In this paper, we establish a residue theorem for Malcev-Neumann series that requires few constraints, and includes previously known combinatorial residue theorems as special cases. Our residue theorem identifies the residues of two formal series that are related by a change of variables. We obtain simple conditions for when a change of variables is possible, and find that the two related forma...

We consider the division of a finite number of homogeneous divisible items among three players. Under the assumption that each player assigns a positive value to every item, we characterize the optimal allocations and we develop two exact algorithms for its search. Both the characterization and the algorithm are based on the tight relationship two geometric objects of fair division: the Individ...

There has been a tendency to use the theory of finite Galois fields, or GF(2n), in cryptographic ciphers (AES, Kuznyechik) and digital signal processing (DSP) systems. It is advisable modular codes polynomial residue number system (PRNS). Modular PRNS are arithmetic which addition, subtraction multiplication operations performed parallel on bases code, irreducible polynomials. In this case, ope...

We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by means of an RNS Montgomery multiplication algorithm that uses the RNS on the layer below. As a result, the actual arithmetic is deferred to the bottom layer. The...

This thesis tackles the problem of data conversion in the Residue Number System (RNS). The RNS has been considered as an interesting theoretical topic for researchers in recent years. Its importance stems from the absence of carry propagation between its arithmetic units. This facilitates the realization of high-speed, low-power arithmetic. This advantage is of paramount importance in embedded ...

Novel radixmoduloarithmetic units for residue number system (RNS)-based architectures are introduced in this paper. The proposed circuits are shown to require several times less area than previously reported architectures for particular moduli of operation, while also being preferable in the area time complexity sense. The complexity reduction is achieved by extending the carry-ignore property ...

This paper presents mathematical foundtions of the Overlap Resolution Number System (ORNS) which employs signed Continuous Valued Digits (CVD’s). ORNS is a redundant Number System employing residue arithmetic. In contrast to the implementation of arithmetic by binary or multiple-valued logic circuits, arithmetic operations in this novel number system are performed by analog digit manipulation c...