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@ -603,7 +603,6 @@ |
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\newcommand{\ECtoOSPXS}{\mathsf{EC2OSP\mhyphen{}XS}} |
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\newcommand{\ItoOSP}[1]{\mathsf{I2OSP}_{#1}} |
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\newcommand{\ItoBSP}[1]{\mathsf{I2BSP}_{#1}} |
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\newcommand{\BStoIP}[1]{\mathsf{BS2IP}_{#1}} |
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\newcommand{\FEtoIP}{\mathsf{FE2IP}} |
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\newcommand{\BNImpl}{\mathtt{ALT\_BN128}} |
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\newcommand{\vpubOld}{\mathsf{v_{pub}^{old}}} |
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@ -856,6 +855,10 @@ defined either on integers or bit sequences according to context. |
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The notation $\vsum{i=1}{\mathrm{N}} a_i$ means the sum of $a_{\allN{}}$.\; |
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$\vxor{i=1}{\mathrm{N}} a_i$ means the bitwise exclusive-or of $a_{\allN{}}$. |
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The binary relations $<$, $\leq$, $=$, $\geq$, and $>$ have their conventional |
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meanings on integers and rationals, and are defined lexicographically on |
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sequences of integers. |
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The notation $\floor{x}$ means the largest integer $\leq x$. |
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$\ceiling{x}$ means the smallest integer $\geq x$. |
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@ -2927,11 +2930,6 @@ Define $\ItoBSP{} \typecolon (u \typecolon \Nat) \times \range{0}{2^u\!-\!1} \ri |
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such that $\ItoBSP{u}(x)$ is the sequence of $u$ bits representing $x$ in |
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big-endian order. |
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Define $\BStoIP{} \typecolon (u \typecolon \Nat) \times \bitseq{u} \rightarrow \range{0}{2^u\!-\!1}$ |
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such that $\BStoIP{u}$ is the inverse of $\ItoBSP{u}$. |
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Define $\Xi_r(a, b) := \BStoIP{2^{r-1} \mult \ell}(\concatbits(X_{i_{a..b}}))$. |
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A \validEquihashSolution is then a sequence $i \typecolon \range{1}{N}^{2^k}$ that |
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satisfies the following conditions: |
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@ -2944,7 +2942,7 @@ $\vxor{j=1}{2^k} X_{i_j} = 0$. |
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For all $r \in \range{1}{k\!-\!1}$, for all $w \in \range{0}{2^{k-r}\!-\!1}$: |
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\begin{itemize} |
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\item $\vxor{j=1}{2^r} X_{i_{w \mult 2^r + j}}$ has $\frac{n \mult r}{k+1}$ leading zeroes; and |
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\item $\Xi_r(w \mult 2^r + 1, w \mult 2^r + 2^{r-1}) < \Xi_r(w \mult 2^r + 2^{r-1} + 1, w \mult 2^r + 2^r)$. |
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\item $i_{w \mult 2^r + 1 .. w \mult 2^r + 2^{r-1}} < i_{w \mult 2^r + 2^{r-1} + 1 .. w \mult 2^r + 2^r}$ lexicographically. |
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\end{itemize} |
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\pnote{ |
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@ -3001,14 +2999,12 @@ and so the first 7 bytes of $\solution$ would be |
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$[0, 2, 32, 0, 10, 127, 255]$. |
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\pnote{ |
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$\ItoBSP{}$ and $\BStoIP{}$ are big-endian, while the encoding of |
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integer fields in $\powheader$ and in the instantiation of $\EquihashGen{}$ |
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is little-endian. The rationale for this is that little-endian |
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serialization of \blockHeaders is consistent with \Bitcoin, but using |
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little-endian ordering of bits in the solution encoding would require |
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bit-reversal (as opposed to only shifting). The comparison of $\Xi_r$ |
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values obtained by a big-endian conversion is equivalent to lexicographic |
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comparison as specified in \cite[section IV A]{BK2016}. |
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$\ItoBSP{}$ is big-endian, while integer field encodings in $\powheader$ |
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and in the instantiation of $\EquihashGen{}$ are little-endian. |
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The rationale for this is that little-endian serialization of |
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\blockHeaders is consistent with \Bitcoin, but using little-endian |
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ordering of bits in the solution encoding would require bit-reversal |
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(as opposed to only shifting). |
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} |
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\nsubsubsection{Difficulty filter} \label{difficulty} |
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@ -3545,6 +3541,9 @@ The errors in the proof of Ledger Indistinguishability mentioned in |
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\subparagraph{2016.0-beta-1.6} |
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\begin{itemize} |
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\item Fix an error in the definition of the sortedness condition for Equihash: |
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it is the sequences of indices that are sorted, not the sequences of |
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hashes. |
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\item Correct the number of bytes in the encoding of $\solutionSize$. |
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\item Update the section on encoding of \transparent addresses. |
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(The precise prefixes are not decided yet.) |
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