Common Modulus and Chosen-Message Attacks on Public-Key Schemes with Linear Recurrence Relations

We consider the linear recurrence relation Vt (x)=∑mi=1(aix+bi )Vt−i(x)+ cx+f wherem> 1, ai and bi , 16 i 6m, are integers. The RSA and LUC schemes can be defined by this relation. In this paper we show that if the linear recurrence relation has some properties, the public-key scheme based on it cannot withstand the common modulus and chosen-message attacks, no matter what the order m is and what the parameters for ai and bi , 16 i 6m, are. This implies that the LUC cryptosystem cannot withstand the common modulus attack and the LUC digital signature scheme cannot withstand the chosen-message attack.  1999 Elsevier Science B.V. All rights reserved.