scholarly journals Montgomery Reduction for Gaussian Integers

Cryptography ◽  
2021 ◽  
Vol 5 (1) ◽  
pp. 6
Author(s):  
Malek Safieh ◽  
Jürgen Freudenberger
Keyword(s):  
Elliptic Curve ◽  
Initial Point ◽  
Modular Arithmetic ◽  
Complex Numbers ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Point Multiplication ◽  
Prime Fields ◽  
Integer Rings ◽  
Montgomery Reduction

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.

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

Electronics ◽  
2020 ◽  
Vol 9 (12) ◽  
pp. 2050
Author(s):  
Malek Safieh ◽  
Johann-Philipp Thiers ◽  
Jürgen Freudenberger
Keyword(s):  
Elliptic Curve ◽  
Modular Arithmetic ◽  
Complex Numbers ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Point Multiplication ◽  
Hardware Implementations ◽  
Secure Hardware ◽  
Prime Fields ◽  
High Flexibility

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.

scholarly journals The Division Algorithm in Complex Bases

1996 ◽  
Vol 39 (1) ◽  
pp. 47-54 ◽  
Author(s):  
William J. Gilbert
Keyword(s):  
Iterated Function System ◽  
Complex Numbers ◽  
Integer Part ◽  
Division Algorithm ◽  
Fractal Boundary ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Linear Maps ◽  
Iterated Function ◽  
Long Division

AbstractComplex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b,D) ∩ ℤ[i], where T(b,D) is the set of complex numbers with zero integer part in the base. The set T(b,D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph.

Study of Safe Elliptic Curve Cryptography over Gaussian Integer

2020 ◽  
Vol E103.A (12) ◽  
pp. 1624-1628
Author(s):  
Kazuki NAGANUMA ◽  
Takashi SUZUKI ◽  
Hiroyuki TSUJI ◽  
Tomoaki KIMURA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Gaussian Integer

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

Elliptic Curve Cryptography on WISPs

2013 ◽  
pp. 562-583
Author(s):  
Michael Hutter ◽  
Erich Wenger ◽  
Markus Pelnar ◽  
Christian Pendl
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Public Key ◽  
Wireless Sensor ◽  
Binary Field ◽  
Montgomery Ladder ◽  
Sensing Platforms ◽  
Cryptographic Primitive ◽  
Prime Fields ◽  
Dedicated Hardware

In this chapter, the authors explore the feasibility of Elliptic Curve Cryptography (ECC) on Wireless Identification and Sensing Platforms (WISPs). ECC is a public-key based cryptographic primitive that has been widely adopted in embedded systems and Wireless Sensor Networks (WSNs). In order to demonstrate the practicability of ECC on such platforms, the authors make use of the passively powered WISP4.1DL UHF tag from Intel Research Seattle. They implemented ECC over 192-bit prime fields and over 191-bit binary extension fields and performed a Montgomery ladder scalar multiplication on WISPs with and without a dedicated hardware multiplier. The investigations show that when running at a frequency of 6.7 MHz, WISP tags that do not support a hardware multiplier need 8.3 seconds and only 1.6 seconds when a hardware multiplier is supported. The binary-field implementation needs about 2 seconds without support of a hardware multiplier. For the WISP, ECC over prime fields provides best performance when a hardware multiplier is available; binary-field based implementations are recommended otherwise. The use of ECC on WISPs allows the realization of different public-key based protocols in order to provide various cryptographic services such as confidentiality, data integrity, non-repudiation, and authentication.

Elliptic Curve Cryptography on WISPs

2013 ◽  
pp. 120-143
Author(s):  
Michael Hutter ◽  
Erich Wenger ◽  
Markus Pelnar ◽  
Christian Pendl
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Public Key ◽  
Wireless Sensor ◽  
Binary Field ◽  
Montgomery Ladder ◽  
Sensing Platforms ◽  
Cryptographic Primitive ◽  
Prime Fields ◽  
Dedicated Hardware

In this chapter, the authors explore the feasibility of Elliptic Curve Cryptography (ECC) on Wireless Identification and Sensing Platforms (WISPs). ECC is a public-key based cryptographic primitive that has been widely adopted in embedded systems and Wireless Sensor Networks (WSNs). In order to demonstrate the practicability of ECC on such platforms, the authors make use of the passively powered WISP4.1DL UHF tag from Intel Research Seattle. They implemented ECC over 192-bit prime fields and over 191-bit binary extension fields and performed a Montgomery ladder scalar multiplication on WISPs with and without a dedicated hardware multiplier. The investigations show that when running at a frequency of 6.7 MHz, WISP tags that do not support a hardware multiplier need 8.3 seconds and only 1.6 seconds when a hardware multiplier is supported. The binary-field implementation needs about 2 seconds without support of a hardware multiplier. For the WISP, ECC over prime fields provides best performance when a hardware multiplier is available; binary-field based implementations are recommended otherwise. The use of ECC on WISPs allows the realization of different public-key based protocols in order to provide various cryptographic services such as confidentiality, data integrity, non-repudiation, and authentication.

Profiled Attacks Against the Elliptic Curve Scalar Point Multiplication Using Neural Networks

2021 ◽  
pp. 238-257
Author(s):  
Alessandro Barenghi ◽  
Diego Carrera ◽  
Silvia Mella ◽  
Andrea Pace ◽  
Gerardo Pelosi ◽  
...  
Keyword(s):  
Neural Networks ◽  
Elliptic Curve ◽  
Point Multiplication ◽  
Scalar Point Multiplication
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