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(187) (188) (189) Rb(x) ≡ 183+BE(∥x∥)·BE(∥x∥)·x elseif ∥x∥<2 64 ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION LEDGER BERLIN VERSION 19 LLL: The Lisp-like Low-level Language, a human-writable language used for authoring simple contracts and general low-level language toolkit for trans-compiling to. Appendix B. Recursive Length Prefix This is a serialisation method for encoding arbitrarily structured binary data (byte arrays). We define the set of possible structures T: (183) T ≡ L⊎B (184) L ≡ {t:t=(t[0],t[1],...) ∧ ∀n<∥t∥:t[n]∈T} (185) B ≡ {b:b=(b[0],b[1],...) ∧ ∀n<∥b∥:b[n]∈O} Where O is the set of (8-bit) bytes. Thus B is the set of all sequences of bytes (otherwise known as byte arrays, and a leaf if imagined as a tree), L is the set of all tree-like (sub-)structures that are not a single leaf (a branch node if imagined as a tree) and T is the set of all byte arrays and such structural sequences. The disjoint union ⊎ is needed only to distinguish the empty byte array () ∈ B from the empty list () ∈ L, which are encoded differently as defined below; as common, we will abuse notation and leave the disjoint union indices implicit, inferable from context. We define the RLP function as RLP through two sub-functions, the first handling the instance when the value is a byte array, the second when it is a sequence of further values: • If the byte array contains solely a single byte and that single byte is less than 128, then the input is exactly equal to the output. • If the byte array contains fewer than 56 bytes, then the output is equal to the input prefixed by the byte equal to the length of the byte array plus 128. • Otherwise, the output is equal to the input, provided that it contains fewer than 264 bytes, prefixed by the minimal-length byte array which when interpreted as a big-endian integer is equal to the length of the input byte array, which is itself prefixed by the number of bytes required to faithfully encode this length value plus 183. Byte arrays containing 264 or more bytes cannot be encoded. This restriction ensures that the first byte of the encoding of a byte array is always below 192, and thus it can be readily distinguished from the encodings of sequences in L. (186) RLP(x) ≡ If the value to be serialised is a byte array, the RLP serialisation takes one of three forms: Formally, we define Rb: BE(x) ≡ (x1, ..., xn) · (y1, ..., ym) = ∥b∥−1 n=0 x (128+∥x∥)·x if ∥x∥=1∧x[0]<128 else if ∥x∥ < 56 Rb(x) if x ∈ B Rl (x) otherwise ∅ (b0,b1,...) : b0 ̸= 0 ∧ x = bn · 256∥b∥−1−n (x1, ..., xn, y1, ..., ym) Thus BE is the function that expands a non-negative integer value to a big-endian byte array of minimal length and the dot operator performs sequence concatenation. If instead, the value to be serialised is a sequence of other items then the RLP serialisation takes one of two forms: • If the concatenated serialisations of each contained item is less than 56 bytes in length, then the output is equal to that concatenation prefixed by the byte equal to the length of this byte array plus 192. • Otherwise, the output is equal to the concatenated serialisations, provided that they contain fewer than 264 bytes, prefixed by the minimal-length byte array which when interpreted as a big-endian integer is equal to the length of the concatenated serialisations byte array, which is itself prefixed by the number of bytes required to faithfully encode this length value plus 247. Sequences whose concatenated serialized items contain 264 or more bytes cannot be encoded. This restriction ensures that the first byte of the encoding does not exceed 255 (otherwise it would not be a byte). (190) (191) Rl(x) ≡ s(x) ≡ 247 + BE(∥s(x)∥) · BE(∥s(x)∥) · s(x) ∅ Thus we finish by formally defining Rl: (192 + ∥s(x)∥) · s(x) if s(x) ̸= ∅ ∧ ∥s(x)∥ < 56 else if s(x) ̸= ∅ ∧ ∥s(x)∥ < 264 otherwise RLP(x[0]) · RLP(x[1]) · ... ∅ if ∀i : RLP(x[i]) ̸= ∅ otherwise otherwisePDF Image | ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION
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