Generalized cryptanalysis of RSA with small public exponent
In this paper, we demonstrate that there exist weak keys in the RSA public key cryptosystem with the public exponent e Na N0.5. In 1999, Boneh and Durfee showed that when a 1 and the private exponent d Nb N0.292, the system is insecure. Moreover, their attack is still effective for 0.5 a 1.875. We propose a generalized cryptanalytic method to attack the RSA cryptosystem with a 0.5. For $c \left\lfloor \frac{{1 \alpha \alpha \right\rfloor and egc d (mod ec) when g, b satisfy \gamma 1 \frac{1}{c} \frac{1}{2\alpha c}}and\beta \alpha c \frac{7}{6} \alpha \gamma c \frac{1}{3}\sqrt {6\alpha 6\alpha c 1 6\alpha \gamma c} we can perform cryptanalytic attacks based on the LLL algorithm. The basic idea is an application of Coppersmith's techniques and we further adapt the technique of unravelled linearization, which leads to an optimized lattice. Our advantage is that we achieve new attacks on RSA with a 0.5 and consequently, there exist weak keys in RSA for most a.Zhai Yao Chuang Xin Dian Ben Wen Fen Xi Liao RSASuan Fa Zhong Dang Gong Yao eXiao Yu Deng Yu NDe 0.5Ci Mi Shi Ke Neng Cun Zai De Ruo Mi Yao Gong Ji Yi Fang Mian Gai Jin Liao Zhi Qian Dang dDa Yu eShi De Gong Ji Fang Fa Ti Chu Liao Ke Yi Yong Yu Fen Xi eXiao Yu Deng Yu NDe 0.5Ci Mi Shi De Yan Yi Gong Ji Ling Yi Fang Mian Ying Yong Zhan Kai Xian Xing Hua De Ji Qiao Jin Yi Bu Ti Chu Liao Ji Yu You Hua De Ge Gou Zao Fang Fa Xia De Yan Yi Gong Ji Ji Ke Yi Suo Xiao Ge De Wei Shu Jian Shao LLLSuan Fa De Yun Xing Shi Jian Ye Ke Yi Ti Gao Li Lun Fen Xi Zhong Si Yao dYing Man Zu De Shang Jie Yu Zhi Qian Yi You De Gong Ji Fang Fa Dui Bi Ke Yi Kan Chu Wo Men De Fang Fa Bu Jin Kuo Da Liao eDe Gua Yong Fan Wei Ye Ti Gao Liao dDe Gua Yong Fan Wei
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