Problems in Cryptography and Cryptanalysis

The integer factorization problem used in the RSA cryptosystem, the discrete logarithm problem used in Diffie-Hellman Key Exchange protocol and the Elliptic Curve Discrete Logarithm problem used in Elliptic Curve Cryptography are traditionally considered the difficult problems and used extensively in the design of cryptographic algorithms. We provide a number of other computationally difficult problems in the areas of Cryptography and Cryptanalysis. A class of problems called the Search problems, Group membership problems, and the Discrete Optimization problems are examples of such problems. A number of computationally difficult problems in Cryptanalysis have also been identified including the Cryptanalysis of Block ciphers, Pseudo-Random Number Generators and Hash functions.

Membership Problem for Two-Dimensional General Row Jumping Finite Automata

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

On the Membership Problem of Permutation Grammars — A Direct Proof of NP-Completeness

One of the most essential classes of problems related to formal languages is the membership problem (also called word problem), i.e., to decide whether a given input word belongs to the language specified, e.g., by a generative grammar. For context-free languages the problem is solved efficiently by various well-known parsing algorithms. However, there are several important languages that are not context-free. The membership problem of the context-sensitive language class is PSPACE-complete, thus, it is believed that it is generally not solvable in an efficient way. There are various language classes between the above mentioned two classes having membership problems with various complexity. One of these classes, the class of permutation languages, is generated by permutation grammars, i.e., context-free grammars extended with permutation rules, where a permutation rule allows to interchange the position of two consecutive nonterminals in the sentential form. In this paper, the membership problem for permutation languages is studied. A proof is presented to show that this problem is NP-complete.

Problems in Cryptography and Cryptanalysis

The integer factorization problem used in the RSA cryptosystem, the discrete logarithm problem used in Diffie-Hellman Key Exchange protocol and the Elliptic Curve Discrete Logarithm problem used in Elliptic Curve Cryptography are traditionally considered the difficult problems and used extensively in the design of cryptographic algorithms. We provide a number of other computationally difficult problems in the areas of Cryptography and Cryptanalysis. A class of problems called the Search problems, Group membership problems, and the Discrete Optimization problems are examples of such problems. A number of computationally difficult problems in Cryptanalysis have also been identified including the Cryptanalysis of Block ciphers, Pseudo-Random Number Generators and Hash functions.