Course Notes on Complexity and Cryptography

The main purpose of this course is an introductory study of computational complexity and cryptography. The rst part introduces the concept of computational complexity by looking at the basic arithmetic operations, i.e., addition, subtraction, multiplication and division. Then matrix multiplication is touched. In order to prepare everything we need later for public-key cryptography, we continue with number theoretic problems and study several algorithms including modular exponentiation, primality testing and taking discrete roots. In the following, we introduce well-known complexity classes, look at complete problems and nish with probabilistic complexity classes. There is also an appendix comprising additional material that had to be omitted due to the introductory character of this course, but may be worth to be known. The second part is devoted to cryptography. After a short historical sketch we deal with public-key cryptography in some more detail, look at authentication and cryptographic protocols, and nish with a more detailed study of digital signatures. Again, there is an appendix containing material for further reading. There will be a midterm problem set and a nal report problem set each worth 100 points. So your grade will be based on these 200 points. Note that the course is demanding. But this is just in line with William S. Clark's encouragement Boys, be ambitious ! Of course, nowadays, we would reformulate this encouragement as Girls and Boys, be ambitious ! c ©Thomas Zeugmann, Hokkaido University, 2008 Course Notes on Complexity and Cryptography iii Recommended Literature The references given below are mandatory. (1) 8¡à : — ÖhaüÈÞÈó ž Ö Information & Computing 106, μ¤ ̈ó1>, 2005 ISBN 978-4-7819-1104-5 (2) Ò¤ ̈ >bKj ûPÅ1 áì ̄Áãü·êüoD-8 þã—÷n ú p ûPÅ1 áf è (3) J ÛÃ× ̄íÕÈ J ¦ëÞó : aüÈÞÈó ž Ö —Ö I, Information & Computing 3, μ¤ ̈ó1>, 1984 ISBN 4-7819-0374-6 (4) J ÛÃ× ̄íÕÈ J ¦ëÞó : aüÈÞÈó ž Ö —Ö II, Information & Computing 3, μ¤ ̈ó1>, 1984 ISBN 4-7819-0432-7 There are some additional references to the literature given in some lectures. So please look there, too. c ©Thomas Zeugmann, Hokkaido University, 2008