From e90d14f88ce0dcafbd57b05ddfbc84e5a605e11b Mon Sep 17 00:00:00 2001 From: Mikko Rantanen Date: Mon, 18 Oct 2010 05:52:50 +0300 Subject: [PATCH] Described varint decode function in documentation --- doc/mumble-protocol.pdf | Bin 387090 -> 401011 bytes doc/mumble-protocol.tex | 47 +++++++++++++++++++++++++++++----------- 2 files changed, 34 insertions(+), 13 deletions(-) diff --git a/doc/mumble-protocol.pdf b/doc/mumble-protocol.pdf index e7d3b51c3fd47df8abbce93ee0b0afeae72e67d9..a091fb4ef495c95bc6c95fa4e6cedf8db9f2ae2a 100644 GIT binary patch delta 93391 zcmX_`V{m3`6Ru<1wr$(i#C9gO@y51oOl+GI+qP}z?EUR?&hM*wb#>LM=kB|^6K63W zy0Gdh!9X~P(u@>9r~t0Y$zBzH-8*$8edNQOxtZP8~Yj;?bXrZx>p-T7zzHy+Lz;ASa{P^vdJFmCI=ZTg}t0GwCX}G z)?vO*$4DsA;h`_Zxo=52`vlXRXiAH7 zbd%Jlz@U$xd=ev)s0qa?q}~{qqsGqh*TW#%ztbDj;bfdNmo4CMR#tCq?-mNt)Wqit z$qKR=S*RyxD+4Y%+*DGfYl{DbnoH%HKdByLisG=Bqf_6DB#1+Zv1rsg^{FQCvxnL;K)? zzlHeHeXGravCij7Y9VUq{i=+_(?#Z~aZ)Bxn|?s_LI(`+uJIPY&7{(Dzt}y~QAs_+ 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zzvZH$ur+}Hs`F-V07rxu@1v;nf1lAU@y;wp-$b^%qvWFM)OH+lv%uVjC#H!#7mX7%e8HxW36;^)fz zF&&BBG_$<2Y3q66_}Vp-SM1Ht+zRQeIVkvpnFIu}b$p)rkR-&~cTyZT!ug`D7=?`o zg26B*M9i`?G(w#MXu=U07snKmjU*Z8wRf$HSL<|G`$H&X5VVy=nC(*LdkvR?6LI9i zK6C)RkrlEC48Np6!8vEsWATij9(rHUwZTuO6#WZH7N8C$j1wenAEA9E!saU5$Jjm` zV|2KTJo@44Ny=sKU*T5pgl(=TKub4rBFp;zGA791f5aT82Bw%DB+-k^Q&K95AycIs z=h7`GkG>Uy+p|4yj<_Cb^eulYJT%Z3f1(uVfx7P`)jJAxp&D@?%``?%8BcE>4gAW~FE7}D6#h#cREHHI`cInfV zgIOFCZ>An79WaPuts=U+J4f7Z>UP0yngtM*!|H)QwTPdG$9eU>} +\DeclareMathOperator{\append}{\triangleright} % Configure fancyvrb for our listings \usepackage{fancyvrb} @@ -418,27 +419,47 @@ Normal talking can be heard by the users of the current channel and all linked c \subsection {64-bit integer encoding} \label{sect:varint} -The variable length integer encoding is used to encode long (64-bit) integers so that short values do not need 4 bytes to be transferred. +The variable length integer encoding is used to encode long (64-bit) integers so that short values do not need the full 8 bytes to be transferred. The encoding function is given below. While it might seem complex it is worth noting that the $(a_v, a_p) \append (b_v, b_p)$ function equals appending the $a_p$ bits long value $a_v$ to a byte stream that already has the $b_p$ bits long value $b_v$. -%\begin{displaymath} \begin{align*} - e &: \mathbb{N} \rightarrow \mathbb{N}_{\geq0} \\ + (a_v, a_p) \append (b_v, b_p) &= (2^{b_p} a_v + b_v, a_p + b_p) \\ +% + e &: \mathbb{N} \rightarrow \mathbb{N}_{\geq0}^2 \\ e(x) &= \begin{dcases*} - e_+(x, 1, x) & when $ 0 \leq x \leq \text{0xFFFFFFF} $ \\ - x + \text{0xF0} \cdot {2^8}^4 & when $ \text{0xFFFFFFF} < x \leq \text{0xFFFFFFFF} $ \\ - x + \text{0xF4} \cdot {2^8}^8 & when $ \text{0xFFFFFFFFF} < x $ \\ - \text{0xFC} - x & when $ 0 < -x \leq \text{0x03} $ \\ - \text{0xF8} 2^{\lfloor \log_2 e(-x) \rfloor + 1} + e(-x) & when $ \text{0x03} < -x $ + e_+(x, 1) & when $ 0 \leq x < 2^{28} $ \\ + \left((2^8 - 2^4) \cdot {2^8}^4 + x, 2^{40}\right) & when $ 2^{28} \leq x < 2^{32} $ \\ + \left((2^8 - 2^4 + 2^2) \cdot {2^8}^8 + x, 2^{72}\right) & when $ 2^{32} \leq x $ \\ + (2^8 - 2^2 - x, 8) & when $ -4 < x < 0 $ \\ + (2^8 - 2^3, 8) \append e(-x) & when $ x \leq -4 $ \\ \end{dcases*} \\ % - e_+(x, b, r) &= \begin{dcases*} - p_+(b) + x & when $ r < 2^(8-b) $ \\ - e_+(x, b_l + 1, \lfloor r / 2^8 \rfloor) & when $ r \geq 2^(8-b) $ + e_+(x, b) &= \begin{dcases*} + (p(b) + x, 8) & when $ r < 2^(8-b) $ \\ + e_+\left(\left\lfloor \frac{x}{2^8} \right\rfloor, b + 1\right) \append (x \bmod 2^8, 8) & when $ r \geq 2^(8-b) $ \end{dcases*} \\ % - p_+(b) &= 2^{8b} \frac{(2^{b-1} - 1)}{2^{b-1}} \\ + p(b) &= 2^8 - 2^{9-b} +\end{align*} + +Essentially the first byte contains the number range information. For positive numbers this tells how many bytes the number takes. For numbers in the range $[0, 2^{28})$ the header bits keep growing steadily, starting from 1 and growing by one each time the number that should be encoded does not fit in the free space, as shown by $e_+(x, b)$. Numbers in the range $[2^{28}, 2^{32})$ have header byte \texttt{1111 0000} followed by the full 4 bytes long binary presentation of the number. Numbers greater than $2^{32}$ have header byte \texttt{1111 0100} followed by the full 8 bytes long binary presentation of the number. Negative numbers in the range $(-4, 0)$ are encoded by performing a bitwise or operation between \texttt{1111 1100} and the positive presentation of the number. Negative numbers less or equal to $-4$ have a header byte \texttt{1111 1000} followed by the varint encoding of their opposite number. + +Decoding is performed by analyzing the first byte after which the rest of the number can be read from the byte stream. + +\begin{align*} + s_0(x) &= 8 - \left\lfloor log_2(2^8-1 - x) \right\rfloor \\ +% + f_x &: \mathbb{N}_{\geq0} \rightarrow [0, 2^8) \\ + d &: f \rightarrow \mathbb{N}, f = \{ f_1, f_2, f_3, ... \} \\ + d(f) &= \begin{dcases*} + d_+\Big(f, s_0\big(f(0)\big)\Big) & when $f(0) \leq 2^8 - 2^4 $ \\ + \sum_{i=0}^4 2^{32-8i}f(i) & when $f(0) = 2^8 - 2^4 $ \\ + \sum_{i=0}^8 2^{64-8i}f(i) & when $f(0) = 2^8 - 2^4 + 2^2 $ \\ + -d(g : g(n) = f(n+1)) & when $f(0) = 2^8 - 2^3 $ \\ + (2^8 - 2^2) - f(0) & when $f(0) \geq 2^8 - 2^2 $ \\ + \end{dcases*} \\ +% + d_+(f, z) &= -2^{8z - 7z} + \sum_{i=1}^z 2^{8z-8i}f(i-1) \end{align*} -%\end{displaymath} \subsection{TCP tunnel} \label{sect:udptunnel}