The authors would like to extend their apologies for the inadvertent inclusion of an erroneous index of the matrix${M}$for DIZY-80 in[1]. We sincerely regret any inconvenience caused by this typographical error and appreciate the chance to rectify it.

We introduce a new construction method of diffusion layers for Substitution Permutation Network (SPN) structures along with its security proofs. The new method can be used in block ciphers, stream ciphers, hash functions, and sponge constructions. Moreover, we define a new stream cipher mode of operation through a fixed pseudorandom permutation and provide its security proofs in the indistinguishability model. We refer to a stream cipher as a Small Internal State Stream (SISS) cipher if its internal state size is less than twice its key size. There are not many studies about how to design and analyze SISS ciphers due to the criterion on the internal state sizes, resulting from the classical tradeoff attacks. We utilize our new mode and diffusion layer construction to design an SISS cipher having two versions, which we call DIZY. We further provide security analyses and hardware implementations of DIZY. In terms of area cost, power, and energy consumption, the hardware performance is among the best when compared to some prominent stream ciphers, especially for frame-based encryptions that need frequent initialization. Unlike recent SISS ciphers such as Sprout, Plantlet, LILLE, and Fruit; DIZY does not have a keyed update function, enabling efficient key changing.

As the need for lightweight cryptography has grown even more due to the evolution of the Internet of Things, it has become a greater challenge for cryptographers to design ultra lightweight stream ciphers in compliance with the rule of thumb that the internal state size should be at least twice as the key size to defend against generic Time-Memory-Data Tradeoff (TMDT) attacks. However, Recently in 2015, Armknecht and Mikhalev sparked a new light on designing keystream generators (KSGs), which in turn yields stream ciphers, with small internal states, called KSG with Keyed Update Function (KSG with KUF), and gave a concrete construction named Sprout. But, currently, security analysis of KSGs with KUF in a general setting is almost non-existent. Our contribution in this paper is two-fold. 1) We give a general mathematical setting for KSGs with KUF, and for the first time, analyze a class of such KSGs, called KSGs with Boolean Keyed Feedback Function (KSG with Boolean KFF), generically. In particular, we develop two generic attack algorithms applicable to any KSG with Boolean KFF having almost arbitrary output and feedback functions where the only requirement is that the secret key incorporation is biased. We introduce an upper bound for the time complexity of the first algorithm. Our extensive experiments validate our algorithms and assumptions made thereof. 2) We study Sprout to show the effectiveness of our algorithms in a practical instance. A straightforward application of our generic algorithm yields one of the most successful attacks on Sprout.