Solving a Class of Modular Polynomial Equations and its Relation to Modular Inversion Hidden Number Problem and Inversive Congruential Generator

In this paper we revisit the modular inversion hidden number problem (MIHNP) and the inversive congruential generator (ICG) and consider how to attack them more efficiently. We consider systems of modular polynomial equations of the form a_{ij}+b_{ij}x_i+c_{ij}x_j+x_ix_j=0 (mod p) and show the relation between solving such equations and attacking MIHNP and ICG. We present three heuristic strategies using Coppersmith\u27s lattice-based root-finding technique for solving the above modular equations. In the first strategy, we use the polynomial number of samples and get the same asymptotic bound on attacking ICG proposed in PKC 2012, which is the best result so far. However, exponential number of samples is required in the work of PKC 2012. In the second strategy, a part of polynomials chosen for the involved lattice are linear combinations of some polynomials and this enables us to achieve a larger upper bound for the desired root. Corresponding to the analysis of MIHNP we give an explicit lattice construction of the second attack method proposed by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001. We provide better bound than that in the work of PKC 2012 for attacking ICG. Moreover, we propose the third strategy in order to give a further improvement in the involved lattice construction in the sense of requiring fewer samples

SAFE-NET: Secure and Fast Encryption using Network of Pseudo-Random Number Generators

We propose a general framework to design a general class of random number generators suit- able for both computer simulation and computer security applications. It can include newly pro- posed generators SAFE (Secure And Fast Encryption) and ChaCha, a variant of Salsa, one of the four finalists of the eSTREAM ciphers. Two requirements for ciphers to be considered se- cure is that they must be unpredictable with a nice distributional property. Proposed SAFE-NET is a network of n nodes with external pseudo-random number generators as inputs nodes, several inner layers of nodes with a sequence of random variates through ARX (Addition, Rotation, XOR) transformations to diffuse the components of the initial state vector. After several rounds of transformations (with complex inner connections) are done, the output layer with n nodes are outputted via additional transformations. By utilizing random number generators with desirable empirical properties, SAFE-NET injects randomness into the keystream generation process and constantly updates the cipher’s state with external pseudo-random numbers during each iteration. Through the integration of shuffle tables and advanced output functions, extra layers of security are provided, making it harder for attackers to exploit weaknesses in the cipher. Empirical results demonstrate that SAFE-NET requires fewer operations than ChaCha while still producing a sequence of uniformly distributed random numbers

Neural Networks as Pseudorandom Number Generators

This thesis brings two disparate fields of research together; the fields of artificial neural networks and pseudorandom number generation. In it, we answer variations on the following question: can recurrent neural networks generate pseudorandom numbers? In doing so, we provide a new construction of an $n$-dimensional neural network that has period $2^n$, for all $n$. We also provide a technique for constructing neural networks based on the theory of shift register sequences. The randomness capabilities of these networks is then measured via the theoretical notion of computational indistinguishability and a battery of statistical tests. In particular, we show that neural networks cannot be pseudorandom number generators according to the theoretical definition of computational indistinguishability. We contrast this result by providing some neural networks that pass all of the tests in the SmallCrush battery of tests in the TestU01 testing suite