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   1 %%% -*-latex-*-
   2 %%%
   3 %%% $Id: rand.tex,v 1.3 1999/10/15 21:05:56 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 %%%----- Revision history ---------------------------------------------------
  30 %%%
  31 %%% $Log: rand.tex,v $
  32 %%% Revision 1.3  1999/10/15 21:05:56  mdw
  33 %%% Add a little more explanatory text for the pool and buffer sizes.
  34 %%%
  35 %%% Revision 1.2  1999/10/12 21:00:34  mdw
  36 %%% Updated.  Almost finished, in fact. ;-)
  37 %%%
  38 %%% Revision 1.1  1999/09/03 08:41:13  mdw
  39 %%% Initial import.
  40 %%%
  41
  42 %%%----- Header -------------------------------------------------------------
  43
  44 \documentclass[a4paper, article, 10pt, notitlepage, numbering]{strayman}
  45 \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts}
  46 \usepackage{mdwtab, mathenv}
  47 \usepackage[T1]{fontenc}
  48 \usepackage{cmtt, url}
  49 \usepackage[tpic, all]{xy}
  50 \usepackage{mathbbol}
  51 % \usepackage{crypto}
  52
  53 \def\mdw{{\normalfont[{\bfseries\itshape mdw}]}}
  54 \urlstyle{tt}
  55 \def\email{\begingroup\urlstyle{rm}\Url}
  56 \urldef\myemail\email{mdw@nsict.org}
  57 \def\Z{\mathbb{Z}}
  58 \let\assign\leftarrow
  59 \let\xor\oplus
  60 \let\bigxor\bigoplus
  61 \def\cat{\mathbin{\|}}
  62
  63 \title{The Catacomb random number generator}
  64 \author{Mark Wooding, \myemail}
  65
  66 %%%----- The main document --------------------------------------------------
  67
  68 \begin{document}
  69
  70 \maketitle
  71
  72 \begin{abstract}
  73   The author describes the random number generator used in the
  74   Straylight/\-Edgeware `Catacomb' library.  While the generator is
  75   superficially similar to (for example) the Linux and OpenBSD random number
  76   generators, it introduces a number of its own innovations which improve
  77   both security and performance.
  78
  79   The Catacomb generator uses an optional secret key, which can provide
  80   additional security against forward state compromise extension.  It uses a
  81   catastrophic reseeding operation to prevent a compromise yielding
  82   information about past generator states.  This operation works on
  83   arbitrary-sized blocks of data, so the generator's output buffer can be
  84   large.  This minimizes the effect of the reseeding overhead.
  85 \end{abstract}
  86
  87 \tableofcontents
  88
  89
  90 \section{The one-way transformation}
  91 \label{sec:oneway}
  92
  93 The most novel part of the generator\footnote{I believe this construction to
  94 be novel.  If I'm wrong, let me know.} is the one-way transformation which is
  95 used to allow pooled input data to affect the output buffer.
  96
  97 Let $H$ be some collision-resistant hash function, and let $E_k$ be a
  98 symmetric cipher with key $k$.  Then I can define the one-way transformation
  99 $T$ by
 100 \[ T(x) = E_{H(x)}(x) \]
 101
 102 I believe, although formal proof seems difficult, that an adversary in
 103 posession of $T(x)$ and a portion of the original $x$ cannot reconstruct the
 104 remainder of $x$ without breaking one of the cryptographic primitives (which
 105 I assume is `difficult') or performing an exhaustive search of one of: the
 106 space of the unknown portion of $x$, the range of the hash function $H$, or
 107 the keyspace of the cipher $E$.
 108
 109 A similar feat of cryptanalysis or exhaustive search seems necessary to work
 110 in a forwards direction: given partial knowledge of both $x$ and $T(x)$, the
 111 adversary cannot work out the remainder of either without trying every
 112 possibility for one or the other unknown portions, or working through the
 113 hash- or keyspace.
 114
 115 A keyed version of $T$ may be defined, given a keyed hash (or MAC) $H_k$:
 116 \[ T_k(x) = E_{H_k(x)}(x) \]
 117 If this is done, the adversary cannot work forwards even with \emph{complete}
 118 knowledge of $B$, or performing one of the obvious exhaustive searches.
 119
 120
 121 \section{Abstract description of the generator}
 122
 123 The generator is divided into two parts: an \emph{input pool} which
 124 accumulates random input data from the environment, and an \emph{output
 125 buffer} which contains data to be passed to clients of the generator on
 126 request.
 127
 128 \subsection{The input pool and mixing function}
 129
 130 New information is contributed to the generator by mixing it with the input
 131 pool, using a mixing function derived from the Linux random number generator
 132 \cite{linux:devrandom}.  The mixing function views the input pool as eight
 133 parallel shift registers.  Input data is added one octet at a time.  Each bit
 134 of an input octet is mixed with a different shift register.
 135
 136 Formally, let $I$ be the input pool, with size $N_I$ bytes; let $P(x) = a_0 +
 137 a_1 x + a_2 x^2 + \cdots + a_{N_I} x^{N_I}$ be a primitive polynomial in
 138 $\mathrm{GF}(2^{N_I})$ with degree $N_I$; let $i$ be an integer such that $0
 139 \le i < N_I$, and $r$ be an integer such that $0 \le r < 8$; and let $x$ be
 140 an input byte.  The result of mixing $x$ with the pool $I$ is calculated as
 141 follows:
 142 \begin{eqlines*}
 143   \begin{spliteqn*}
 144     I'[8j + b] =
 145     \begin{cases}
 146       x\bigl[(r + b) \bmod 8\bigr] \xor
 147         \bigxor_{0 \le k < N_I}
 148           a_k I\bigl[8\bigl((j + k) \bmod N_I\bigr) + b\bigr] & if $i = j$ \\
 149       I[j + b] & otherwise
 150     \end{cases} \\
 151     \textrm{for all integers $j$ and $b$ where $0 \le j < N_I$ and
 152       $0 \le b < 8$}
 153   \end{spliteqn*}
 154   \\
 155   I \assign I' \qquad
 156   i \assign (i + 1) \bmod N_I \qquad
 157   r \assign (r + 5) \bmod 8
 158 \end{eqlines*}
 159 Initially, $i$ and $r$ are both zero.  The use of 8-bit bytes above is
 160 arbitrary but convenient for modern computers.
 161
 162 The mixing function isn't intended to be cryptographically strong.  Its
 163 purpose is just to hold data without letting too much of the randomness get
 164 away.
 165
 166 \subsection{The output buffer}
 167
 168 Newly added data doesn't affect the output buffer until a `gating' operation
 169 is performed.  This uses the one-way transformation described earlier over
 170 the entire generator state.
 171
 172 Data requested by clients of the generator is read from the output buffer
 173 $O$.  Initially the buffer contains zeroes.  The output buffer is large
 174 enough for $N_O$ bits.
 175
 176 When the generator estimates that there's enough entropy in the input pool, a
 177 \emph{gating} operation is performed, using the one-way function described in
 178 section~\ref{sec:oneway}.  Hash both the input pool and output buffer, and
 179 then encrypt both using the hash as the key:
 180 \begin{eqlines*}
 181   h = H(I \cat O) \\
 182   I \assign E_h(I) \qquad O \assign E_h(O)
 183 \end{eqlines*}
 184 If the output buffer is exhausted before the next gating operation, it is
 185 \emph{stretched} using the one-way function: $O \assign E_{H(O)}(O)$.
 186
 187 \subsection{Other tweaks}
 188
 189 The first $N_S$ bits of this buffer take part in the output transformation
 190 but are never actually output.  They're there to make predicting further
 191 output from the generator difficult.
 192
 193 Also, optionally, the one-way functions can be keyed.  This does, of course,
 194 beg the question as to where the key comes from.  This might be one of those
 195 things best done the old-fashioned way with a bunch of coins or dice or
 196 something.
 197
 198
 199 \section{The actual implementation}
 200
 201 The Catacomb implementation of the generator uses the following parameters:
 202 \begin{itemize}
 203 \item The hash function used in the one-way transformation is RIPEMD-160
 204   \cite{rmd160}; the block cipher is Blowfish, using a 160-bit key.
 205 \item The input pool size $N_I$ is 128 bytes.  The output buffer size $N_O$
 206   is 512 bytes.  The size $N_S$ of the secret part of the output buffer
 207   is 160 bits (20 bytes).
 208 \item The polynomial $P(x)$ used for mixing in new input is
 209   $1 + x + x^2 + x^7 + x^{128}$.
 210 \end{itemize}
 211 The hash and block cipher are well-known and respected cryptographic
 212 primitives.
 213
 214 The input pool is rater larger than it strictly needs to be to contain
 215 `enough' entropy to bring the generator up to the strength of its
 216 cryptographic primitives.  The pool is large to reduce the effect of
 217 asymptotic behaviour in the amount of entropy in the pool.
 218
 219 The output buffer is large simply to improve performance: Blowfish has a
 220 heavy key schedule, so it pays to perform fewer rekeyings per byte of data.
 221 The precise size of 512 bytes was chosen empirically as being about where the
 222 performance improvement stops being linear with the buffer size on my
 223 machine.
 224
 225 \begin{thebibliography}{99}
 226
 227 \bibitem{cp:rand}
 228   J.~Kelsey, B.~Schneier, D.~Wagner, and C.~Hall, ``Cryptographic Attacks on
 229   Pseudorandom Number Generators'', \emph{Fast Software Encryption, Fifth
 230   International Workshop Proceedings (March 1998)}, Springer-Verlag, 1998,
 231   pp. 168--188, \url{http://www.counterpane.com/pseudorandom_number.html}
 232
 233 \bibitem{linux:devrandom}
 234   T.~Ts'o, ``A string random number generator'', Linux sources,
 235   \path{drivers/char/random.c}.
 236
 237 \bibitem{mdw:devrandom}
 238   M.~Wooding, ``Linux \path{/dev/random} generator security'', Usenet article
 239   posted to \mtt{sci.crypt}, July 1998.
 240
 241 \end{thebibliography}
 242
 243 %%%----- That's all, folks --------------------------------------------------
 244
 245 \end{document}