Commit | Line | Data |
---|---|---|
e6e0e332 MW |
1 | %%% -*-latex-*- |
2 | %%% | |
31b692a0 | 3 | %%% $Id: storin.tex,v 1.2 2000/05/21 21:43:26 mdw Exp $ |
e6e0e332 MW |
4 | %%% |
5 | %%% Definition of the cipher | |
6 | %%% | |
7 | %%% (c) 2000 Mark Wooding | |
8 | %%% | |
9 | ||
10 | %%%----- Revision history --------------------------------------------------- | |
11 | %%% | |
12 | %%% $Log: storin.tex,v $ | |
31b692a0 MW |
13 | %%% Revision 1.2 2000/05/21 21:43:26 mdw |
14 | %%% Fix a couple of typos. | |
15 | %%% | |
e6e0e332 MW |
16 | %%% Revision 1.1 2000/05/21 11:28:30 mdw |
17 | %%% Initial check-in. | |
18 | %%% | |
19 | ||
20 | %%%----- Preamble ----------------------------------------------------------- | |
21 | ||
22 | \documentclass[a4paper]{article} | |
23 | \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts} | |
24 | \usepackage{mdwtab} | |
25 | \usepackage{mathenv} | |
26 | \usepackage{amsfonts} | |
27 | \usepackage{mdwmath} | |
28 | \usepackage[all, dvips]{xy} | |
29 | ||
30 | \def\ror{\mathbin{>\!\!>\!\!>}} | |
31 | \def\rol{\mathbin{<\!\!<\!\!<}} | |
32 | \def\lsr{\mathbin{>\!\!>}} | |
33 | \def\lsl{\mathbin{<\!\!<}} | |
34 | \def\xor{\oplus} | |
35 | \def\seq#1{{\langle #1 \rangle}} | |
36 | ||
37 | \sloppy | |
38 | ||
39 | \title{Storin: A block cipher for digital signal processors} | |
40 | \author{Mark Wooding (\texttt{mdw@nsict.org})} | |
41 | ||
42 | %% --- The cipher diagrams --- | |
43 | ||
44 | \def\figkeymix#1#2#3#4{% | |
45 | \ar "a"; p-(0, 0.5)*{\xor} ="a" \ar "a"+(1, 0) *+[r]{k_{#1}}; "a"% | |
46 | \ar "b"; p-(0, 0.5)*{\xor} ="b" \ar "b"+(1, 0) *+[r]{k_{#2}}; "b"% | |
47 | \ar "c"; p-(0, 0.5)*{\xor} ="c" \ar "c"+(1, 0) *+[r]{k_{#3}}; "c"% | |
48 | \ar "d"; p-(0, 0.5)*{\xor} ="d" \ar "d"+(1, 0) *+[r]{k_{#4}}; "d"% | |
49 | } | |
50 | ||
51 | \def\figmatrix{% | |
52 | \POS "a"+(3, -1) *++=(7, 0)[F]u\txt{Matrix multiply} ="m"% | |
53 | \ar "a"; "m"+U-(3, 0) \ar "b"; "m"+U-(1, 0)% | |
54 | \ar "c"; "m"+U+(1, 0) \ar "d"; "m"+U+(3, 0)% | |
55 | } | |
56 | ||
57 | \def\figlintrans{% | |
58 | \ar "m"+D-(3, 0); "a"-(0, 2.25)*{\xor} ="a"% | |
59 | \POS "a"+(1, 0) *+[F]{{} \lsr 12} ="x"% | |
60 | \ar `r "a"+(0, 0.5); p+(1, 0) "x" \ar "x"; "a"% | |
61 | \ar "m"+D-(1, 0); "b"-(0, 2.25)*{\xor} ="b"% | |
62 | \POS "b"+(1, 0) *+[F]{{} \lsr 12} ="x"% | |
63 | \ar `r "b"+(0, 0.5); p+(1, 0) "x" \ar "x"; "b"% | |
64 | \ar "m"+D+(1, 0); "c"-(0, 2.25)*{\xor} ="c"% | |
65 | \POS "c"+(1, 0) *+[F]{{} \lsr 12} ="x"% | |
66 | \ar `r "c"+(0, 0.5); p+(1, 0) "x" \ar "x"; "c"% | |
67 | \ar "m"+D+(3, 0); "d"-(0, 2.25)*{\xor} ="d"% | |
68 | \POS "d"+(1, 0) *+[F]{{} \lsr 12} ="x"% | |
69 | \ar `r "d"+(0, 0.5); p+(1, 0) "x" \ar "x"; "d"% | |
70 | } | |
71 | ||
72 | \def\figilintrans{% | |
73 | \ar "a"; "a"-(0, 1)*{\xor} ="a"% | |
74 | \POS "a"+(1, 0) *+[F]{{} \lsr 12} ="x"% | |
75 | \ar `r "a"+(0, 0.5); p+(1, 0) "x" \ar "x"; "a"% | |
76 | \ar "b"; "b"-(0, 1)*{\xor} ="b"% | |
77 | \POS "b"+(1, 0) *+[F]{{} \lsr 12} ="x"% | |
78 | \ar `r "b"+(0, 0.5); p+(1, 0) "x" \ar "x"; "b"% | |
79 | \ar "c"; "c"-(0, 1)*{\xor} ="c"% | |
80 | \POS "c"+(1, 0) *+[F]{{} \lsr 12} ="x"% | |
81 | \ar `r "c"+(0, 0.5); p+(1, 0) "x" \ar "x"; "c"% | |
82 | \ar "d"; "d"-(0, 1)*{\xor} ="d"% | |
83 | \POS "d"+(1, 0) *+[F]{{} \lsr 12} ="x"% | |
84 | \ar `r "d"+(0, 0.5); p+(1, 0) "x" \ar "x"; "d"% | |
85 | } | |
86 | ||
31b692a0 | 87 | \def\figstart#1{% |
e6e0e332 MW |
88 | \POS 0;<1cm,0cm>:% |
89 | \turnradius{4pt}% | |
31b692a0 MW |
90 | \ar @{-} (0, 0) *+{a#1}; p-(0, 0.5) ="a" |
91 | \ar @{-} (2, 0) *+{b#1}; p-(0, 0.5) ="b" | |
92 | \ar @{-} (4, 0) *+{c#1}; p-(0, 0.5) ="c" | |
93 | \ar @{-} (6, 0) *+{d#1}; p-(0, 0.5) ="d" | |
e6e0e332 MW |
94 | } |
95 | ||
96 | \def\figround#1#2#3#4#5{% | |
97 | \ar @{.} "a"-(0.5, 0); p+(8, 0)% | |
98 | \POS "a"+(8, -1.75) *[r]\txt{#5}% | |
99 | \figkeymix{#1}{#2}{#3}{#4}% | |
100 | \figmatrix% | |
101 | \figlintrans% | |
102 | \ar @{-} "a"; p-(0, .5) ="a" | |
103 | \ar @{-} "b"; p-(0, .5) ="b" | |
104 | \ar @{-} "c"; p-(0, .5) ="c" | |
105 | \ar @{-} "d"; p-(0, .5) ="d" | |
106 | } | |
107 | ||
108 | \def\figiround#1#2#3#4#5{% | |
109 | \ar @{.} "a"-(0.5, 0); p+(8, 0)% | |
110 | \POS "a"+(8, -1.75) *[r]\txt{#5}% | |
111 | \figkeymix{#1}{#2}{#3}{#4}% | |
112 | \figilintrans% | |
113 | \figmatrix% | |
114 | \ar @{-} "m"+D-(3, 0); p-(0, .5) ="a" | |
115 | \ar @{-} "m"+D-(1, 0); p-(0, .5) ="b" | |
116 | \ar @{-} "m"+D+(1, 0); p-(0, .5) ="c" | |
117 | \ar @{-} "m"+D+(3, 0); p-(0, .5) ="d" | |
118 | } | |
119 | ||
120 | \def\figgap{% | |
121 | \ar @{.} "a"-(0.5, 0); p+(8, 0) | |
122 | \POS "a"+(8, -1)*[r]\txt{Six more rounds} | |
123 | \ar @{--} "a"; "a"-(0, 2) ="a" | |
124 | \ar @{--} "b"; "b"-(0, 2) ="b" | |
125 | \ar @{--} "c"; "c"-(0, 2) ="c" | |
126 | \ar @{--} "d"; "d"-(0, 2) ="d" | |
127 | } | |
128 | ||
31b692a0 | 129 | \def\figwhite#1#2#3#4#5{% |
e6e0e332 MW |
130 | \ar @{.} "a"-(0.5, 0); p+(8, 0) |
131 | \POS "a"+(8, -1)*[r]\txt{Postwhitening} | |
132 | \figkeymix{#1}{#2}{#3}{#4} | |
31b692a0 MW |
133 | \ar "a"; p-(0, 1) *+{a#5} |
134 | \ar "b"; p-(0, 1) *+{b#5} | |
135 | \ar "c"; p-(0, 1) *+{c#5} | |
136 | \ar "d"; p-(0, 1) *+{d#5} | |
e6e0e332 MW |
137 | } |
138 | ||
139 | \begin{document} | |
140 | \maketitle | |
141 | ||
142 | %%%----- The main text ------------------------------------------------------ | |
143 | ||
144 | \begin{abstract} | |
145 | We present Storin: a new 96-bit block cipher designed to play to the | |
146 | strengths of current digital signal processors (DSPs). In particular, DSPs | |
147 | tend to provide single-cycle multiply-and-accumulate operations, making | |
148 | matrix multiplications very cheap. Working in an environment where | |
149 | multiplication is as fast as exclusive-or changes the usual perceptions | |
150 | about which operations provide good cryptographic strength cheaply. The | |
151 | scarcity of available memory, for code and for tables, and a penalty for | |
152 | nonsequential access to data also make traditional block ciphers based | |
153 | around substitution tables unsuitable. | |
154 | \end{abstract} | |
155 | ||
156 | \tableofcontents | |
157 | ||
158 | \section{Definition of the cipher} | |
159 | ||
160 | \subsection{Overview} | |
161 | ||
162 | Storin is an eight-round SP network operating on 96-bit blocks. The block | |
163 | cipher uses 36 24-bit subkey words, derived from a user key by the key | |
164 | schedule. | |
165 | ||
166 | The 96-bit input is split into four 24-bit words. Each round then processes | |
167 | these four words, using the following three steps: | |
168 | \begin{enumerate} | |
169 | \item Mixing in of some key material. Four 24-bit subkey words are XORed | |
170 | with the four data words. | |
171 | \item A matrix multiplication mod $2^{24}$. The four words are treated as a | |
172 | column vector and premultiplied by a $4 \times 4$ vector using addition and | |
173 | multiplication mod $2^{24}$. This is the main nonlinear step in the | |
174 | cipher, and it also provides most of the cipher's diffusion. | |
175 | \item A simple linear transformation, which replaces each word $x$ by $x \xor | |
176 | (x \lsr 12)$. | |
177 | \end{enumerate} | |
178 | The four data words output by the final round are XORed with the last four | |
179 | subkey words in a final postwhitening stage and combined to form the 96-bit | |
180 | ciphertext. | |
181 | ||
182 | The cipher structure is shown diagrammatically in figure~\ref{fig:cipher}. | |
183 | ||
184 | \begin{figure} | |
185 | \centering | |
186 | \leavevmode | |
187 | \begin{xy} | |
188 | \xycompile{ | |
31b692a0 | 189 | \figstart{} |
e6e0e332 MW |
190 | \figround{0}{1}{2}{3}{Round 1} |
191 | \figround{4}{5}{6}{7}{Round 2} | |
192 | \figgap | |
31b692a0 | 193 | \figwhite{32}{33}{34}{35}{'}} |
e6e0e332 MW |
194 | \end{xy} |
195 | \caption{The Storin encryption function} | |
196 | \label{fig:cipher} | |
197 | \end{figure} | |
198 | ||
199 | Since the matrix used in step 2 is chosen to be invertible, the cipher can be | |
200 | inverted readily, simply by performing the inverse steps in the reverse | |
201 | order. Since the postwhitening stage is the same as a key mixing stage, | |
202 | decryption can be viewed as eight rounds consisting of key mixing, linear | |
203 | transformation and matrix multiplication, followed by a postwhitening stage. | |
204 | Thus, the structure of the inverse cipher is very similar to the forwards | |
205 | cipher, and uses the same components. The decryption function is shown | |
206 | diagrammatically in figure~\ref{fig:decipher}. | |
207 | ||
208 | \begin{figure} | |
209 | \centering | |
210 | \leavevmode | |
211 | \begin{xy} | |
212 | \xycompile{ | |
31b692a0 | 213 | \figstart{'} |
e6e0e332 MW |
214 | \figiround{32}{33}{34}{35}{Round 1} |
215 | \figiround{28}{29}{30}{31}{Round 2} | |
216 | \figgap | |
31b692a0 | 217 | \figwhite{0}{1}{2}{3}{}} |
e6e0e332 MW |
218 | \end{xy} |
219 | \caption{The Storin decryption function} | |
220 | \label{fig:decipher} | |
221 | \end{figure} | |
222 | ||
223 | The key schedule is designed to be simple and to reuse the cipher components | |
224 | already available. Given a user key, which is a sequence of one or more | |
225 | 24-bit words, it produces the 36 subkey words required by the cipher. The | |
226 | key schedule is very similar to Blowfish \cite{blowfish}. The subkey array | |
227 | is assigned an initial constant value derived from the matrix used in the | |
228 | cipher. Words from the user key are XORed into the array, starting from the | |
229 | beginning, and restarting from the beginning of the user key when all the | |
230 | user key words are exhausted. A 96-bit block is initialized to zero, and | |
231 | enciphered with Storin, using the subkeys currently in the array. The first | |
232 | four subkey words are then replaced with the resulting ciphertext, which is | |
233 | then encrypted again using the new subkeys. The next four subkey words are | |
234 | replaced with the ciphertext, and the process continues, nine times in all, | |
235 | until all of the subkey words have been replaced. | |
236 | ||
237 | The Storin key schedule can accept user keys up to 36 words (864 bits) long. | |
238 | It is unrealistic, however, to expect this much strength from the cipher. We | |
239 | recommend against using keys longer than 5 words (120 bits). | |
240 | ||
241 | ||
242 | \subsection{Encryption} | |
243 | ||
244 | We define $\mathcal{W} = \mathbb{Z}_{2^{24}}$ to be set of 24-bit words, and | |
245 | $\mathcal{P} = \mathcal{W}^4$ to be the set of four-entry column vectors over | |
246 | $\mathcal{W}$. Storin plaintext blocks are members of $\mathcal{P}$. | |
247 | ||
248 | The Storin encryption function uses 36 24-bit words of key material $k_0$, | |
249 | $k_1$, \ldots, $k_{35}$, which are produced from the user key by the key | |
250 | schedule, described below. The key-mixing operation $K_i: \mathcal{P} | |
251 | \rightarrow \mathcal{P}$ is defined for $0 \le i < 9$ by: | |
252 | \[ | |
253 | K_i \begin{pmatrix} a \\ b \\ c \\d \end{pmatrix} | |
254 | = | |
255 | \begin{pmatrix} | |
256 | a \xor k_{4i} \\ b \xor k_{4i+1} \\ c \xor k_{4i+2} \\ d \xor k_{4i+3} | |
257 | \end{pmatrix} | |
258 | \] | |
259 | ||
260 | The matrix multiplication operation $M: \mathcal{P} \to \mathcal{P}$ | |
261 | is described by $M(\mathbf{x}) = \mathbf{M} \mathbf{x}$, where $\mathbf{M}$ | |
262 | is a fixed invertible $4 \times 4$ matrix over $\mathcal{W}$. The value of | |
263 | $\mathbf{M}$ is defined below. | |
264 | ||
265 | The linear transformation $L: \mathcal{P} \to \mathcal{P}$ is defined by: | |
266 | \[ | |
267 | L \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} | |
268 | = | |
269 | \begin{pmatrix} | |
270 | a \xor (a \lsr 12) \\ | |
271 | b \xor (b \lsr 12) \\ | |
272 | c \xor (c \lsr 12) \\ | |
273 | d \xor (d \lsr 12) | |
274 | \end{pmatrix} | |
275 | \] | |
276 | ||
277 | The round function $R_i: \mathcal{P} \to \mathcal{P}$ is defined for $0 \le i | |
278 | < 8$ by | |
279 | \[ R_i(\mathbf{x}) = L\bigl(\mathbf{M} K_i(\mathbf{x}) \bigr) \] | |
280 | ||
281 | The cipher $C: \mathcal{P} \to \mathcal{P}$ is defined in terms of $R_i$ and | |
282 | $K_i$. Let $\mathbf{x}_0 \in \mathcal{P}$ be a plaintext vector. Let | |
283 | $\mathbf{x}_{i+1} = R_i(\mathbf{x}_i)$ for $0 \le i < 8$. Then we define | |
284 | $C(\mathbf{x})$ by setting $C(\mathbf{x}_0) = K_8(\mathbf{x}_8)$. | |
285 | ||
286 | ||
287 | \subsection{Key schedule} | |
288 | ||
289 | The key schedule converts a user key, which is a sequence of 24-bit words, | |
290 | into the 36 subkeys required by the cipher. | |
291 | ||
292 | For $i \ge 0$, we define that | |
293 | \[ | |
294 | \begin{pmatrix} | |
295 | m_{16i + 0} & m_{16i + 1} & m_{16i + 2} & m_{16i + 3} \\ | |
296 | m_{16i + 4} & m_{16i + 5} & m_{16i + 6} & m_{16i + 7} \\ | |
297 | m_{16i + 8} & m_{16i + 9} & m_{16i + 10} & m_{16i + 11} \\ | |
298 | m_{16i + 12} & m_{16i + 13} & m_{16i + 14} & m_{16i + 15} | |
299 | \end{pmatrix} | |
300 | = \mathbf{M}^{i + 2} | |
301 | \] | |
302 | ||
303 | Let the user-supplied key be $u_0$, $u_1$, \ldots, $u_{n-1}$, for some $n > | |
304 | 0$. We define the sequence $z_0$, $z_1$, \ldots\ by | |
305 | \[ z_i = m_i \xor u_{i \bmod n} \] | |
306 | for $i \ge 0$. | |
307 | ||
308 | Denote the result of encrypting vector $\mathbf{x}$ using subkeys from the | |
309 | sequence $\seq{w} = w_0, w_1, \ldots, w_{35}$ as $C_{\seq{w}}(\mathbf{x})$. | |
310 | We define the key schedule to be $k_0$, $k_1$, \ldots, $k_{35}$, where: | |
311 | \begin{eqlines*} | |
312 | \seq{p^{(i)}} = k_0, k_1, \ldots, k_{4i-1}, z_{4i}, z_{4i+1}, \ldots \\ | |
313 | \mathbf{x}_0 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}; \qquad | |
314 | \begin{pmatrix} k_{4i} \\ k_{4i+1} \\ k_{4i+2} \\ k_{4i+3} \end{pmatrix} | |
315 | = \mathbf{x}_{i+1} = C_{\seq{p^{(i)}}}(\mathbf{x}_i) | |
316 | \end{eqlines*} | |
317 | ||
318 | ||
319 | \subsection{Decryption} | |
320 | ||
321 | The individual operations used during encryption are all invertible. Key | |
322 | mixing is inverted by taking keys from the other end of the array: | |
323 | \[ K^{-1}_i(\mathbf{x}) = K_{8-i}(\mathbf{x}) \] | |
324 | The matrix multiplication may be inverted simply by using the inverse matrix | |
325 | $\mathbf{M}^{-1}$: | |
326 | \[ M^{-1}(\mathbf{x}) = \mathbf{M}^{-1} \mathbf{x} \] | |
327 | Finally, the linear transformation is its own inverse: | |
328 | \[ L^{-1}(\mathbf{x}) = L(\mathbf{x}) \] | |
329 | The inverse round function can now be defined as: | |
330 | \[ R^{-1}_i(\mathbf{x}) = | |
331 | \mathbf{M}^{-1} L\bigl(K^{-1}_i(\mathbf{x})\bigr) \] | |
332 | ||
333 | The decryption function $C^{-1}: \mathcal{P} \to \mathcal{P}$ is defined | |
334 | in terms of $R^{-1}$ and $K^{-1}$ in a very similar way to encryption. Let | |
335 | $\mathbf{x}_0$ be a ciphertext vector. Let $\mathbf{x}_{i+1} = | |
336 | R^{-1}_i(\mathbf{x}_i)$ for $0 \le i < 8$. Then we define | |
337 | $C^{-1}(\mathbf{x}_0) = K^{-1}_8(\mathbf{x}_8)$. | |
338 | ||
339 | ||
340 | \subsection{Constants} | |
341 | ||
342 | The matrix $\mathbf{M}$ and its inverse $\mathbf{M}^{-1}$ are: | |
343 | \begin{eqnarray*}[rl] | |
344 | \mathbf{M} = & | |
345 | \begin{pmatrix}[[>{\hbox\bgroup\ttfamily}c<{\egroup}] | |
346 | f7a413 & 54bd81 & 447550 & ff4449 \\ | |
347 | f31e87 & d85388 & de32cb & 40e3d7 \\ | |
348 | d9db1d & 551b45 & e9d19f & e443de \\ | |
349 | 4b949a & 4d435d & ef0a17 & b784e1 | |
350 | \end{pmatrix} \\ | |
351 | \mathbf{M}^{-1} = & | |
352 | \begin{pmatrix}[[>{\hbox\bgroup\ttfamily}c<{\egroup}] | |
353 | 17391b & fafb4b & a66823 & f2efb6 \\ | |
354 | 13e0e5 & 2ed5e4 & b2cfff & d9cdb5 \\ | |
355 | 2af462 & 33826d & de66a1 & eb6c85 \\ | |
356 | c2f423 & e904a3 & e772d8 & d791f1 | |
357 | \end{pmatrix} | |
358 | \end{eqnarray*} | |
359 | ||
360 | ||
361 | ||
362 | \section{Rationale and analysis} | |
363 | ||
364 | \subsection{Design decisions} | |
365 | ||
366 | The initial objective was to produce a cipher which played to the particular | |
367 | strengths of digital signal processors. DSPs tend to have good multipliers, | |
31b692a0 | 368 | and are particularly good at matrix multiplication. The decision to use a |
e6e0e332 MW |
369 | matrix multiplication over $\mathbb{Z}_{2^{24}}$ seemed natural, given that |
370 | 24 bits is a commonly offered word size. | |
371 | ||
372 | The choice of a 96-bit block is also fairly natural. A 2 word (48-bit) block | |
373 | is clearly too small, and a 3 word (72-bit) block is a little on the small | |
374 | side too. | |
375 | ||
376 | ||
377 | \subsection{Matrix multiplication over $\mathbb{Z}_{2^{24}}$} | |
378 | ||
379 | Integer multiplication on a DSP is a cheap source of nonlinearity. Note that | |
380 | bit $i$ of the result depends on all of the bits in the operands of lesser or | |
381 | equal significance.position $i$ downwards. | |
382 | ||
383 | The decision to make the $4 \times 4$ matrix fixed was taken fairly early on. | |
384 | Generating invertible matrices from key material seemed like too much work to | |
385 | expect from the DSP. | |
386 | ||
387 | The matrix is generated pseudorandomly from a seed string, using SHA-1. The | |
388 | criteria we used to choose the matrix are: | |
389 | \begin{enumerate} | |
390 | \item The matrix must be invertible. | |
391 | \item Exactly one entry in each row and column of the matrix must be even. | |
392 | \end{enumerate} | |
393 | Criterion 1 is obvious. Criterion 2 encourages diffusion between the entries | |
394 | in the block vector. Note that if a matrix satisfies the second criterion, | |
395 | its inverse also does. | |
396 | ||
397 | Consider a vector $\mathbf{x}$ and its product with the matrix $\mathbf{M} | |
398 | \mathbf{x}$. Whether the top bit of entry $i$ in $\mathbf{x}$ affects | |
399 | entry $j$ in the product depends on whether the entry in row $j$, column $i$ | |
400 | of $\mathbf{M}$ is even. Criterion 2 ensures the following: | |
401 | \begin{itemize} | |
402 | \item A top-bit change in a single word or three words affects three words in | |
403 | the output. | |
404 | \item A top-bit change in two words affects two words in the output. | |
405 | \end{itemize} | |
406 | ||
407 | The seed string used is \texttt{matrix-seed-string}. The program which | |
408 | generates the matrix is included with the Storin example source code. | |
409 | ||
410 | \subsection{The linear transformation} | |
411 | ||
412 | A bit change in one of the inputs to the matrix can only affect bits at that | |
413 | position and higher in the output. The linear transformation at the end of | |
414 | the round aims to provide diffusion from the high-order bits back to the | |
415 | low-order bits. | |
416 | ||
417 | A single high-order bit change in the input to a round will affect the | |
418 | high-order bits of three words in the output of the matrix multiply. The | |
419 | linear transformation causes it to affect bits in the low halves of each of | |
420 | these words. The second round's multiplication causes these bits to affect | |
421 | the whole top halves of all of the output words. The linear transformation | |
422 | propagates this change to the bottom halves. Complete avalanche is therefore | |
423 | achieved after three rounds of Storin. | |
424 | ||
425 | ||
426 | \subsection{Key schedule notes} | |
427 | ||
428 | The key schedule is intended to be adequate for bulk encryption; it doesn't | |
429 | provide good key agility, and isn't intended to. The key schedule accepts an | |
430 | arbitrary number of 24-bit words, although expecting 864 bits of security | |
431 | from the cipher is not realistic. The suggested maximum of 5 words (120 | |
432 | bits) seems more sensible. This maximum can be raised easily when our | |
433 | understanding of the cipher increases our confidence in it. | |
434 | ||
435 | The key schedule is strongly reminiscent of Blowfish \cite{blowfish}. Use of | |
436 | existing components of the cipher, such as the matrix multiplication and the | |
437 | cipher itself, help reduce the amount of code required in the implementation. | |
438 | ||
439 | There is an interesting feature of the key schedule: the output of the first | |
440 | round of the second encryption is zero. To see why this is so, it is enough | |
441 | to note that the first round key has just been set equal to what is now the | |
442 | plaintext; the result of the key mixing stage is zero, which is unaffected by | |
443 | the matrix and linear transformation. | |
444 | ||
445 | ||
446 | \subsection{Attacking Storin} | |
447 | ||
448 | A brief\footnote{About three days' worth on a 300MHz Pentium II.} | |
449 | computerized analysis of the matrix multiplication failed to turn up any | |
450 | high-probability differential characteristics. While an exhaustive search | |
451 | was clearly not possible, the program tested all differentials of Hamming | |
452 | weight 5 or less, and then random differentials, applying each to a suite of | |
453 | $2^{13}$ different 96-bit inputs chosen at random. No output difference was | |
454 | noted more than once. | |
455 | ||
456 | One possible avenue of attack worth exploring is to attempt to cause zero | |
457 | words to be input into the first-round matrix by choosing plaintext words | |
458 | identical to subkey words for the first round. Causing $n$ matrix input | |
459 | words to be zero clearly takes $O(2^{24n})$ time. If a method can be found | |
460 | to detect when zero words have been input to the matrix, this can be used to | |
461 | discover the subkey words rather more rapidly than exhaustive search. We | |
462 | can't see a way to exploit this at the moment, but it could be a fruitful | |
463 | place to look for cryptanalysis. | |
464 | ||
465 | ||
466 | \section{Conclusion} | |
467 | ||
468 | We have presented a new block cipher, Storin. Any cryptanalysis will be | |
469 | received with interest. | |
470 | ||
471 | ||
472 | \begin{thebibliography}{99} | |
473 | \bibitem{idea}{X. Lai, `On the Design and Security of Block Ciphers', ETH | |
474 | Series in Informatics Processing, J. L. Massey (editor), vol. 1, | |
475 | Hartung-Gorre Verlag Konstanz, Technische Hochschule (Zurich), 1992} | |
476 | \bibitem{blowfish}{B. Schneier, `The Blowfish Encryption Algorithm', | |
477 | \textit{Dr Dobb's Journal}, vol. 19 no. 4, April 1994, pp. 38--40} | |
478 | \end{thebibliography} | |
479 | ||
480 | %%%----- That's all, folks -------------------------------------------------- | |
481 | ||
482 | \end{document} | |
483 | ||
484 | %%% Local Variables: | |
485 | %%% mode: latex | |
486 | %%% TeX-master: t | |
487 | %%% End: |