Time Memory Trade O (TMTO) is a cryptanalytic method that aims to develop an attack which has a lower memory complexity than lookup table and a lower online time complexity than exhaustive search. TMTO methods are widely studied in the literature and used for inverting various cryptosystems. We focus on the design and the analysis of TMTO on symmetric ciphers in this thesis. Firstly, the summary of the random mapping statistics from the view point of TMTO is presented. We also recalculate some expected values with a simpler approach than the existing proofs. Then, we propose some variant constructions and also present three new distinguishers based on random mappings. Next, we provide a detailed analysis of the success rate of two main improvements of the attack / Distinguished Point Method and Rainbow Method. Finally, we discuss the adjustment of the parameters to achieve a high success rate. To support our theoretical framework, we also present empirical results of our analysis to actual ciphers.

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