A novice cryptosystem based on n<sup>th</sup> root of Gaussian integers

Bhargava, Kriti; Soni, Vineeta

doi:10.1109/comptelix.2017.8003977

Cited by 2 publications

(3 citation statements)

References 7 publications

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“…Without loss of generality we consider two elements x = c + di and y = e + f i from the first quadrant for the product in (19). This implies To determine the maximum value, we consider the bivariate function g(e, c) = eN + cN − 2ce and its partial derivatives ∂g(e, c) ∂e = N − 2c = 0, (22) ∂g(e, c)…”

Section: Algorithm 3 Precision Reduction For Gaussian Integersmentioning

confidence: 99%

“…Due to the isomorphism between Gaussian integer rings and ordinary integer rings, this arithmetic is suitable for many cryptography systems. The Rabin cryptography system and elliptic curve cryptography over Gaussian integers were considered in [19][20][21][22]. Moreover, coding applications over Gaussian integers use Gaussian integer arithmetic [23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

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Montgomery Reduction for Gaussian Integers

Safieh

Freudenberger

2021

Cryptography

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Modular arithmetic over integers is required for many cryptography systems. Montgomery reduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Montgomery reduction is used for rings of ordinary integers. In contrast, we investigate the modular reduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real and imaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings. In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Two algorithms for the precision reduction are presented. We demonstrate that the proposed Montgomery reduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptography. In particular, we consider the elliptic curve point multiplication according to the randomized initial point method which is protected against side-channel attacks. The implementation of this protected point multiplication is significantly faster than comparable algorithms over ordinary prime fields.

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Section: Algorithm 3 Precision Reduction For Gaussian Integersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Montgomery Reduction for Gaussian Integers

Safieh

Freudenberger

2021

Cryptography

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“…It was shown in [9][10][11][12] that a significant complexity reduction can be achieved for the RSA cryptographic system using Gaussian integers. Similarly, Rabin cryptographic system was previously considered over Gaussian integers in [13,14]. In the context of ECC, arithmetic over complex numbers was proposed to speed up the point multiplication [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers

2020

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This work presents a new concept to implement the elliptic curve point multiplication (PM). This computation is based on a new modular arithmetic over Gaussian integer fields. Gaussian integers are a subset of the complex numbers such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this arithmetic is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of secure hardware implementations, which are robust against attacks. Furthermore, an area-efficient coprocessor design is proposed with an arithmetic unit that enables Montgomery modular arithmetic over Gaussian integers. The proposed architecture and the new arithmetic provide high flexibility, i.e., binary and non-binary key expansions as well as protected and unprotected PM calculations are supported. The proposed coprocessor is a competitive solution for a compact ECC processor suitable for applications in small embedded systems.

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A novice cryptosystem based on n^th root of Gaussian integers

Cited by 2 publications

References 7 publications

Montgomery Reduction for Gaussian Integers

Montgomery Reduction for Gaussian Integers

A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers

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A novice cryptosystem based on nth root of Gaussian integers

Cited by 2 publications

References 7 publications

Montgomery Reduction for Gaussian Integers

Montgomery Reduction for Gaussian Integers

A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers

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A novice cryptosystem based on n^th root of Gaussian integers