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1 | %%% -*-latex-*- |
2 | %%% |
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3 | %%% $Id: rand.tex,v 1.3 1999/10/15 21:05:56 mdw Exp $ |
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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 $ |
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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 | %%% |
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35 | %%% Revision 1.2 1999/10/12 21:00:34 mdw |
36 | %%% Updated. Almost finished, in fact. ;-) |
37 | %%% |
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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 |
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61 | \def\cat{\mathbin{\|}} |
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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} |
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91 | \label{sec:oneway} |
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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 | |
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121 | \section{Abstract description of the generator} |
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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 | |
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128 | \subsection{The input pool and mixing function} |
129 | |
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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 | |
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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 |
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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 |
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147 | \bigxor_{0 \le k < N_I} |
148 | a_k I\bigl[8\bigl((j + k) \bmod N_I\bigr) + b\bigr] & if $i = j$ \\ |
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149 | I[j + b] & otherwise |
150 | \end{cases} \\ |
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151 | \textrm{for all integers $j$ and $b$ where $0 \le j < N_I$ and |
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152 | $0 \le b < 8$} |
153 | \end{spliteqn*} |
154 | \\ |
155 | I \assign I' \qquad |
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156 | i \assign (i + 1) \bmod N_I \qquad |
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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 |
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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} |
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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 |
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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 $N_O$ |
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206 | is 512 bytes. The size $N_S$ of the secret part of the output buffer |
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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 |
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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. |
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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} |