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1 | %%% -*-latex-*- |
2 | %%% |
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3 | %%% $Id: rand.tex,v 1.4 2004/04/08 01:36:15 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 | |
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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 |
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48 | \def\cat{\mathbin{\|}} |
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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} |
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78 | \label{sec:oneway} |
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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 | |
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108 | \section{Abstract description of the generator} |
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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 | |
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115 | \subsection{The input pool and mixing function} |
116 | |
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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 | |
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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 |
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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 |
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134 | \bigxor_{0 \le k < N_I} |
135 | a_k I\bigl[8\bigl((j + k) \bmod N_I\bigr) + b\bigr] & if $i = j$ \\ |
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136 | I[j + b] & otherwise |
137 | \end{cases} \\ |
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138 | \textrm{for all integers $j$ and $b$ where $0 \le j < N_I$ and |
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139 | $0 \le b < 8$} |
140 | \end{spliteqn*} |
141 | \\ |
142 | I \assign I' \qquad |
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143 | i \assign (i + 1) \bmod N_I \qquad |
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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 |
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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} |
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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 |
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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 $N_O$ |
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193 | is 512 bytes. The size $N_S$ of the secret part of the output buffer |
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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 |
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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. |
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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} |