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1 | %%% -*-latex-*- |
2 | %%% |
3 | %%% $Id: rand.tex,v 1.1 1999/09/03 08:41:13 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 1.1 1999/09/03 08:41:13 mdw |
33 | %%% Initial import. |
34 | %%% |
35 | |
36 | %%%----- Header ------------------------------------------------------------- |
37 | |
38 | \documentclass[a4paper, article, 10pt, notitlepage, numbering]{strayman} |
39 | \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts} |
40 | \usepackage{mdwtab, mathenv} |
41 | \usepackage[T1]{fontenc} |
42 | \usepackage{cmtt, url} |
43 | \usepackage[tpic, all]{xy} |
44 | \usepackage{mathbbol} |
45 | % \usepackage{crypto} |
46 | |
47 | \def\mdw{{\normalfont[{\bfseries\itshape mdw}]}} |
48 | \urlstyle{tt} |
49 | \def\email{\begingroup\urlstyle{rm}\Url} |
50 | \urldef\myemail\email{mdw@nsict.org} |
51 | \def\Z{\mathbb{Z}} |
52 | \let\assign\leftarrow |
53 | \let\xor\oplus |
54 | \let\bigxor\bigoplus |
55 | |
56 | \title{The Catacomb random number generator} |
57 | \author{Mark Wooding, \myemail} |
58 | |
59 | %%%----- The main document -------------------------------------------------- |
60 | |
61 | \begin{document} |
62 | |
63 | \maketitle |
64 | |
65 | \begin{abstract} |
66 | The author describes the random number generator used in the |
67 | Straylight/\-Edgeware `Catacomb' library. While the generator is |
68 | superficially similar to (for example) the Linux and OpenBSD random number |
69 | generators, it introduces a number of its own innovations which improve |
70 | both security and performance. |
71 | |
72 | The Catacomb generator uses an optional secret key, which can provide |
73 | additional security against forward state compromise extension. It uses a |
74 | catastrophic reseeding operation to prevent a compromise yielding |
75 | information about past generator states. This operation works on |
76 | arbitrary-sized blocks of data, so the generator's output buffer can be |
77 | large. This minimizes the effect of the reseeding overhead. |
78 | \end{abstract} |
79 | |
80 | \tableofcontents |
81 | |
82 | |
83 | \section{The one-way transformation} |
84 | |
85 | The most novel part of the generator\footnote{I believe this construction to |
86 | be novel. If I'm wrong, let me know.} is the one-way transformation which is |
87 | used to allow pooled input data to affect the output buffer. |
88 | |
89 | Let $H$ be some collision-resistant hash function, and let $E_k$ be a |
90 | symmetric cipher with key $k$. Then I can define the one-way transformation |
91 | $T$ by |
92 | \[ T(x) = E_{H(x)}(x) \] |
93 | |
94 | I believe, although formal proof seems difficult, that an adversary in |
95 | posession of $T(x)$ and a portion of the original $x$ cannot reconstruct the |
96 | remainder of $x$ without breaking one of the cryptographic primitives (which |
97 | I assume is `difficult') or performing an exhaustive search of one of: the |
98 | space of the unknown portion of $x$, the range of the hash function $H$, or |
99 | the keyspace of the cipher $E$. |
100 | |
101 | A similar feat of cryptanalysis or exhaustive search seems necessary to work |
102 | in a forwards direction: given partial knowledge of both $x$ and $T(x)$, the |
103 | adversary cannot work out the remainder of either without trying every |
104 | possibility for one or the other unknown portions, or working through the |
105 | hash- or keyspace. |
106 | |
107 | A keyed version of $T$ may be defined, given a keyed hash (or MAC) $H_k$: |
108 | \[ T_k(x) = E_{H_k(x)}(x) \] |
109 | If this is done, the adversary cannot work forwards even with \emph{complete} |
110 | knowledge of $B$, or performing one of the obvious exhaustive searches. |
111 | |
112 | |
113 | \section{Description of the generator} |
114 | |
115 | The generator is divided into two parts: an \emph{input pool} which |
116 | accumulates random input data from the environment, and an \emph{output |
117 | buffer} which contains data to be passed to clients of the generator on |
118 | request. |
119 | |
120 | New information is contributed to the generator by mixing it with the input |
121 | pool, using a mixing function derived from the Linux random number generator |
122 | \cite{linux:devrandom}. The mixing function views the input pool as eight |
123 | parallel shift registers. Input data is added one octet at a time. Each bit |
124 | of an input octet is mixed with a different shift register. |
125 | |
126 | Formally, let $I$ be the input pool, with size $n_I$ bytes; let $P(x) = a_0 + |
127 | a_1 x + a_2 x^2 + \cdots + a_{n_I} x^{n_I}$ be a primitive polynomial in |
128 | $\mathrm{GF}(2^{n_I})$ with degree $n_I$; let $i$ be an integer such that $0 |
129 | \le i < n_I$, and $r$ be an integer such that $0 \le r < 8$; and let $x$ be |
130 | an input byte. The result of mixing $x$ with the pool $I$ is calculated as |
131 | follows: |
132 | \begin{eqlines*} |
133 | \begin{spliteqn*} |
134 | I'[8j + b] = |
135 | \begin{cases} |
136 | x\bigl[(r + b) \bmod 8\bigr] \xor |
137 | \bigxor_{0 \le k < n_I} |
138 | a_k I\bigl[8\bigl((j + k) \bmod n_I\bigr) + b\bigr] & if $i = j$ \\ |
139 | I[j + b] & otherwise |
140 | \end{cases} \\ |
141 | \textrm{for all integers $j$ and $b$ where $0 \le j < n_I$ and |
142 | $0 \le b < 8$} |
143 | \end{spliteqn*} |
144 | \\ |
145 | I \assign I' \qquad |
146 | i \assign (i + 1) \bmod n_I \qquad |
147 | r \assign (r + 5) \bmod 8 |
148 | \end{eqlines*} |
149 | Initially, $i$ and $r$ are both zero. The use of 8-bit bytes above is |
150 | arbitrary. |
151 | |
152 | Newly added data doesn't affect the output buffer until a `gating' operation |
153 | is performed. This uses the one-way transformation described earlier over |
154 | the entire generator state. |
155 | |
156 | Data requested by clients of the generator is read from the output buffer |
157 | $O$. Initially the buffer contains zeroes. |
158 | |
159 | \begin{thebibliography}{99} |
160 | |
161 | \bibitem{cp:rand} |
162 | J.~Kelsey, B.~Schneier, D.~Wagner, and C.~Hall, ``Cryptographic Attacks on |
163 | Pseudorandom Number Generators'', \emph{Fast Software Encryption, Fifth |
164 | International Workshop Proceedings (March 1998)}, Springer-Verlag, 1998, |
165 | pp. 168--188, \url{http://www.counterpane.com/pseudorandom_number.html} |
166 | |
167 | \bibitem{linux:devrandom} |
168 | T.~Ts'o, ``A string random number generator'', Linux sources, |
169 | \path{drivers/char/random.c}. |
170 | |
171 | \bibitem{mdw:devrandom} |
172 | M.~Wooding, ``Linux \path{/dev/random} generator security'', Usenet article |
173 | posted to \mtt{sci.crypt}, July 1998. |
174 | |
175 | \end{thebibliography} |
176 | |
177 | %%%----- That's all, folks -------------------------------------------------- |
178 | |
179 | \end{document} |