# Construction of RSA Cryptosystem over the Algebraic Field Using Ideal Theory and Investigation of Its Security

Tsuyoshi Takagi, Shozo Naito

### 抄録

The RSA cryptosystem is extended to the algebraic field by using ideal theory. In this paper, we describe the generation algorithm of prime ideals, how to select a representative class using an ideal as modulus, and the algorithm to compute a representative element, for cyclotomic fields and quadratic fields. From here, an RSA cryptosystem can be constructed on cyclotomic fields and quadratic fields. To break completely the proposed cryptosystem, when the public key is a product of inert prime numbers, is the same as for the conventional RSA cryptosystem and its security is better against the Håstad attacks.

元の言語 英語 19-29 11 Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi) 83 8 https://doi.org/10.1002/(SICI)1520-6440(200008)83:8<19::AID-ECJC3>3.0.CO;2-0 出版済み - 1 1 2000

Cryptography

### All Science Journal Classification (ASJC) codes

• Electrical and Electronic Engineering

### これを引用

@article{a14403b4c3614c19bacda7ec5b4ebbfd,
title = "Construction of RSA Cryptosystem over the Algebraic Field Using Ideal Theory and Investigation of Its Security",
abstract = "The RSA cryptosystem is extended to the algebraic field by using ideal theory. In this paper, we describe the generation algorithm of prime ideals, how to select a representative class using an ideal as modulus, and the algorithm to compute a representative element, for cyclotomic fields and quadratic fields. From here, an RSA cryptosystem can be constructed on cyclotomic fields and quadratic fields. To break completely the proposed cryptosystem, when the public key is a product of inert prime numbers, is the same as for the conventional RSA cryptosystem and its security is better against the H{\aa}stad attacks.",
author = "Tsuyoshi Takagi and Shozo Naito",
year = "2000",
month = "1",
day = "1",
doi = "10.1002/(SICI)1520-6440(200008)83:8<19::AID-ECJC3>3.0.CO;2-0",
language = "English",
volume = "83",
pages = "19--29",
journal = "Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)",
issn = "1042-0967",
publisher = "John Wiley and Sons Inc.",
number = "8",

}

TY - JOUR

T1 - Construction of RSA Cryptosystem over the Algebraic Field Using Ideal Theory and Investigation of Its Security

AU - Takagi, Tsuyoshi

AU - Naito, Shozo

PY - 2000/1/1

Y1 - 2000/1/1

N2 - The RSA cryptosystem is extended to the algebraic field by using ideal theory. In this paper, we describe the generation algorithm of prime ideals, how to select a representative class using an ideal as modulus, and the algorithm to compute a representative element, for cyclotomic fields and quadratic fields. From here, an RSA cryptosystem can be constructed on cyclotomic fields and quadratic fields. To break completely the proposed cryptosystem, when the public key is a product of inert prime numbers, is the same as for the conventional RSA cryptosystem and its security is better against the Håstad attacks.

AB - The RSA cryptosystem is extended to the algebraic field by using ideal theory. In this paper, we describe the generation algorithm of prime ideals, how to select a representative class using an ideal as modulus, and the algorithm to compute a representative element, for cyclotomic fields and quadratic fields. From here, an RSA cryptosystem can be constructed on cyclotomic fields and quadratic fields. To break completely the proposed cryptosystem, when the public key is a product of inert prime numbers, is the same as for the conventional RSA cryptosystem and its security is better against the Håstad attacks.

UR - http://www.scopus.com/inward/record.url?scp=0042730598&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042730598&partnerID=8YFLogxK

U2 - 10.1002/(SICI)1520-6440(200008)83:8<19::AID-ECJC3>3.0.CO;2-0

DO - 10.1002/(SICI)1520-6440(200008)83:8<19::AID-ECJC3>3.0.CO;2-0

M3 - Article

AN - SCOPUS:0042730598

VL - 83

SP - 19

EP - 29

JO - Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)

JF - Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)

SN - 1042-0967

IS - 8

ER -