【作者】 张斌；

【导师】 肖国镇；

【作者基本信息】 西安电子科技大学 ， 密码学， 2002， 硕士

【摘要】 本文对RSA型公钥密码体制进行了研究。第一章介绍了公钥密码学的研究背景、意义、现状以及公钥密码体制的基本概念和标准RSA体制；第二章对TauyoshiTakagi关于n-adic多块RSA型公钥体制的工作做了详细介绍；在第三章里详细介绍了标准RSA低指数攻击方面的研究成果，包括D.Coppersmith等提出的对低指数相关消息RSA的攻击及最新的由Dan Boneh与Glenn Durfee证明的私钥d<n~0.292的RSA是不安全的重要结果以及作者对D.Coppersmith的分析所做的注记；第四章介绍了孙琦、曹珍富等在多项式环上构造模拟RSA型公钥密码体制的工作以及作者在这方面的工作。 本文的主要结果如下： 1．指出了D.Coppersmith在文[31]中存在的一些错误和混乱，证明了环Zn上两个加密多项式在Zn上只有唯一公共根，这虽然不能说明用Euclidean算法求环Zn上两个加密多项式的最大公因式从而恢复明文的方法的确定性，但也反映了问题的一个方面。 2．改进了曹珍富在文[50]中提出的模拟RSA型公钥密码体制，解决了其中存在的密文扩展问题，并讨论了新体制的安全性；提出了一个新的模拟RSA型公钥密码体制，该体制无密文数据扩展，而且在概念上更接近于标准RSA。更多还原

【Abstract】 This thesis mainly has a study on the RSA-type public key cryptosystems. InChapter One, the research background, significance and state of public-key cryptologyare introduced along with the introduction to elementary concepts of public-keycryptosystems and the standard RSA scheme. In Chapter Two, the n-adic multi-blockRSA-type public key cryptosystem presented by Tauyoshi Takagi in Crypto’97 has beenintroduced. In Chapter Three, the achievements on the low-exponent attack of thestandard RSA scheme are introduced, including the low-exponent attack with relatedmassages presented by D.Coppersmith and the new important result finished by DanBoneh and Glenn Durfee that the standard RSA scheme with private key d less thanN is insecure, a remark on D.Coppersmith’s analysis is also given in this chapter. InChapter Four, the work on construction of the RSA analogue over polynomial rings bySunQi and CaoZhenfu is introduced, and the author’s related work is also included inthis chapter.The main results of this thesis are as follows:1 .Sonie mistakes and confusion made by D.Coppersmith in [31] are pointed out; a proofthat the two encryption polynomials only have one common root over Z~ is given,which may reflect one aspect of the problem though cannot sufficiently demonstrate thedefinity of the attack using Euclidean algorithm to find the greatest common divisor ofthe two encryption polynomials over Zn.2.A modification of the RSA analogue presented by CaoZhenfu in [50] is made, vhichhas solved the ciphertext extension problem in the above analogue, and the security ofthe new scheme is also discussed; another new RSA analogue has been proposed, whichhas no ciphertext extension and is much similar to the standard RSA scheme inconception.更多还原