Tree-Based Symmetric Key Broadcast Encryption A thesis submitted to Indian Statistical Institute in partial fulfillment of the thesis requirements for the degree of Doctor of Philosophy in Computer Science Author: Supervisor: Sanjay Bhattacherjee Prof. Palash Sarkar Applied Statistics Unit Indian Statistical Institute 203, B. T. Road, Kolkata, West Bengal, India - 700 108. To my family. Acknowledgements I have always believed that life has been very kind to me. I have probably almost always got more than I deserved. This thesis is a culmination of all that I have earned through my associations with the people around me. I take up this opportunity to account for all that I have received from the innumerable contributors in my life till date. This is more of a remembral and hence is in no way exhaustive accounting. After my graduation, I worked in the industry for two years. I joined the M. Tech. (CS) course in 2007 at the Indian Statistical Institute, Kolkata. Prof. Palash Sarkar was one of the faculty members teaching in the first semester. Within the first few days of attending Palash-da's classes, I knew that he was one of those people I would aspire to be like. It has been a privilege being his student and then an even more enriching experience to have him as my thesis supervisor first during M. Tech. and then during PhD. His inspiration has played a consistent role in shaping up my academic contours. While working together, he knew almost every time how to get the best out of me. His meticulous discipline, academic depth and breadth, clarity of thought and consistent effort are few of his many qualities I would like to succeed in emulating some day. Most cherished of all that I have received from him is his unending support, guidance and encouragement - academic and otherwise. My PhD days would not have been as enjoyable without the presence of Dr. Kishan Chand Gupta. Academic discussions with Kishan-da are always very revealing because of his practice of digging things up to insane depths. He has been a patient teacher, an ever supportive friend, and a guide in many ways of life. I extend my sincerest gratitude to Prof. Bimal Roy who has been providing relentless support in driving forward the research on Cryptology happening in India and ISI Kolkata becoming its largest hub. I shall be grateful to Prof. Rana Barua and Dr. Mridul Nandi for their courses and talks that shaped my understanding of Cryptology in many ways. Rana-da's humble ways and incredible capability of abstraction and Mridul-da's emphasis on details of concepts are things I would love to inculcate. I have also learnt many aspects of the subject from Prof. Subhamoy Maitra. I adore his capability to portray difficult concepts in a very lucid manner. I would like to thank Subhamoy-da for his support during my years at ISI. Dr. Sanjit Chatterjee provided me the opportunity for a short academic visit to IISc. During the three weeks I spent at IISc, I underwent intense reading and had very fruitful i discussions with Sanjit-da. I would love to pursue those directions of research in near future. I would also like to thank Dr. Soumitra Sanadhyay, Dr. Goutam Paul, Dr. Sushmita Ruj, Dr. Sumanta Sarkar for the useful discussions I have had with them during the course of my PhD. Some people are so much a part and parcel of one's life that it becomes hard to separate out their contributions. Somindu C. R. and Subhabrata Samajder are two such people. We have been friends since our M. Tech. days and their continuous support has enabled me keep my chin up even during the most difficult times. My return to academics is largely contributed to two individuals - Srimanta Bhattacharya and Dinesh Layek. They were my colleagues at my workplace in Kolkata who inspired me to pursue higher studies. I will always be thankful to them that they found me worthy of their advice. Having seniors like Dr. Rishiraj Bhattacharya and Dr. Sumit Pandey has been a privilege all through. I got one of my first insights of Cryptology from Sumit-da. His unending support and helping nature has earned him reverence from all his friends and juniors including me. Rishi-da has come up with crucial advices at important junctures of my ISI life. Starting from the first day we sat on the football ground together to this day, he has always pushed me ahead and helped me take longer strides. Having friends like Avik Chakraborty, Binanda Sengupta, Indranil Ghosh Ray, Nilanjan Dutta, Raju Maiti and Tapas Pandit at one's workplace makes life very enjoyable. A few words with friends like Dr. Sourav Sengupta, Dr. Santanu Sarkar, Kaushik Chakraborty, Mrinal Nandi, Samiran Bag, Satrajit Ghosh, Shashank Singh and Subhadeep Banik feels refreshing. Dr. Ashish Coudhury and Dr. Arpita Patra have visited ISI on many occasions. Knowing them has been a pleasure all along. During my travel to the Seventh International Workshop on Coding and Cryptography, Paris in April 2011, Ashish-da ensured that I had very little to worry about. I would like to thank the Cryptology Research Society of India for all its support. A special thanks to Amitabha Sinha for his tireless cooperation in making life at ISI very smooth. I am thankful to the Director's Office, the Dean's Office, the ASU Office and other administrative offices at ISI for their cooperation all through. Dr. Mandar Mitra is one of those teachers in ISI who have influenced me most. His technical soundness was just the first of the numerous impressions he has left on me. His ii eloquence and depth of knowledge made me enjoy his classes to the brim. But above all, I would be ever indebted to him for being a living personification of humility and modesty in my life. Prof. Sandip Das is another such teacher. I have seen no one try harder to get the best out of even the most inattentive student as Sandip-da used to. His untiring efforts and dedication as a teacher would be a model to follow. I have been extremely lucky to get teachers like Prof. Aditya Bagchi, Prof. Amitabha Bandopadhyay, Prof. Bhabani P. Sinha, Prof. Bhargab B. Bhattacharya, Prof. Krishnendu Mukhopadhyay, Prof. Subhash Nandy, Dr. Arijit Bishnu, Dr. Pinakpani Pal and Dr. Utpal Garain. Although I did not get an opportunity to attend the classes of Prof. Bhabotosh Chanda, my first academic tryst with ISI was due to him. During my graduation, I thoroughly enjoyed doing a summer training at his lab. It was a pleasure having academic discussions with him. I would like to extend a special note of thanks to Prof. Sushmita Sur Kolay and Nandakumar Narayanmangalam for helping me get an internship opportunity at ARM during my M. Tech. It was there I had my first practical experience of doing research work. I have had pleasing associations with quite a few other faculty members of ISI. Prof. Anup Diwanji, Prof. Indraneel Dasgupta, Prof. Mousumi Bose, Prof. Pradipta Bandopadhyay, Prof. Smarajit Bose, Prof. Sumitra Purkayastha, Prof. Tapas Samanta, Dr. Ansuman Baner- jee, Dr. Atanu Ghosh, Dr. Diganta Mukherjee, Dr. Indranil Mukhopadhyay, Dr. Samarjit Das, Dr. Sourabh Ghosh and Dr. Swagatam Das are amongst them. I would like to thank ISI, its faculty members and other staff, students, research scholars and others for providing and maintaining an amicable environment for studies and research. Talking of ISI and the time I spent here, an absolutely inseparable thing would be the friends I have made here. It all started on the days during admission to M. Tech. and then during the first few classes. Amit Tripathi, Ayyappadas A. M., Dr. Aritra Banik, Dr. Chiranjit Chakraborty, Dr. Sandeep Kumar Dey, Dr. Kalikinkar Mandal, Dr. Nargis Pervin, Dr. Pulak Purkait, Mrinmoy Ghorai, Rajnish Kumar, Santanu Bhowmick, Swarup Chattopadhyay and all other M. Tech. batch-mates, seniors and juniors were an integral part in shaping up my life then on. From group-study sessions to the Bonhooghly excursions to the most adventurous campus endeavors - we have been accomplices in everything. Being part of the numerous extra-curriculars especially our annual cultural event \All-Go-Rhythm" (a part of which was the play \Bheem Badh") was one of the most memorable times at ISI. I have spent hours at a stretch at the research scholars' hostel sharing very memorable times with Dr. Debasmita Basu, Dr. Navonil De Sarkar, Dr. Sayantan Dutta, Dr. Srikanta iii Kundu and Debojyoti Mazumdar. I have had an absolutely enjoyable hostel life of more than five years, that has endowed me with very dear friends. Some of them are Aleya Bose, Amiya Bhowmick, Anindita Chatterjee, Apurba Das, Arnab Hazra, Arvind Nam- biar, Bidesh Das, Biman Chakraborty, Bipul Islam, Chintan Parmar, Debanjan Majumdar, Dr. Abhijit Mandal, Dr. Aparajita Bakshi, Dr. Buddhananda Banerjee, Dr. Kaushik Kundu, Dr. Kushal Banik Choudhury, Dr. Minati De, Dr. Prosenjit Das, Dr. Rajat Subhra Hazra, Dr. Rituparna Basak, Dr. Sabyasachi Mukhopadhyay, Dr. Shamsher Singh, Dr. Sourav Rana, Dr. Sriparna Ganguly, Dr. Subhendu Chakraborty, Dr. Suchismita Roy, Dr. Sudip Samanta, Dr. Sutripta Sarkar, Dr. Trishita Ray Barman, Gopakumar Achuthankutty, Hi- rak Sarkar, Kavita Suresh Parab, Keya Dutta, Koustav Sarkar, Mahamitra Das, Mannu Dwivedi, Manusree Mahato, Minerva Mukhopadhyay, Mithun Bhowmik, Narayan Rakshit, Parikshit De, Paromita Dubey, Pramita Bagchi, Prasenjit Ghosh, Priyam Biswas, Rajashree Bhattacharyya, Reshmi Mitra, Ritwik Bhattacharya, Roshni Roy, Ruchira Biswas, Sandip Sarkar, Sayan Roy, Sebanti Sengupta, Sedigheh Mirzaei Salehabadi, Shivani Santosh, Som- nath Ghatak, Sourav Kumar Sasmal, Suman Sarkar, Tanmay Das, Tomojoy Ghosh and Tridip Sardar.