Multiple modes of operation have been a popular method to improve upon the security of a blockcipher. Especially triple modes of operation are widely suggested, the most prominent example being the preferred usage of triple DES over DES in many applications.
In E. Biham [J. Cryptology 11, 45-58 (1998; Zbl 0902.94013), or see Lect. Notes Comput Sci. 917, 278-292 (1995; Zbl 0872.94027)] many multiple modes of operation were proven to be of a strength comparable to single encryption. The current paper determines the strength of all double and triple cascaded modes of operation. It is shown that of all possible cascaded double and triple modes, only the triple ECB mode is significantly more secure than single encryption. Moreover the author proposes some noncascaded modes, and conjectures these to be more secure than cascaded modes. Finally the author describes the modes proposed in the ANSI draft standard X9.52, Triple Data Encryption Algorithm Modes of Operation, revision 6.0, 1996.
The paper introduces interesting cryptanalytic techniques of combined blockciphers, and proofs a result that some might not have expected to hold.