%%% -*-latex-*-
%%%
-%%% $Id: storin.tex,v 1.2 2000/05/21 21:43:26 mdw Exp $
+%%% $Id: storin.tex,v 1.5 2000/07/02 15:22:34 mdw Exp $
%%%
%%% Definition of the cipher
%%%
%%%----- Revision history ---------------------------------------------------
%%%
%%% $Log: storin.tex,v $
+%%% Revision 1.5 2000/07/02 15:22:34 mdw
+%%% Overhaul of differential cryptanalysis, including a new attack.
+%%%
+%%% Revision 1.4 2000/05/28 00:39:32 mdw
+%%% Fix some errors.
+%%%
+%%% Revision 1.3 2000/05/25 19:46:22 mdw
+%%% Improve analysis section.
+%%%
%%% Revision 1.2 2000/05/21 21:43:26 mdw
%%% Fix a couple of typos.
%%%
\def\xor{\oplus}
\def\seq#1{{\langle #1 \rangle}}
+\def\hex#1{\texttt{#1}_{16}}
+
\sloppy
\title{Storin: A block cipher for digital signal processors}
replaced with the ciphertext, and the process continues, nine times in all,
until all of the subkey words have been replaced.
-The Storin key schedule can accept user keys up to 36 words (864 bits) long.
-It is unrealistic, however, to expect this much strength from the cipher. We
-recommend against using keys longer than 5 words (120 bits).
+The Storin key schedule can in theory accept user keys up to 36 words (864
+bits) long. However, there are known problems with keys longer than 28 words
+(672 bits), and these large keys are forbidden. We expect that with long
+keys, attacks will be found which are more efficient than an exhaustive
+search of the keyspace; we therefore (conservatively) recommend 5 word
+(120-bit) keys as a practical maximum.
\subsection{Encryption}
The matrix $\mathbf{M}$ and its inverse $\mathbf{M}^{-1}$ are:
\begin{eqnarray*}[rl]
\mathbf{M} = &
- \begin{pmatrix}[[>{\hbox\bgroup\ttfamily}c<{\egroup}]
- f7a413 & 54bd81 & 447550 & ff4449 \\
- f31e87 & d85388 & de32cb & 40e3d7 \\
- d9db1d & 551b45 & e9d19f & e443de \\
- 4b949a & 4d435d & ef0a17 & b784e1
+ \begin{pmatrix}
+ \hex{f7a413} & \hex{54bd81} & \hex{447550} & \hex{ff4449} \\
+ \hex{f31e87} & \hex{d85388} & \hex{de32cb} & \hex{40e3d7} \\
+ \hex{d9db1d} & \hex{551b45} & \hex{e9d19f} & \hex{e443de} \\
+ \hex{4b949a} & \hex{4d435d} & \hex{ef0a17} & \hex{b784e1}
\end{pmatrix} \\
\mathbf{M}^{-1} = &
- \begin{pmatrix}[[>{\hbox\bgroup\ttfamily}c<{\egroup}]
- 17391b & fafb4b & a66823 & f2efb6 \\
- 13e0e5 & 2ed5e4 & b2cfff & d9cdb5 \\
- 2af462 & 33826d & de66a1 & eb6c85 \\
- c2f423 & e904a3 & e772d8 & d791f1
+ \begin{pmatrix}
+ \hex{17391b} & \hex{fafb4b} & \hex{a66823} & \hex{f2efb6} \\
+ \hex{13e0e5} & \hex{2ed5e4} & \hex{b2cfff} & \hex{d9cdb5} \\
+ \hex{2af462} & \hex{33826d} & \hex{de66a1} & \hex{eb6c85} \\
+ \hex{c2f423} & \hex{e904a3} & \hex{e772d8} & \hex{d791f1}
\end{pmatrix}
\end{eqnarray*}
entry $j$ in the product depends on whether the entry in row $j$, column $i$
of $\mathbf{M}$ is even. Criterion 2 ensures the following:
\begin{itemize}
-\item A top-bit change in a single word or three words affects three words in
- the output.
+\item A top-bit change in a single word affects three words in the output.
\item A top-bit change in two words affects two words in the output.
\end{itemize}
\subsection{Key schedule notes}
The key schedule is intended to be adequate for bulk encryption; it doesn't
-provide good key agility, and isn't intended to. The key schedule accepts an
-arbitrary number of 24-bit words, although expecting 864 bits of security
-from the cipher is not realistic. The suggested maximum of 5 words (120
-bits) seems more sensible. This maximum can be raised easily when our
-understanding of the cipher increases our confidence in it.
+provide good key agility, and isn't intended to. The key schedule accepts up
+to 28 words of user key, although expecting 672 bits of security from the
+cipher is not realistic. The suggested maximum of 5 words (120 bits) seems
+more sensible. This maximum can be raised easily when our understanding of
+the cipher increases our confidence in it.
The key schedule is strongly reminiscent of Blowfish \cite{blowfish}. Use of
existing components of the cipher, such as the matrix multiplication and the
cipher itself, help reduce the amount of code required in the implementation.
-There is an interesting feature of the key schedule: the output of the first
-round of the second encryption is zero. To see why this is so, it is enough
-to note that the first round key has just been set equal to what is now the
-plaintext; the result of the key mixing stage is zero, which is unaffected by
-the matrix and linear transformation.
+The restriction of the key schedule to 28 words is due to an interesting
+property, also shared by Blowfish \cite{blowfish} (see
+figure~\ref{fig:bfkeysched}): the output of the first round of the second
+encryption doesn't depend on the previous round. To see why this is so, it
+is enough to note that the first round key has just been set equal to what is
+now the plaintext; the result of the key mixing stage is zero, which is
+unaffected by the matrix and linear transformation.
+
+A limit of 28 words is chosen to ensure that the round-1 key affects the
+round-2 key in a part of the cipher earlier than the postwhitening stage.
+\begin{figure}
+\centering
+\leavevmode
+\begin{xy}
+ \xycompile{
+ \POS 0; <0.7cm, 0cm>:
+ \POS (0, 0) ="o" +(3, 0) ="w"
+ \ar "o" *+{P[0]}; p-(0, 1) *{\xor} ="x"
+ \ar "x" -(1, 0) *+[l]{P[0]}; "x"
+ \ar@{-} "x"; p-(0, 2) ="as"
+ \ar "w" *+{P[1]}; p-(0, 2) *{\xor} ="x"
+ \ar "o"-(0, 2); "x" |-*+[F]{F}
+ \ar@{-} "x"; p-(0, 1) ="bs"
+ \ar@{-} "as"; "bs"-(0, 1) ="w"
+ \ar@{-} "bs"; "as"-(0, 1) ="o"
+ \ar "o"; p-(0, 1) *+{P[1] \xor F(0)} ="x"
+ \ar "x"; p-(0, 1) *{\xor} ="x"
+ \ar "x" -(1, 0) *+[l]{P[1]}; "x"
+ \ar "x"; p-(0, 2) *+{F(0)}
+ \ar "w"; p-(0, 1) *+{0} ="x"
+ \ar "x"; p-(0, 2) *{\xor} ="x"
+ \ar "o"-(0, 3); "x" |-*+[F]{F}
+ \ar "x"; p-(0, 1) *+{F^2(0)}}
+\end{xy}
+\caption{Blowfish key schedule: $P[2]$ and $P[3]$ don't depend on $P[0]$ and
+ $P[1]$.}
+\label{fig:bfkeysched}
+\end{figure}
\subsection{Attacking Storin}
-A brief\footnote{About three days' worth on a 300MHz Pentium II.}
-computerized analysis of the matrix multiplication failed to turn up any
-high-probability differential characteristics. While an exhaustive search
-was clearly not possible, the program tested all differentials of Hamming
-weight 5 or less, and then random differentials, applying each to a suite of
-$2^{13}$ different 96-bit inputs chosen at random. No output difference was
-noted more than once.
+\subsubsection{Differential cryptanalysis}
+
+There is a two-round truncated differential \cite{storin-tdiff}, which can be
+used to break Storin reduced to only 2 rounds. The differential
+\[ \begin{pmatrix}
+ 1 \lsl 23 \\ 1 \lsl 23 \\ 1 \lsl 23 \\ 0
+ \end{pmatrix} \to
+ \begin{pmatrix}
+ 0 \\ 0 \\ 1 \lsl 23 \\ 0
+ \end{pmatrix}
+\]
+holds with probability 1 through the matrix multiplication.
+Differentials in the linear transform are easy to find; for example:
+\[ \begin{pmatrix}
+ 0 \\ 0 \\ 1 \lsl 23 \\ 0
+ \end{pmatrix} \to
+ \begin{pmatrix}
+ 0 \\ 0 \\ (1 \lsl 23) \xor (1 \lsl 11) \\ 0
+ \end{pmatrix}
+\]
+We can continue through the second round's matrix multiplication with a
+truncated differential, again with probability 1:
+\[ \begin{pmatrix}
+ 0 \\ 0 \\ (1 \lsl 23) \xor (1 \lsl 11) \\ 0
+ \end{pmatrix} \to
+ \begin{pmatrix}
+ \delta_0 \lsl 12 \\
+ (\delta_1 \lsl 12) \xor (1 \lsl 11) \\
+ (\delta_2 \lsl 12) \xor (1 \lsl 11) \\
+ (\delta_3 \lsl 12) \xor (1 \lsl 11) \\
+ \end{pmatrix}
+\]
+where the $\delta_i$ are unknown 12-bit values. Applying the linear
+transformation to this output difference gives us
+\[ \begin{pmatrix}
+ \delta_0 \lsl 12 \\
+ (\delta_1 \lsl 12) \xor (1 \lsl 11) \\
+ (\delta_2 \lsl 12) \xor (1 \lsl 11) \\
+ (\delta_3 \lsl 12) \xor (1 \lsl 11) \\
+ \end{pmatrix} \to
+ \begin{pmatrix}
+ (\delta_0 \lsl 12) \xor \delta_0 \\
+ (\delta_1 \lsl 12) \xor \delta_1 \xor (1 \lsl 11) \\
+ (\delta_2 \lsl 12) \xor \delta_2 \xor (1 \lsl 11) \\
+ (\delta_3 \lsl 12) \xor \delta_3 \xor (1 \lsl 11) \\
+ \end{pmatrix}
+\]
+A subsequent key-mixing or postwhitening stage won't affect the difference.
+We can therefore combine the differentials above to construct a probability-1
+truncated differential for a 2-round variant of Storin:
+\[ \begin{pmatrix}
+ 1 \lsl 23 \\ 1 \lsl 23 \\ 1 \lsl 23 \\ 0
+ \end{pmatrix} \to
+ \begin{pmatrix}
+ (\delta_0 \lsl 12) \xor \delta_0 \\
+ (\delta_1 \lsl 12) \xor \delta_1 \xor (1 \lsl 11) \\
+ (\delta_2 \lsl 12) \xor \delta_2 \xor (1 \lsl 11) \\
+ (\delta_3 \lsl 12) \xor \delta_3 \xor (1 \lsl 11) \\
+ \end{pmatrix}
+\]
+This characteristic is non-iterative, and can't be extended to more rounds.
+
+The differential can be converted into a key-recovery attack against $n$
+rounds fairly easily, by obtaining the ciphertext for an appropriate
+plaintext pair and guessing the $n - 2$ round keys, testing the guesses by
+working backwards and finding out whether the expected output difference is
+visible. The attack requires a pair of chosen plaintexts, and
+$O(2^{96(n - 2)})$ work. It is only more efficient than exhaustive search
+when the key is longer than $96(n - 2)$ bits.
+
+This attack can be improved. Consider a 3-round variant of Storin, where the
+objective is to discover the postwhitening keys. The postwhitening stage can
+be commuted with the linear transform simply by applying the transform to the
+postwhitening keys. We do this, and guess the least significant 12 bits of
+each of the (transformed) postwhitening key words. Working through the
+matrix multiplication mod $2^{12}$ rather than mod $2^{24}$ then gives us the
+12 least significant bits of the state words on input to the matrix. Further
+key bits can then be guessed and tested, four at a time, to recover the
+remaining postwhitening key bits, by ensuring that the differences in the
+more significant bits of the third round matrix input obey the characteristic
+described above. This requires only about $2^{48}$ work, and may be extended
+to further rounds by exhaustively searching for the extra round keys.
+
+This attack can break Storin with $n$ rounds ($n \ge 3$) with minimal chosen
+plaintext and $O(2^{48 + 96(n - 3)})$ work. This is the best attack known
+against Storin.
+
+\subsubsection{Other attacks}
+
+In \cite{storin-collide}, Matthew Fisher speculates on breaking 2 rounds of
+Storin by forcing collisions in the matrix multiplication outputs. This
+attack doesn't extend to more than two rounds either.
One possible avenue of attack worth exploring is to attempt to cause zero
words to be input into the first-round matrix by choosing plaintext words
\begin{thebibliography}{99}
-\bibitem{idea}{X. Lai, `On the Design and Security of Block Ciphers', ETH
- Series in Informatics Processing, J. L. Massey (editor), vol. 1,
- Hartung-Gorre Verlag Konstanz, Technische Hochschule (Zurich), 1992}
-\bibitem{blowfish}{B. Schneier, `The Blowfish Encryption Algorithm',
- \textit{Dr Dobb's Journal}, vol. 19 no. 4, April 1994, pp. 38--40}
+\bibitem{storin-collide}
+ M. Fisher,
+ `Yet another block cipher: Storin',
+ sci.crypt article, message-id \texttt{<8gjctn\$9ct\$1@nnrp1.deja.com>}
+\bibitem{idea}
+ X. Lai,
+ `On the Design and Security of Block Ciphers',
+ ETH Series in Informatics Processing, J. L. Massey (editor), vol. 1,
+ Hartung-Gorre Verlag Konstanz, Technische Hochschule (Zurich), 1992
+\bibitem{blowfish}
+ B. Schneier,
+ `The Blowfish Encryption Algorithm',
+ \textit{Dr Dobb's Journal}, vol. 19 no. 4, April 1994, pp. 38--40
+\bibitem{storin-tdiff}
+ M. D. Wooding,
+ `Yet another block cipher: Storin',
+ sci.crypt article, message-id
+ \texttt{<slrn8iqhaq.872.mdw@mull.ncipher.com>}
\end{thebibliography}
%%%----- That's all, folks --------------------------------------------------
~mdw / storin / blobdiff