Factoring large numbers using Programmable Gate array

1
Factoring large numbers using Programmable Gate
array

• Akhilesh Chaudhary 2002102
• Gaurav Gupta 2002435

2
Use of Factoring in Encryption (RSA)

• In RSA we find
• C Me mod N
• M Cd mod N Med mod N
• Using Eulers theorem
• M(kf(n) 1) M(k(p-1)(q-1) 1)
• ? M mod N
• ed kf(n) 1
• (e, N) is public, (d, N) is private key.

3
Previous work done

• Only one or two FPGA implementations of the MPQS
  algorithm have been registered uptil now. One
  such implementation was done by H.J. Kim and
  W.H.M. Smith of UCLA.
• They found out the speed of the fpga to be 28
  times faster than the 200 MHz Computer.

4
Factoring Algorithms

• The Basic Quadratic Sieve Algorithm was developed
  by C. Pomerance. It was based on the idea of
  finding two numbers X and Y such that X² ? Y² mod
  N where N is the number to be factored.
• (X-Y)(XY) ? mod N. If N pq then it can safely
  be assumed that X-Y is divided by one of them and
  XY by the other. Then the gcd of X-Y and N would
  give us one factor and that of XY and N would
  give us the other.

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An Example

• Let N 2041.
• Floor (vN) 45
• Q(x) (x45)² - 2041
• Q(1) 75 35²
• Q(2) 168 2³ 3 7
• Q(4) 360 2³ 3² 5
• Q(6) 560 24 5 7
• Q(1)Q(2)Q(4)Q(6) 210 34 54 72

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• (46474951)² ? (2532527)² mod 2041
• 46474951 5402838 X
• 2532527 50400 Y
• XY 5453238
• X-Y 5352438
• GCD (XY,N) 157
• GCD(X-Y,N) 13
• 2041 157 13

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Implementation on FPGA

• Parallel Memory Access On a single FPGA external
  memory bags can be installed and accessed
  simultaneously.
• Each memory bag will be used as a sieve.