owners of tokens are addresses. In Solidity, these are untyped pointers to externally owned accounts (sort of bank accounts controlled by an external app or human) or to contracts (objects geared by their code). Although in principle contracts can hold tokens, this could be problematic if their code is not programmed to deal with such tokens. In such a case, the tokens could remain stuck forever, since only the contract can transfer them but the code of the contract does not deal with token transfers. Therefore, it is normally assumed that only externally owned accounts own tokens, but the implementations of ERC-20 do not check this constraint and do not forbid to transfer tokens to contracts, even inadvertently. Section 3 will show that the same problem occurs for ERC-721 tokens, whose implementations have tried to solve the issue in a cumbersome and finally ineffective way. The functions of the ERC-20 standard are for: (1) Direct transfers: totalSupply() yields the integer total amount of tokens in circulation. bal-anceOf(address owner) yields the amount of tokens that owner owns. transfer(address to, uint value) transfers value tokens from the balance of the caller to the balance of to (uint is an unsigned integer of 256 bits). This function must emit a Transfer event.
(2) Delegated transfers: function approve(address delegate, uint cap) allows delegate to transfer up to cap tokens on behalf of the caller. It must emit an Approval event. trans-ferFrom(address owner, address to, uint value) transfers value tokens from owner to to, but only if owner has approved the caller to do so. This function must emit a Transfer event. allowance(address owner, address delegate) yields the amount of tokens that delegate has been approved to transfer on behalf of owner. The first part of this interface is just the API of a dynamic ledger of token balances. Not surprisingly, OpenZeppelin's code, shown in Fig. 1, stores the user's balance in a field _balances 2 of type mapping (address ¼ [ uint), that binds each address to the amount of tokens it holds, and with an integer field _totalSupply, assigned at contract creation time. The second part of the interface allows token owners to delegate, to other participants, the transfer of a capped amount of tokens. OpenZeppelin implements this through a field _allowances of type mapping (address =[ mapping (address =[ uint)): a map from each token owner to another map from each delegate to its allowed cap. The third, optional part is just manifest information about the tokens. Both transfer and transferFrom use an internal function _transfer, that shifts the tokens from the owner to the destination to, calling the handler _be-foreTokenTransfer. This does not do anything by default, but subclasses can redefine it to add extra functionalities to the contract. Function _transfer checks, defensively, for missing values (address(0)) that might arise from incorrect use of the contract. Function transferFrom additionally checks if the owner of the tokens has actually delegated msg.sender (the caller of the function) to transfer at least value tokens on its behalf. This check occurs after the call to _transfer, which is fine since Solidity's functions do not commit their side-effects if they fail. The code of transferFrom ends with a call to _approve (not shown), which reduces the allowance. OpenZeppelin adds a _mint function that initializes the total supply of the token: it is internal since it is meant to be called from the constructors of subclasses that deploy actual instances of the contract. This function uses address(0) to represent the fact that minted tokens come from nowhere.

ERC-721 and its OpenZeppelin implementation
The ERC-721 standard [13] defines an interface with ten functions and three events. As for the ERC-20 standard, token owners can be both externally owned accounts and contracts, but contracts should be avoided, unless they have been explicitly programmed to deal with ERC-721 tokens. We will be back on this issue in a moment. The functions of the ERC-721 standard are for: (1) Direct transfers: balanceOf(address owner) yields the amount of tokens that owner owns. ownerOf(uint tokenId) yields the owner of the given token, if any. transferFrom(address from, address to, uint tokenId) transfers the given token from from to to. In general, the caller of this function must coincide with from, or at least be authorized to transfer the given token on behalf of from (see later). This function does not even try to check that to is an externally owned account or a contract that will be able to deal with the token. If that is not the case, the token will be transferred to to and stuck forever. Because of that, this function is considered to be unsafe. This function must emit a Transfer event. safeTransferFrom(address from, address to, uint tokenId) behaves like transferFrom, but additionally tries to ensure that to is an externally owned account or a contract able to deal with the token. In this sense, it is considered to be safe.
(2) Delegation: function approve(address delegate, uint tokenId) allows delegate to transfer the given token on behalf of the caller of the function, that must be the owner of the token or itself an authorized operator for the token. The previous delegate (if any) loses its delegation after this function has been called. This function emits an Approval event.
The function setApprovalForAll(address operator, bool approved) allows operator to transfer all tokens owned by the caller of the function (if approved is true) or removes that right (if approved is false). It is possible to allow more operators per token owner. This function emits an ApprovalForAll event. getApproved(uint tokenId) yields the delegate for the given token, if any. isApprovedForAll(address owner, address operator) determines if operator has been authorized to transfer all tokens owned by owner.
(3) Optional info: name() yields the name of the tokens. symbol() yields the symbol of the tokens.
OpenZeppelin's implementation of the ERC-721 standard is relatively long, so we only report a portion of the code in Fig. 2. Most information is kept in four maps: _owners specifies who is the owner of each given token; _balances tells how many tokens each given owner owns; _tokenApprovals specifies which delegate has been authorized for each given token (if any); and _opera-torApprovals yields the set of approved operators for each token owner. Note that mapping (address =[ bool) is actually a set of approved operators: Solidity has no set type, hence sets are encoded as their characteristic map. Figure 2 shows that transferFrom calls an auxiliary function _transfer that decreases the balance of the sender, increases the balance of the receiver and assigns the token to the receiver (to). There is no check on the fact that to is actually an externally owned account, or a contract, able to deal with the token it receives. This check exists for function safeTransferFrom (not shown in Fig. 2). The idea is that contracts ready to receive ERC-721 tokens must be explicitly labeled by their programmer as implementing an IERC721Receiver interface, whose only method onReceive is called when the contracts receive an ERC-721 token. In general, it would be enough to check that to instanceof IERC721Receiver in order to be sure that the programmer was actually expecting the contract to receive ERC-721 tokens and to call onReceive in that case. But this is not possible in Solidity, since that language lacks the instanceof operator and, in general, it misses any way to check the dynamic type of values. This is not just a missed feature: it is actually impossible to implement such a check in Solidity, since Ethereum implements data as unboxed values, so that their dynamic type is not available and no instanceof operator can ever be implemented. Because of this limitation, Solidity programmers use a very cumbersome technique, based on the ERC-165 standard [30], consisting in adding a function that yields a hash of the signatures of the methods implemented by a contract. By calling that function, it is possible, at run time, to guess the interfaces implemented by a contract. This technique (that we have highly simplified but is much more complicated than what we could express here) is very weak, since contracts are free to cheat and pretend to implement an interface that they actually do not implement. However, it is the best that a programmer can do in Solidity. There is an even weaker approach to cope with this problem. Namely, the ERC-223 token standard [12] requires to cast the token receiver to an interface IERC223Recipient and then call its tokenReceived method. If the receiver does not implement such method, the transaction fails. This is even weaker than ERC-165 since it makes no attempt to guarantee that the receiver was actually declared to implement IERC223Recipient: casts are unchecked in Solidity, they are pure decorations to make the compiler accept the code, but they are not verified at run time.

From Solidity to Takamaka
OpenZeppelin's implementations of ERC-20 ( Fig. 1) and of ERC-721 ( Fig. 2) are only around a few hundred noncomment lines of Solidity. Their code is not particularly complex, although their correctness has never been proved formally, in particular against overflows and underflows, by using formal techniques such as abstract interpretation, as already possible in Java [33]. Bugs are not a theoretical possibility, as the iToken incident shows [21] (that, however, did not affect OpenZeppelin's, but another ERC-20 implementation). Bugged contracts cannot be patched and replaced in blockchain, but only redeployed at another address. Their correctness is hence of major importance. Years of exposure to the open-source community and 35 Github contributors give some confidence in OpenZeppelin's code. Hence, if an ERC-20 or ERC-721 implementation must be provided in another programming language, a literal translation of OpenZeppelin's code is a more reliable starting point than a complete rewriting from scratch.
However, code migration between different programming languages can be tricky, also for relatively simple code. There is no formal way that one can follow to perform such a translation. Therefore, we are not going here to provide any formal proof of equivalence between the original Solidity code and its translation into Takamaka, but only a re-engineering approach and some translation patterns.
Languages might have different semantics for apparently similar constructs or might require different coding styles, for efficiency, which is more often the case if they compile towards different virtual machines. For instance, Vyper [40] and Solidity compile for the same EVM and the translation from Solidity to Vyper [41] is almost immediate. Takamaka compiles for the JVM and the translation from Solidity to Takamaka is more difficult. In many cases, different programming languages have specific solutions that cannot be translated literally: for instance, Java has an instanceof operator, hence it is pointless to translate the ERC-165-based technique used in Solidity to allow contracts to hold ERC-721 tokens only if they explicitly declare to implement a specific interface. Just use instanceof in Java instead. Nevertheless, our analysis of both languages highlights some translation patterns from Solidity to Takamaka, as shown below.
Visibility modifiers. Solidity's public and private have direct Java equivalents. Solidity's internal corresponds to Java's protected, but the latter grants access also to code in the same package of the class C where protected is used, which is not the case for internal (Solidity has no packages). This might be dangerous since an attacker might place a new class in C's package and get access to C's methods that were meant to be C's implementation details. To avoid this scenario, the verifier of Takamaka code, that Hotmoka runs before installing code in blockchain, rejects split packages, i.e. does not allow two classes in the same package to occur in different jars (Java archives) in the classpath (Java enforces the same constraint only from Java 9). Thanks to this constraint, internal can be safely translated into Java's protected. Solidity's external grants access to a function only to other contracts and, in this sense, it is used to specify the public API of a contract. There is no such visibility notion in Java. However, Takamaka introduces the @FromContract annotation, which restricts the callers of a method or constructor to be contracts. Hence external can be translated into public @FromContract.
The following view modifier. In Solidity, this states that a function (such as balanceOf in Fig. 1) has no side-effects and can consequently be executed outside of transactions, in every single node of the blockchain. This translates into Takamaka's @View annotation, with the same semantics.
override and virtual modifiers. Solidity and Java take opposite approaches to non-private methods redefinition. Namely, methods can be redefined in Solidity only if they are marked with virtual and redefinitions must be marked with override. In Java, methods can always be redefined unless they are marked with final and redefinitions do not need any special syntactical mark, although the @Override annotation has become customary. Consequently, the translation of these modifiers from Solidity to Takamaka is the following: uint type. Solidity uses uint (short form of uin-t256) to represent unsigned, potentially very large integers (up to 2 256 À 1). For instance, ERC-20 implementations use uint to represent token balances (Fig. 1). This type suffers from (silent) underflows and overflows. To cope with this problem, Solidity code can use the SafeMath library that provides arithmetic functions with defensive checks against underflows, overflows and divisions by zero. The latest versions of Solidity implement such checks in the language, natively, at an increased gas cost. Takamaka code can use UnsignedBigInteger for that, a wrapper of Java's BigInteger class, from Takamaka's support library, whose operations include defensive checks, with the extra advantage that they are unbounded unsigned integers, hence do not suffer from overflows.
mapping type: Solidity uses the mapping type for maps between values, as for field _balances in Fig. 1. These are not data structures, but rather an algorithm that spreads the bindings of the mapping in the key/value store of Ethereum (with an unlikely risk of hash collision). Takamaka can use an actual, generic data structure StorageTreeMap\Key,Value[ instead, an implementation of the interface StorageMap\Key,Value[, from Takamaka's support library. Solidity's maps default to 0, hence one must use getOrDefault(index, 0) calls on StorageTreeMap in Takamaka. If mapping is used in Solidity as a trick to implement a set (as in the codomain of _operatorApproval in Fig. 2), then in Takamaka it is simpler and more efficient to use a StorageTreeSet\Value[ instead, that is an implementation of the interface StorageSet\Value[ from Takamaka's support library.
msg.sender. This Solidity expression refers to the contract that calls a function. In Takamaka, this corresponds to caller() inside a @FromContract method.
address(0). This Solidity expression refers to a contract or account at address 0. It is assumed that nobody controls that contract or account. Hence, traditionally, it stands for a missing value or for the sign of missing information in a transaction request. In Takamaka, the same can be achieved with null. Figure 3 shows our manual translation in Takamaka of the Solidity code for ERC-20 in Fig. 1, by following the heuristics above. The translation is almost literal, with a few exceptions. For instance, function transferFrom in Fig. 1 enforces a non-negative allowance through a require assertion. In Fig. 3, that same check is moved inside the subtract method of the UnsignedBigInteger class. Figure 4 shows our manual translation in Takamaka of the Solidity code for ERC-721 in Fig. 2. Also this translation is almost literal. We observe that the _opera-torApprovals field uses a StorageSet in Takamaka, instead of the Solidity trick of using a map to represent a set. Token instances are represented as BigInteger in Takamaka, hence they are more general than in Solidity, where they are limited to be uint, hence 256 bits only. The _balances field uses BigInteger to represent the balance of each token holder. This is cheaper than UnsignedBigInteger and has been preferred in this case since the code of the contract guarantees such values to be non-negative, hence the run-time checks of UnsignedBigInteger are not useful here. Maps in Takamaka cannot use the handy indexing notation of Solidity and do not use null to represent a missing binding. This explains why the Takamaka code is sometimes a bit more verbose (see for instance the methods isApprovedForAll and _approve). In Takamaka, both methods transferFrom and safeTrans-ferFrom have been collapsed into a single method transferFrom that safely checks if the receiver of the token is an externally owned account or a contract that implements IERC721Receiver. In this latter case, its onReceive method is called. The check on the type of the receiver is sound in Takamaka and doesn't need the tricky and fragile ERC-165 machinery, since Java has an instanceof operator that fails if the test is false.

Snapshots of ERC-20 ledgers
OpenZeppelin has subclassed its ERC20 implementation (Sect. 2) to provide extra functionalities, for instance for tokens that can be (further) minted, burned, capped or paused. Among them, this paper focuses on the ERC20Snapshot subclass only, which supports snapshots, shown in Fig. 5. Namely, it adds a _snapshot function that performs a snapshot of the ledger and yields its progressive identifier (starting at 1). Then it overloads methods balanceOf and totalSupply from Fig. 1 with variants that receive a snapshot identifier and yield the balance and the total supply at the time of that snapshot (Fig. 5). For that, it stores the modification history of an integer variable by using the following data structure: For instance, if a variable v is associated with a Snapshots structure with fields ids={5,8,15} and values={6,7,20}, then the value of v was 20 for snapshot identifiers from 9 to 15; it was 7 for snapshot identifiers from 6 to 8; it was 6 for snapshot identifiers from 1 to 5; for snapshot identifiers after 20, the value of v is v's current value in the ledger. A function _valueAt (not shown in Fig. 5) reconstructs the value of a variable at a snapshot. There is one Snapshots instance for each address that takes part in the token, inside a new field mapping (address =[ Snapshots) private _balancesSnapshots, and for _totalSupply, with a new field Snapshots private _totalSup-plySnapshots. Such structures are allocated and populated whenever a balance gets updated or the total supply changes (the latter situation occurs if mints or burns are allowed). This is achieved by overriding the internal function _beforeTokenTransfer (see Fig. 5).
The code of ERC20Snapshot (that is very technical and consequently we do not show) has good computational complexity: it creates snapshots in O(1); since the ids fields are sorted, it retrieves balances and total supply at each given snapshot in Oðlog nÞ, by binary search, where n is the number of snapshots already performed. Nevertheless, it has some drawbacks: (1) It is complex and tricky. We found it very hard to reach a sufficient trust in its correctness. It is so complicated and specific to ERC-20 that its extension from ERC-20 to ERC-721 tokens has never been done. Fig. 4 A portion of our ERC-721 implementation in Takamaka. Its full code is available at https://github.com/Hotmoka/hotmoka/blob/master/iotakamaka-code/src/main/java/io/takamaka/code/tokens/ERC721.java (2) It induces a significant overhead for the manipulation of the Snapshots, also because it needs the extra _balancesSnapshots map. (3) All participants pay the overhead of the previous point when they transfer tokens, not just those who create snapshots. That is, if a participant creates a snapshot, then the other participants will later pay the overhead during transfers, even though they were not interested in the snapshot. (4) If a large number of snapshots is generated, arrays ids and values might become so long that their manipulation exceeds the maximal gas (metering of code execution) allowed for Ethereum transactions, which is the perfect surface for a denial of service attack. That is why function _snapshot is internal: subclasses must implement some security policy to control its access.
We have translated in Takamaka the Solidity code from Fig. 5. The result of this translation is at https://github.com/ Hotmoka/hotmoka/blob/master/io-hotmoka-examples/src/ main/java/io/hotmoka/examples/tokens/ERC20OZSnap shot.java. It works perfectly but suffers from the same issues highlighted above for its Solidity counterpart. Hence, it is interesting to investigate whether a better implementation of ERC-20 contracts with snapshots exists, at least in Takamaka, which is our target language. Moreover, it is interesting to see if that implementation can also work for ERC-721 tokens, currently missing the snapshot feature in Solidity. 6 An efficient algorithm for snapshots By looking at OpenZeppelin's code in Fig. 1, it would be convenient to implement the _snapshot function in a way completely different from that described in Sect. 5: it should return an actual snapshot (not its identifier), i.e. a data structure containing an immutable view of the ledger. This new implementation does not increase the length of any array and can be safely public. In Solidity-like code, this would look like in Fig. 6. However, this code cannot be written in Solidity. The main reason is that Solidity maps cannot be cloned, since they are not data structures, but just an algorithm for distributing key/value pairs in the storage of Ethereum. Solidity maps do not even know their set of keys, whose iteration would at least allow a (very expensive) clone of the map. Moreover, at the time we conducted the analysis and experiments, Solidity functions could not return a struct (from Solidity v0.8, ABIEncoderV2 implements that feature). This is why we talk about pseudocode in Fig. 6: it does not really compile. Figure 7 shows that the corresponding code can well be written in Takamaka instead. The local inner class SnapshotImpl plays the role of the struct in Solidity. At creation time, it clones fields _totalSupply and _balances from the outer ERC20 object. Class SnapshotImpl actually implements a new superinterface IERC20View of IERC20, that has only the readonly methods of ERC-20, i.e. totalSupply and bal-anceOf. Figure 8 shows the UML diagram of these interfaces and classes. It shows that there is no special class for ERC-20 contracts with snapshots anymore: all ERC-20 contracts can be snapshotted.
The magic of this Java code is that, in Takamaka, an immutable clone of _balances is simply _balances.snapshot() (the snapshot method of StorageMap), that runs in O(1). Therefore, the problem is now to understand how the class StorageTreeMap and its snapshot method work. They exploit the same idea used, for instance, in the Git version control system and in the storage of Ethereum, allowing one to check out their full history of states, by simply swapping a root pointer. They favor the re-creation of immutable data structures instead of updates to mutable data structures.
More in detail, in our case class Stor-ageTreeMap\K,V[ implements red/black trees [31], a special kind of balanced binary search trees that orders keys of type K by their storage reference, i.e. a machineindependent pointer to the keys in the memory of the blockchain [35]. Such references are 32 bytes long, i.e. 256 bits. Since a red/black tree is balanced, the length of a path from root to leaf is 256 at most and get and put operations run in O(256), i.e. in O(1). Figure 9(a) shows a Stor-ageTreeMap\Contract,UnsignedBigInteger[ _balances that implements the mapping with the following insertion order: 81af7 !14, 77b17 !18, da897 !14, 71a07 !19, fa317 !35 and 91007 !5 (for simplicity, this example assumes that storage references are only 2 bytes long, i.e. four hexadecimal digits or 16 bits). We remember that the O notation states a worst-case scenario. Namely, the cost for get and put is often smaller than 256 operations, being in general dependent on the number of elements in the tree. We are not stating that get and put cost always exactly 256 operations, which would need the H notation instead. What we are stating is that it is never higher than 256, which is the meaning of the O notation. The fact that get and put run in constant worst-case time is made possible by the choice of a particular kind of keys, whose size is fixed a priori. The situation here is similar to the use of Merkle-Patricia tries for implementing the storage of Ethereum, whose get and put operations are considered to run in constant time as well, since their cost increases with the size of the trie but is bounded from above by a constant [4,6]. Also, in that case, constant worst-case time is possible since keys are Ethereum addresses, hence of fixed size. Figure 9a shows also the computation of a clone of _balances: it is another StorageTreeMap whose root is the same root of _balances. The independence between _balances and its clones is obtained by making the nodes of the trees immutable data structures: destructive updates of the tree actually create new nodes instead of modifying old nodes. For instance, Fig. 9b shows an update to _balances, that changes the value bound to da89, from 14 to 30. It shows that both nodes for 81af and da89 are recreated (darkened in the figure), and the root of _balances is updated. The clone's root remains unchanged instead and points to the old tree. Note that Fig. 6 The pseudocode of an alternative implementation of snapshots in Solidity, that its compiler does not accept computing a clone means just creating a new root cell that points to the current root of the tree. Hence, a clone is computed in O(1). The idea of creating independent clones of a tree by using immutable nodes and a new root pointer is not new. We have borrowed this idea from the way the Git version control system works internally. Git allows very inexpensive creation of branches of a repository in O(1), since a branch is just a moving reference to the root of the repository.
The code in Fig. 7 has the same asymptotical complexity as OpenZeppelin's ERC-20 contracts with snapshots, but overcomes all its drawbacks reported at the end of Sect. 5: (1) It is simple and intuitive. Class StorageTreeMap might look complex but comes with the support library of Takamaka and needn't be re-implemented. (2) It has no overhead because of snapshots and no _balancesSnapshots map exists anymore. (3) Who creates a snapshot pays gas. The other participants can transfer coins without paying any overhead because of that snapshot. (4) There are no arrays that grow in size when snapshots are created, hence a denial of service attack is not possible.
Moreover, the same technique can be used to implement a snapshot of an ERC-721 token ledger as well. There is no extra difficulty in comparison with ERC-20 ledgers. The only difference is that the snapshot must be performed for two maps this time: for the _balances and for the _owners maps of the implementation in Fig. 4. The snapshot method added to the code in Fig. 4 is shown in Fig. 10. Also, in this case, there is an IERC721View interface that collects the read-only methods of IERC721.
The same technique can be applied to generate snapshots of other data structures. The idea is always that described above, based on the snapshot method of the underlying components of the data structure. For instance, we have implemented snapshots also for shared entities [5], that represents an object divided in many, dynamically changing shares, such as a private company, a DAO or the set of validators of a blockchain.

Performance evaluation
This section compares the performance of the literal translation into Takamaka of OpenZeppelin's ERC-20 contracts with snapshots (Sect. 5) against that of our implementation in Takamaka that uses a more efficient snapshot algorithm (Figs. 3, 7), which we call Native, in terms of gas consumed for code execution. Gas is the standard cost measure for smart contracts, since it reflects the actual number of resources (CPU cycles, RAM allocations, storage slots) that each node of a blockchain must consume. However, gas is a low-level, bytecode-specific measure and Solidity and Takamaka use two completely different bytecode languages. Because of that, what we are actually going to compare is OpenZeppelin's ERC-20 contract with snapshots translated in Takamaka (end of Sect. 5), that we call OpenZeppelin, against our Native. Both are written in Takamaka and both are compiled into Java bytecode. Hence, the comparison gives a measure of the relative efficiency of the two algorithmic solutions, which is what we are looking for. Instead, this is not a comparison between OpenZeppelin's Solidity code and our Takamaka code, or more generally between Solidity and Takamaka, that would be meaningless and that we cannot provide, since they compile into distinct bytecode languages, have different gas models and do not allow the same algorithmic solutions: maps can be cloned in Takamaka but not in Solidity.
We have written a JUnit test case that simulates a typical usage scenario for an ERC-20 contract: it creates the contract in blockchain, spreads its tokens among a set of investors (some random externally owned accounts), play for some time with the ERC-20 contract (we assumed for 10 days), performing random token transfers between them, burning some random tokens or minting new random tokens. At the end of each day, it takes a snapshot. The test case is implementation-agnostic: given an implementation of ERC-20 with snapshots (such as OpenZeppelin or Native), the test case will reproduce the scenario and report the gas consumption. Moreover, in order to be deterministic and fair, the test case uses a fixed seed for random choices. Hence its execution is exactly the same at each run, with both OpenZeppelin and Native. Similarly, the number and kind of transactions executed by the test case do not change. The interested reader can inspect and run the test case by cloning the repository of Hotmoka (git clone https://github.com/Hotmoka/hotmoka.git-berc20-comparison) and following the instructions in the file README.txt. Figure 11 8 The UML class diagram of the IERC20View and IERC20 interfaces, implemented by the ERC20 class. The IERC20View interface is a very abstract view of a ledger: it has methods for read-only access and for creating snapshots This experiment shows that our Native solution with efficient snapshots (Fig. 7) saves gas units (hence money) and reduces the overall time for the execution of the test case. This time reduction is more apparent when there are many investors, as the overhead of OpenZeppelin's solution consequently grows.

Related work
The ERC-20 standard [39] for fungible tokens was originally defined for initial coin offers and for the definition of new kinds of tokens supported by the underlying, native token of the blockchain. The ERC-721 standard [13] for non-fungible tokens has experienced an impressive success, mainly for the definition of NFTs for art and, in general, for representing things having a specific value. Their OpenZeppelin implementations [26] and [27], respectively, are currently the de facto standard implementations for Ethereum-like blockchains. The importance of such standards is growing with the progressive application of blockchain technology beyond its original context of cryptocurrency.
Blockchain technology is nowadays involved in different fields of application, such as tracking energy consumption [20], transportation systems [11], supply chain management [1], IoV [32], IoT [2,23], cloud computing [15], to support the decentralization of the Internet [42], and also it includes the technology for decentralized identities [29]. Different application contexts also involve different programming languages, systems and platforms, which must implement and support these standards. For instance, Hyperledger Fabric proposes some sample implementation in Java, Go, Javascript [17,18]. Instead, for Cosmos, there are implementations written in Rust [19]. However, they limit themselves to proposing minimal versions of these standards by omitting valuable features, like snapshots, without offering improvements by exploiting the target languages.

Conclusion
This paper has shown some patterns for the code migration from Solidity to Takamaka, applied to the specific examples of ERC-20 and ERC-721 contracts. It has shown an improvement of the literal code translation of ERC-20 contracts with snapshots, by using maps with immutable clones, not available in Solidity but implemented in other programming languages. It has shown that the same technique applies to ERC-721 contracts, where snapshots were previously missing. The result has been validated with a test case that shows the reduced gas and time costs of our implementation w.r.t. OpenZeppelin's within the JVM.
The possibility of creating immutable clones of maps in O(1) is useful to simplify the code of other contracts as well, where Solidity must use tricky code instead (as that described in Sect. 5) or recur to events, to mark the historical evolution of data and allow its recovery (with extra gas costs). For instance, we have added snapshots also to shared entities, that are used to implement DAOs and the set of validators of a proof-of-stake blockchain [5]. Data availability Enquiries about data availability should be directed to the authors.

Decalarations
Conflict of interest The authors have not disclosed any competing interests.
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