Study On The RSA-Type Public-Key Cryptosystems

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.