# A Study of the Quadratic Sieve Algorithm with the Large Prime Variation for Factoring an Integer.

## Date of Submission

December 1995

## Date of Award

Winter 12-12-1996

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

## Subject Name

Computer Science

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Roy, Bimal Kumar (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

The question of divisibility is arguably the oldest problem in Mathematics. Ancient peoples observed the cycles of nature: the day, lunar month, and the year, and assumed that each divided evenly into the next. Civilizations as separate as the Egyptians of ten thousand years ago and the Central American Mayans adopted a month of thirty days and a year of twelve months. Even when the inaccuracy of a 360-day year became apparent, they preferred to retain it and add five intercalary days. The number 360 retains its psychological appeal today because it is divisible by many small integers. The teclnical term for such a mumber reflects this appeal. It is called a snooth mumber.1.2 MotivationIt is therefore surprising that a subject that is so very old should at the same time be so very new. Factorization and primality testing is a very hot area of current rescarch. Among the factors creating interest in factorization, one cannot. omit the advent of the RSA public key cryptosystem. The RSAC, as we shall henceforth address it, is an example of a public key or asymmetric encryption scheme.In a public key scheme, the encryption and decryption keys are distinct. It is computationally infeasible or extremely difficult (though not impossible !) to obtain the decryption key from the encryption key. It is called a public key encryption scheme because not only does it eliminate the danger of the encryption scheme being stolen, but the encryption scheme can now be freely published so that anyone can send in a coded message.Main Idea of RSAC:Choosen = p,q, where p.q are large primes not close to one another.e s.t gcd(e, Ã¸(n)) =1 this is the encryption key.d s.t e x d = 1 (mod n) this is the decryption key.(n,e) : public(p.q) : concealedd: privateEncoding Scheme :M = message to be sent.E = encrypted message.E = Me (mod n)Decoding Scheme :M = Ed (mod n)It can be shown that once p.q are also made public thenÃ¸ (n) = (p- 1) x (q- 1)and d can then be calculated by a method similar to the Euclidean algorithm. Thus the problem of finding d, the decryption key, has been reduced to finding the factorization of n. This gives us enough motivation why the subject of primality is a hot topic.The success of the RSAC is based on the fact that it is very difficult to factor a number into ts prime factors. For further information on the RSAC refer KÃ¶blitz[8].

## Control Number

ISI-DISS-1995-20

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/6195

## Recommended Citation

Sridhar, M. S., "A Study of the Quadratic Sieve Algorithm with the Large Prime Variation for Factoring an Integer." (1996). *Master’s Dissertations*. 366.

https://digitalcommons.isical.ac.in/masters-dissertations/366

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843455