MarriageTheoremのこと

_ 今日は筑波大学のリスク工学研究会という催しで、暗号と数学の関わりをテーマに１時間ほど話してきた。学生さんの反応はよくわからなかった（←追記：某氏によれば学生さんにも面白がってもらえたらしい。よかったよかった）のだけど、先生方や司会進行の某871氏にはそれなりに面白がってもらえた（色々な意味で）みたいなのでその点はよかったなぁと思っている。ただ、喉の具合を鑑みて途中で水分補給しながら喋るつもりが、熱が入りすぎてぶっ通しで喋り続けてしまったので、明日喉が無事であることを切に願う。

_ 何の話かは書かないが、書面で強い表現を使いつつ口頭で「あまり固く受け取らないように」と言うぐらいなら、書面の段階でもっと緩い表現に改めてほしいものである。検証可能な形で後に残るのは書面だけなのだから。

_ arXiv:math 11月6日分まで、IACR ePrint 2012/687まで確認済み

_ 気になった論文：The k-BDH Assumption Family: Bilinear Map Cryptography from Progressively Weaker Assumptions, Karyn Benson and Hovav Shacham and Brent Waters, http://eprint.iacr.org/2012/687

Over the past decade bilinear maps have been used to build a large variety of cryptosystems. In addition to new functionality, we have concurrently seen the emergence of many strong assumptions. In this work, we explore how to build bilinear map cryptosystems under progressively weaker assumptions. We propose $k$-BDH, a new family of progressively weaker assumptions that generalizes the decisional bilinear Diffie-Hellman (DBDH) assumption. We give evidence in the generic group model that each assumption in our family is strictly weaker than the assumptions before it. DBDH has been used for proving many schemes secure, notably identity-based and functional encryption schemes; we expect that our $k$-BDH will lead to generalizations of many such schemes. To illustrate the usefulness of our $k$-BDH family, we construct a family of selectively secure Identity-Based Encryption (IBE) systems based on it. Our system can be viewed as a generalization of the Boneh-Boyen IBE, however, the construction and proof require new ideas to fit the family. We then extend our methods to produces hierarchical IBEs and CCA security; and give a fully secure variant. In addition, we discuss the opportunities and challenges of building new systems under our weaker assumption family.