**Two New Examples of TTM**

*T.Moh*

**Abstract: **We will review the past history of the attacks and defenses of TTM. The main tool of the past attacks is
linear algebra, while the defenses rely on algebraic geometry and commutative algebra. It is hard for
attackers to completely succeed against the formidable castle of modern mathematics. It is out of the common
sense that problems of algebraic geometry can always be solved by linear algebra. It repeatly happens that the
attackers find some points which could be exploited by linear algebra using complicated computations, usually
the attackers overexaggerate the power of linear algebra and illusional believe that they succeed totally, then
the points are disappearing by a simple twist in algebraic geometry and commutative algebra. All attacks in
the past simply strengthen the structures of TTM. For these facts we are very grateful to the attackers.

Last year there is a paper entitled "{\it Breaking a New Instance of TTM Cryptosystem}" by Xuyun Nie, Lei Hu, Jianyu Li, Crystal Updegrove and Jintai Ding [11] claiming a successive attack on the scheme of TTM presented in [7]. In our previous article [8], we show that their claim is a {\bf misunderstanding}.

The discussions of [11] and [8] center on if in [11] the authors really just use the {\it public keys}. Right aft er we post [8], to settle the discrepancy of [11] and [8], we have sent the public keys of a new example (which is attached as the {\bf Appendix I} of this article) to the authors of [11] to test their claim in the {\it abstract} of [11], i.e., they will be able to crack TTM using only the public keys (in 20 minutes as stated in the abstract of [11]). After two weeks, Mr Nie asks the private keys of the new example for his {\it theoretical analysis} and we will consider his request only if he concedes that he is unable to crack the new example by the method of [11]. Since there is no definite answer from them after 4 months, we will publish the example in this article to give other people chances to attack. Furthermore, we publish a second example as {\bf Appendix II}.

**Category / Keywords: **public-key cryptography /

**Date: **received 20 Apr 2007

**Contact author: **ttm at math purdue edu

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

**Version: **20070423:083738 (All versions of this report)

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