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   1 %%% -*-latex-*-
   2 %%%
   3 %%% $Id: rand.tex,v 1.4 2004/04/08 01:36:15 mdw Exp $
   4 %%%
   5 %%% Description of Catacomb's random number generator
   6 %%%
   7 %%% (c) 1999 Straylight/Edgeware
   8 %%%
   9
  10 %%%----- Licensing notice ---------------------------------------------------
  11 %%%
  12 %%% This file is part of Catacomb.
  13 %%%
  14 %%% Catacomb is free software; you can redistribute it and/or modify
  15 %%% it under the terms of the GNU Library General Public License as
  16 %%% published by the Free Software Foundation; either version 2 of the
  17 %%% License, or (at your option) any later version.
  18 %%%
  19 %%% Catacomb is distributed in the hope that it will be useful,
  20 %%% but WITHOUT ANY WARRANTY; without even the implied warranty of
  21 %%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  22 %%% GNU Library General Public License for more details.
  23 %%%
  24 %%% You should have received a copy of the GNU Library General Public
  25 %%% License along with Catacomb; if not, write to the Free
  26 %%% Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
  27 %%% MA 02111-1307, USA.
  28
  29 %%%----- Header -------------------------------------------------------------
  30
  31 \documentclass[a4paper, article, 10pt, notitlepage, numbering]{strayman}
  32 \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts}
  33 \usepackage{mdwtab, mathenv}
  34 \usepackage[T1]{fontenc}
  35 \usepackage{cmtt, url}
  36 \usepackage[tpic, all]{xy}
  37 \usepackage{mathbbol}
  38 % \usepackage{crypto}
  39
  40 \def\mdw{{\normalfont[{\bfseries\itshape mdw}]}}
  41 \urlstyle{tt}
  42 \def\email{\begingroup\urlstyle{rm}\Url}
  43 \urldef\myemail\email{mdw@nsict.org}
  44 \def\Z{\mathbb{Z}}
  45 \let\assign\leftarrow
  46 \let\xor\oplus
  47 \let\bigxor\bigoplus
  48 \def\cat{\mathbin{\|}}
  49
  50 \title{The Catacomb random number generator}
  51 \author{Mark Wooding, \myemail}
  52
  53 %%%----- The main document --------------------------------------------------
  54
  55 \begin{document}
  56
  57 \maketitle
  58
  59 \begin{abstract}
  60   The author describes the random number generator used in the
  61   Straylight/\-Edgeware `Catacomb' library.  While the generator is
  62   superficially similar to (for example) the Linux and OpenBSD random number
  63   generators, it introduces a number of its own innovations which improve
  64   both security and performance.
  65
  66   The Catacomb generator uses an optional secret key, which can provide
  67   additional security against forward state compromise extension.  It uses a
  68   catastrophic reseeding operation to prevent a compromise yielding
  69   information about past generator states.  This operation works on
  70   arbitrary-sized blocks of data, so the generator's output buffer can be
  71   large.  This minimizes the effect of the reseeding overhead.
  72 \end{abstract}
  73
  74 \tableofcontents
  75
  76
  77 \section{The one-way transformation}
  78 \label{sec:oneway}
  79
  80 The most novel part of the generator\footnote{I believe this construction to
  81 be novel.  If I'm wrong, let me know.} is the one-way transformation which is
  82 used to allow pooled input data to affect the output buffer.
  83
  84 Let $H$ be some collision-resistant hash function, and let $E_k$ be a
  85 symmetric cipher with key $k$.  Then I can define the one-way transformation
  86 $T$ by
  87 \[ T(x) = E_{H(x)}(x) \]
  88
  89 I believe, although formal proof seems difficult, that an adversary in
  90 posession of $T(x)$ and a portion of the original $x$ cannot reconstruct the
  91 remainder of $x$ without breaking one of the cryptographic primitives (which
  92 I assume is `difficult') or performing an exhaustive search of one of: the
  93 space of the unknown portion of $x$, the range of the hash function $H$, or
  94 the keyspace of the cipher $E$.
  95
  96 A similar feat of cryptanalysis or exhaustive search seems necessary to work
  97 in a forwards direction: given partial knowledge of both $x$ and $T(x)$, the
  98 adversary cannot work out the remainder of either without trying every
  99 possibility for one or the other unknown portions, or working through the
 100 hash- or keyspace.
 101
 102 A keyed version of $T$ may be defined, given a keyed hash (or MAC) $H_k$:
 103 \[ T_k(x) = E_{H_k(x)}(x) \]
 104 If this is done, the adversary cannot work forwards even with \emph{complete}
 105 knowledge of $B$, or performing one of the obvious exhaustive searches.
 106
 107
 108 \section{Abstract description of the generator}
 109
 110 The generator is divided into two parts: an \emph{input pool} which
 111 accumulates random input data from the environment, and an \emph{output
 112 buffer} which contains data to be passed to clients of the generator on
 113 request.
 114
 115 \subsection{The input pool and mixing function}
 116
 117 New information is contributed to the generator by mixing it with the input
 118 pool, using a mixing function derived from the Linux random number generator
 119 \cite{linux:devrandom}.  The mixing function views the input pool as eight
 120 parallel shift registers.  Input data is added one octet at a time.  Each bit
 121 of an input octet is mixed with a different shift register.
 122
 123 Formally, let $I$ be the input pool, with size $N_I$ bytes; let $P(x) = a_0 +
 124 a_1 x + a_2 x^2 + \cdots + a_{N_I} x^{N_I}$ be a primitive polynomial in
 125 $\mathrm{GF}(2^{N_I})$ with degree $N_I$; let $i$ be an integer such that $0
 126 \le i < N_I$, and $r$ be an integer such that $0 \le r < 8$; and let $x$ be
 127 an input byte.  The result of mixing $x$ with the pool $I$ is calculated as
 128 follows:
 129 \begin{eqlines*}
 130   \begin{spliteqn*}
 131     I'[8j + b] =
 132     \begin{cases}
 133       x\bigl[(r + b) \bmod 8\bigr] \xor
 134         \bigxor_{0 \le k < N_I}
 135           a_k I\bigl[8\bigl((j + k) \bmod N_I\bigr) + b\bigr] & if $i = j$ \\
 136       I[j + b] & otherwise
 137     \end{cases} \\
 138     \textrm{for all integers $j$ and $b$ where $0 \le j < N_I$ and
 139       $0 \le b < 8$}
 140   \end{spliteqn*}
 141   \\
 142   I \assign I' \qquad
 143   i \assign (i + 1) \bmod N_I \qquad
 144   r \assign (r + 5) \bmod 8
 145 \end{eqlines*}
 146 Initially, $i$ and $r$ are both zero.  The use of 8-bit bytes above is
 147 arbitrary but convenient for modern computers.
 148
 149 The mixing function isn't intended to be cryptographically strong.  Its
 150 purpose is just to hold data without letting too much of the randomness get
 151 away.
 152
 153 \subsection{The output buffer}
 154
 155 Newly added data doesn't affect the output buffer until a `gating' operation
 156 is performed.  This uses the one-way transformation described earlier over
 157 the entire generator state.
 158
 159 Data requested by clients of the generator is read from the output buffer
 160 $O$.  Initially the buffer contains zeroes.  The output buffer is large
 161 enough for $N_O$ bits.
 162
 163 When the generator estimates that there's enough entropy in the input pool, a
 164 \emph{gating} operation is performed, using the one-way function described in
 165 section~\ref{sec:oneway}.  Hash both the input pool and output buffer, and
 166 then encrypt both using the hash as the key:
 167 \begin{eqlines*}
 168   h = H(I \cat O) \\
 169   I \assign E_h(I) \qquad O \assign E_h(O)
 170 \end{eqlines*}
 171 If the output buffer is exhausted before the next gating operation, it is
 172 \emph{stretched} using the one-way function: $O \assign E_{H(O)}(O)$.
 173
 174 \subsection{Other tweaks}
 175
 176 The first $N_S$ bits of this buffer take part in the output transformation
 177 but are never actually output.  They're there to make predicting further
 178 output from the generator difficult.
 179
 180 Also, optionally, the one-way functions can be keyed.  This does, of course,
 181 beg the question as to where the key comes from.  This might be one of those
 182 things best done the old-fashioned way with a bunch of coins or dice or
 183 something.
 184
 185
 186 \section{The actual implementation}
 187
 188 The Catacomb implementation of the generator uses the following parameters:
 189 \begin{itemize}
 190 \item The hash function used in the one-way transformation is RIPEMD-160
 191   \cite{rmd160}; the block cipher is Blowfish, using a 160-bit key.
 192 \item The input pool size $N_I$ is 128 bytes.  The output buffer size $N_O$
 193   is 512 bytes.  The size $N_S$ of the secret part of the output buffer
 194   is 160 bits (20 bytes).
 195 \item The polynomial $P(x)$ used for mixing in new input is
 196   $1 + x + x^2 + x^7 + x^{128}$.
 197 \end{itemize}
 198 The hash and block cipher are well-known and respected cryptographic
 199 primitives.
 200
 201 The input pool is rater larger than it strictly needs to be to contain
 202 `enough' entropy to bring the generator up to the strength of its
 203 cryptographic primitives.  The pool is large to reduce the effect of
 204 asymptotic behaviour in the amount of entropy in the pool.
 205
 206 The output buffer is large simply to improve performance: Blowfish has a
 207 heavy key schedule, so it pays to perform fewer rekeyings per byte of data.
 208 The precise size of 512 bytes was chosen empirically as being about where the
 209 performance improvement stops being linear with the buffer size on my
 210 machine.
 211
 212 \begin{thebibliography}{99}
 213
 214 \bibitem{cp:rand}
 215   J.~Kelsey, B.~Schneier, D.~Wagner, and C.~Hall, ``Cryptographic Attacks on
 216   Pseudorandom Number Generators'', \emph{Fast Software Encryption, Fifth
 217   International Workshop Proceedings (March 1998)}, Springer-Verlag, 1998,
 218   pp. 168--188, \url{http://www.counterpane.com/pseudorandom_number.html}
 219
 220 \bibitem{linux:devrandom}
 221   T.~Ts'o, ``A string random number generator'', Linux sources,
 222   \path{drivers/char/random.c}.
 223
 224 \bibitem{mdw:devrandom}
 225   M.~Wooding, ``Linux \path{/dev/random} generator security'', Usenet article
 226   posted to \mtt{sci.crypt}, July 1998.
 227
 228 \end{thebibliography}
 229
 230 %%%----- That's all, folks --------------------------------------------------
 231
 232 \end{document}