%%% -*-latex-*-
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-%%% $Id: storin.tex,v 1.3 2000/05/25 19:46:22 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.
%%%
until all of the subkey words have been replaced.
The Storin key schedule can in theory accept user keys up to 36 words (864
-bits) long. However, there are possible security problems with keys shorter
-than 28 words (672 bits). We believe that it's unrealistic to expect this
-much strength from the cipher and recommend against using keys longer than 5
-words (120 bits).
+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}
cipher itself, help reduce the amount of code required in the implementation.
The restriction of the key schedule to 28 words is due to an interesting
-property, also shared by Blowfish \cite{blowfish}: 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. See figure~\ref{fig:bfkeysched}.
+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.
\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
-\[ (\hex{800000}, \hex{800000}, \hex{800000}, 0) \to
- (0, 0, \hex{800000}, 0) \]
+\[ \begin{pmatrix}
+ 1 \lsl 23 \\ 1 \lsl 23 \\ 1 \lsl 23 \\ 0
+ \end{pmatrix} \to
+ \begin{pmatrix}
+ 0 \\ 0 \\ 1 \lsl 23 \\ 0
+ \end{pmatrix}
+\]
holds with probability 1 through the matrix multiplication.
Differentials in the linear transform are easy to find; for example:
-\[ (0, 0, \hex{800000}, 0) \to (0, 0, \hex{800800}, 0) \]
+\[ \begin{pmatrix}
+ 0 \\ 0 \\ 1 \lsl 23 \\ 0
+ \end{pmatrix} \to
+ \begin{pmatrix}
+ 0 \\ 0 \\ (1 \lsl 23) \xor (1 \lsl 11) \\ 0
+ \end{pmatrix}
+\]
We can continue through the second round's matrix multiplication with a
truncated differential, again with probability 1:
-\[ (0, 0, \hex{800800}, 0) \to
- (\hex{???000}, \hex{???800}, \hex{???800}, \hex{???800}) \]
-The following linear transform can be commuted with the postwhitening by
-applying a trivial reversible transformation to the postwhitening keys, and
-we can apply it to the ciphertext. If we do this, we can combine the
-differentials above to construct a probability-1 characteristic for a 2-round
-variant of Storin:
-\[ (\hex{800000}, \hex{800000}, \hex{800000}, 0) \to
- (\hex{???000}, \hex{???800}, \hex{???800}, \hex{???800}) \]
+\[ \begin{pmatrix}
+ 0 \\ 0 \\ (1 \lsl 23) \xor (1 \lsl 11) \\ 0
+ \end{pmatrix} \to
+ \begin{pmatrix}
+ \delta_0 \lsl 12 \\
+ (\delta_1 \lsl 12) \xor (1 \lsl 11) \\
+ (\delta_2 \lsl 12) \xor (1 \lsl 11) \\
+ (\delta_3 \lsl 12) \xor (1 \lsl 11) \\
+ \end{pmatrix}
+\]
+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.
\begin{thebibliography}{99}
-\bibitem{storin-collide} M. Fisher, `Yet another block cipher: Storin',
+\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,
+\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',
+\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',
+\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}
~mdw / storin / blobdiff