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(16) LT(T) ≡ (Tn,Tp,Tg,Tt,Tv,Ti,Tw,Tr,Ts) (Tn, Tp, Tg, Tt, Tv, Td, Tw, Tr, Ts) if Tt = ∅ otherwise where (18) Nn ={P :P ∈N∧P <2n} ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION LEDGER BERLIN VERSION 5 to sign. For the present we simply assert that the sender of a given transaction T can be represented with S(T). number: A scalar value equal to the number of an- cestor blocks. The genesis block has a number of zero; formally Hi. gasLimit: A scalar value equal to the current limit of gas expenditure per block; formally Hl. gasUsed: A scalar value equal to the total gas used in transactions in this block; formally Hg. timestamp: A scalar value equal to the reasonable output of Unix’s time() at this block’s inception; formally Hs. extraData: An arbitrary byte array containing data relevant to this block. This must be 32 bytes or fewer; formally Hx. mixHash: A 256-bit hash which, combined with the nonce, proves that a sufficient amount of compu- tation has been carried out on this block; formally Hm. nonce: A64-bitvaluewhich,combinedwiththemix- hash, proves that a sufficient amount of computa- tion has been carried out on this block; formally Hn. The other two components in the block are simply a list of ommer block headers (of the same format as above), BU and a series of the transactions, BT. Formally, we can refer to a block B: (20) B ≡ (BH, BT, BU) 4.3.1. Transaction Receipt. In order to encode information about a transaction concerning which it may be useful to form a zero-knowledge proof, or index and search, we encode a receipt of each transaction containing certain in- formation from its execution. Each receipt, denoted BR[i] for the ith transaction, is placed in an index-keyed trie and the root recorded in the header as He. The transaction receipt, R, is a tuple of four items comprising: the status code of the transaction, Rz, the cumulative gas used in the block containing the transac- tion receipt as of immediately after the transaction has happened, Ru, the set of logs created through execution of the transaction, Rl and the Bloom filter composed from information in those logs, Rb: (21) R ≡ (Rz, Ru, Rb, Rl) We assert that the status code Rz is a non-negative integer: (22) Rz ∈ N We assert that Ru, the cumulative gas used, is a non- negative integer and that the logs Bloom, Rb, is a hash of size 2048 bits (256 bytes): (23) Ru ∈N ∧ Rb ∈B256 The sequence Rl is a series of log entries, (O0,O1,...). A log entry, O, is a tuple of the logger’s address, Oa, a possibly empty series of 32-byte log topics, Ot and some number of bytes of data, Od: Here, we assume all components are interpreted by the RLP as integer values, with the exception of the arbitrary length byte arrays Ti and Td. (17) Tn ∈N256 ∧ Tv ∈N256 ∧ Tg ∈N256 ∧ Tw ∈N256 ∧ Ts∈N256 ∧ Td∈B ∧ Tp ∈N256 ∧ Tr ∈N256 ∧ Ti∈B The address hash Tt is slightly different: it is either a 20-byte address hash or, in the case of being a contract- creation transaction (and thus formally equal to ∅), it is the RLP empty byte sequence and thus the member of B0: (19) Tt ∈ B20 ifTt̸=∅ B0 otherwise 4.3. The Block. The block in Ethereum is the collec- tion of relevant pieces of information (known as the block header), H, together with information corresponding to the comprised transactions, T, and a set of other block headers U that are known to have a parent equal to the present block’s parent’s parent (such blocks are known as ommers2). The block header contains several pieces of information: parentHash: TheKeccak256-bithashoftheparent block’s header, in its entirety; formally Hp. ommersHash: The Keccak 256-bit hash of the om- mers list portion of this block; formally Ho. beneficiary: The 160-bit address to which all fees collected from the successful mining of this block be transferred; formally Hc. stateRoot: The Keccak 256-bit hash of the root node of the state trie, after all transactions are executed and finalisations applied; formally Hr. transactionsRoot: The Keccak 256-bit hash of the root node of the trie structure populated with each transaction in the transactions list portion of the block; formally Ht. receiptsRoot: The Keccak 256-bit hash of the root node of the trie structure populated with the re- ceipts of each transaction in the transactions list portion of the block; formally He. logsBloom: The Bloom filter composed from index- able information (logger address and log topics) contained in each log entry from the receipt of each transaction in the transactions list; formally Hb. difficulty: A scalar value corresponding to the dif- ficulty level of this block. This can be calculated from the previous block’s difficulty level and the timestamp; formally Hd. (24) O ≡ (Oa, (Ot0, Ot1, ...), Od) (25) Oa ∈B20 ∧ ∀x∈Ot :x∈B32 ∧ Od ∈B 2ommer is a gender-neutral term to mean “sibling of parent”; see https://nonbinary.miraheze.org/wiki/Gender_neutral_language_in_ English#Aunt/UnclePDF Image | ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION
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