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(193) (194) HP(x,t) : x ∈ Y ≡ f(t) ≡ if ∥x∥ is even ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION LEDGER BERLIN VERSION 20 If RLP is used to encode a scalar, defined only as a non-negative integer (in N, or in Nx for any x), it must be encoded as the shortest byte array whose big-endian interpretation is the scalar. Thus the RLP of some non-negative integer i is defined as: (192) RLP(i : i ∈ N) ≡ RLP(BE(i)) When interpreting RLP data, if an expected fragment is decoded as a scalar and leading zeroes are found in the byte sequence, clients are required to consider it non-canonical and treat it in the same manner as otherwise invalid RLP data, dismissing it completely. There is no specific canonical encoding format for signed or floating-point values. Appendix C. Hex-Prefix Encoding Hex-prefix encoding is an efficient method of encoding an arbitrary number of nibbles as a byte array. It is able to store an additional flag which, when used in the context of the trie (the only context in which it is used), disambiguates between node types. It is defined as the function HP which maps from a sequence of nibbles (represented by the set Y) together with a boolean value to a sequence of bytes (represented by the set B): (16f (t), 16x[0] + x[1], 16x[2] + x[3], ...) (16(f (t) + 1) + x[0], 16x[1] + x[2], 16x[3] + x[4], ...) otherwise 2 if t̸=0 0 otherwise Thus the high nibble of the first byte contains two flags; the lowest bit encoding the oddness of the length and the second-lowest encoding the flag t. The low nibble of the first byte is zero in the case of an even number of nibbles and the first nibble in the case of an odd number. All remaining nibbles (now an even number) fit properly into the remaining bytes. Appendix D. Modified Merkle Patricia Tree The modified Merkle Patricia tree (trie) provides a persistent data structure to map between arbitrary-length binary data (byte arrays). It is defined in terms of a mutable data structure to map between 256-bit binary fragments and arbitrary-length binary data, typically implemented as a database. The core of the trie, and its sole requirement in terms of the protocol specification, is to provide a single value that identifies a given set of key-value pairs, which may be either a 32-byte sequence or the empty byte sequence. It is left as an implementation consideration to store and maintain the structure of the trie in a manner that allows effective and efficient realisation of the protocol. Formally, we assume the input value I, a set containing pairs of byte sequences with unique keys: (195) I = {(k0 ∈ B,v0 ∈ B),(k1 ∈ B,v1 ∈ B),...} When considering such a sequence, we use the common numeric subscript notation to refer to a tuple’s key or value, thus: (196) ∀I ∈ I : I ≡ (I0,I1) Any series of bytes may also trivially be viewed as a series of nibbles, given an endian-specific notation; here we assume big-endian. Thus: (197) y(I) = {(k′0 ∈ Y,v0 ∈ B),(k′1 ∈ Y,v1 ∈ B),...} ⌊kn[i÷2]÷16⌋ if i is even (198) ∀n : ∀i < 2∥kn∥ : k′n[i] ≡ kn[⌊i ÷ 2⌋] mod 16 otherwise We define the function TRIE, which evaluates to the root of the trie that represents this set when encoded in this structure: (199) TRIE(I) ≡ KECRLP(c(I, 0)) We also assume a function n, the trie’s node cap function. When composing a node, we use RLP to encode the structure. As a means of reducing storage complexity, we store nodes whose composed RLP is fewer than 32 bytes directly; for those larger we assert prescience of the byte array whose Keccak hash evaluates to our reference. Thus we define in terms of c, the node composition function: () (200) n(I, i) ≡ c(I, i) KECRLP(c(I, i)) if I=∅ if ∥ RLPc(I, i)∥ < 32 otherwise In a manner similar to a radix tree, when the trie is traversed from root to leaf, one may build a single key-value pair. The key is accumulated through the traversal, acquiring a single nibble from each branch node (just as with a radix tree). Unlike a radix tree, in the case of multiple keys sharing the same prefix or in the case of a single key having a uniquePDF Image | ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION
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