In this thesis we obtain several interesting algebraic properties of the operations used in the block cipher IDEA which are important for cryptographic analyzes. We view each of these operations
as a function from $mathbb Z_{2}^n times mathbb Z_{2}^n to mathbb Z_{2}^n$. By fixing one of variables $v(z)=mathbf Z$ in $mathbb Z_{2}^n times mathbb Z_{2}^n$, we define functions $mathbf {f}_z$ and $mathbf {g}_z$ from $mathbb Z_{2}^n$ to $mathbb Z_{2}^n$ for the addition $BIGboxplus$ and the multiplication $BIGodot$ operations, respectively. We first show that the nonlinearity of
$mathbf {g}_z$ remains the same under some transformations of $z$. We give an upper bound for the nonlinearity of $mathbf {g}_{2^k}$, where $2leq k < / n-1$. We list all linear relations which make the nonlinearity of $mathbf {f}_z$ and $mathbf {g}_z$ zero and furthermore, we present all linear relations for $mathbf {g}_z$ having a high probability. We use these linear relations to derive many more linear relations for 1-round IDEA. We also devise also a new algorithm to find a set of new linear relations for 1-round IDEA based on known linear relations. Moreover, we extend the largest known linear class of weak keys with cardinality $2^{23}$ to two classes with cardinality $2^{24}$ and $2^{27}$.
Finally, we obtain several interesting properties of the set
$ { ({mathbf X},{mathbf X} BIGoplus {mathbf A}) in mathbb Z_2^n times mathbb Z_2^n ,|, (mathbf {X}BJoin {mathbf Z})BIGoplus( ({mathbf X} BIGoplus {mathbf A} ) BJoin mathbf {Z} ) =
{mathbf B} }$ for varying ${mathbf A}, {mathbf B}$ and ${mathbf Z}$ in $mathbb Z_2^n$, where $BJoin in { BIGodot,BIGboxplus }$. By using some of these properties, we present impossible differentials for 1-round IDEA and Pseudo-Hadamard Transform.
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/3/12608289/index.pdf |
| Date | 01 March 2007 |
| Creators | Yildirim, Hamdi Murat |
| Contributors | Akyildiz, Ersan |
| Publisher | METU |
| Source Sets | Middle East Technical Univ. |
| Language | English |
| Detected Language | English |
| Type | Ph.D. Thesis |
| Format | text/pdf |
| Rights | To liberate the content for public access |
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