Residue arithmetic systems in cryptography: a survey on modern security applications

Schoinianakis, Dimitrios

doi:10.1007/s13389-020-00231-w

Cited by 21 publications

(13 citation statements)

References 60 publications

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“…The Residue Number System (RNS) has been widely applied to efficient carry-free arithmetic operations in classical computers without the carry propagation problem [28][29][30]. RNS-based arithmetic modulo 2 n − 1 computation is one of the most common RNS operations that is used in pseudorandom number generation and various cryptographic algorithms [31,32]. The basic idea for modulo 2 n − 1 operation is to use the constant value −(2 n − 1) to be added by the sum of two numbers A and B, where the two inputs A and B are to be in the range {0, 2 n − 2}.…”

Section: Residue Number Systemmentioning

confidence: 99%

Quantum Modular Adder over GF(2n − 1) without Saving the Final Carry

et al. 2021

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Addition is the most basic operation of computing based on a bit system. There are various addition algorithms considering multiple number systems and hardware, and studies for a more efficient addition are still ongoing. Quantum computing based on qubits as the information unit asks for the design of a new addition because it is, physically, wholly different from the existing frequency-based computing in which the minimum information unit is a bit. In this paper, we propose an efficient quantum circuit of modular addition, which reduces the number of gates and the depth. The proposed modular addition is for the Galois Field GF(2n−1), which is important as a finite field basis in various domains, such as cryptography. Its design principle was from the ripple carry addition (RCA) algorithm, which is the most widely used in existing computers. However, unlike conventional RCA, the storage of the final carry is not needed due to modifying existing diminished-1 modulo 2n−1 adders. Our proposed adder can produce modulo sum within the range 0,2n−2 by fewer qubits and less depth. For comparison, we analyzed the proposed quantum addition circuit over GF(2n−1) and the previous quantum modular addition circuit for the performance of the number of qubits, the number of gates, and the depth, and simulated it with IBM’s simulator ProjectQ.

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Section: Residue Number Systemmentioning

confidence: 99%

Quantum Modular Adder over GF(2n − 1) without Saving the Final Carry

et al. 2021

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“…The three main areas of application are signal processing [9], [10], [11], cryptography but also in theoretical computer science to reach complexity bounds [12], [13]. The work in this paper is relevant for all applications on large numbers including cryptography since the 90's [14] with RSA, DH, ECC [15], [16], pairing [17], Euclidean lattices, homomorphic protocols [18], [19], etc.…”

Section: Introductionmentioning

confidence: 99%

Generating Very Large RNS Bases

Bajard

Fukushima

Plantard³

et al. 2022

IEEE Trans. Emerg. Topics Comput.

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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“…We aim to directly generate randomly distributed integers with the desired precision using the PWLCM. The proposed algorithm combines chaos, modular arithmetic, and lattice-based cryptography [3,8,35]. The latter allows to easily extend the external key length without duplication.…”

Section: Introductionmentioning

confidence: 99%

An extendable key space integer image-cipher using 4-bit piece-wise linear cat map

Nkuigwa

Nzeuga

Fouda

et al. 2022

Multimed Tools Appl

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This paper presents a multiplierless image-cipher, with extendable 2048-bit key-space, based on a 4-dimensional (4D) quantized piece-wise linear cat map (PWLCM). The quantized PWLCM exhibits limit-cycles of 4-bit encoded integers with periods greater than 107. The synthesis of the PWLCM in a finite state space allows to eliminate the undesirable finite precision effect due to the hardware realization. The proposed image-cipher combines chaos, modular arithmetic, and lattice-based cryptography to encrypt a color image by performing pixel permutation and diffusion in a single operation. Further, an image-dependent confusion operation based on an 8-bit 2D-PWLCM is performed on the whole image to enhance security. In order to increase the key-space without key duplication, 16 × 16 sub-images are modified using sub-keys of different lattice length vectors generated from the external key. Both simulations and security analyses confirm that the proposed algorithm can resist common cipher attacks, in addition to its advantages such as simplicity, ease of implementation on low-end processors and extensibility of key-space that allows it to easily adapt even for future post-quantum computing attacks.

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Residue arithmetic systems in cryptography: a survey on modern security applications

Cited by 21 publications

References 60 publications

Quantum Modular Adder over GF(2n − 1) without Saving the Final Carry

Quantum Modular Adder over GF(2n − 1) without Saving the Final Carry

Generating Very Large RNS Bases

An extendable key space integer image-cipher using 4-bit piece-wise linear cat map

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