The Journal of Korean Institute of Communications and Information Sciences (한국통신학회논문지)

The Korean Institute of Commucations and Information Sciences (한국통신학회)
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Square and Cube Root Algorithms in Finite Field and Their Applications

유한체상의 제곱근과 세제곱근을 찾는 알고리즘과 그 응용

  • Received : 2012.10.15
  • Accepted : 2012.12.03
  • Published : 2012.12.28
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Abstract

We study an algorithm that can efficiently find square roots and cube roots by modifying Tonelli-Shanks algorithm, which has an application in Number Field Sieve (NFS). The Number Field Sieve, the fastest known factoring algorithm, is a powerful tool for factoring very large integer. NFS first chooses two polynomials having common root modulo N, and it consists of the following four major steps; 1. Polynomial Selection 2. Sieving 3. Matrix 4. Square Root. The last step of NFS needs the process of square root computation in Number Field, which can be computed via square root algorithm over finite field.

Tonelli-Shanks 알고리즘을 변형한 새로운 알고리즘을 통해 효율적으로 제곱근 및 세제곱근을 찾을 수 있는 방법을 연구하였다. 이 논문에서 소개하는 제곱근을 찾는 알고리즘은 Number Field Sieve에 응용할 수 있다. 큰 합성수를 인수분해하는데 가장 효율적인 알고리즘으로 알려진 Number Field Sieve (NFS)는 법 N에 대하여 공통근을 갖는 두 다항식 선택한 후에, sieving, linear algebra, square root 단계를 차례대로 거친다. NFS의 마지막 단계에서는 수체(Number Field)상에서 제곱근을 구하는 과정이 필요한데 이를 유한체(Finite Field)상으로 내려서 계산한 후 CRT(Chinese Remainder Theorem)을 이용하여 수체 상에서의 제곱근으로 복원하는 과정에서 제안된 알고리즘이 사용될 수 있다.

Keywords

Acknowledgement

Supported by : 한국연구재단

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