{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

gaj_PKC_2008 - An Optimized Hardware Architecture for the...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
An Optimized Hardware Architecture for the Montgomery Multiplication Algorithm Miaoqing Huang 1 , Kris Gaj 2 , Soonhak Kwon 3 , Tarek El-Ghazawi 1 1 The George Washington University, Washington, D.C., U.S.A. 2 George Mason University, Fairfax, VA, U.S.A. 3 Sungkyunkwan University, Suwon, Korea
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Outline Motivation Classical Hardware Architecture for Montgomery Multiplication by Tenca and Koc from CHES 1999 Our Optimized Hardware Architecture Conceptual Comparison Implementation Results Possible Extensions Conclusions
Image of page 2
Motivation Fast modular multiplication required in multiple cryptographic transformations RSA, DSA, Diffie-Hellman Elliptic Curve Cryptosystems ECM, p-1, Pollard’s rho methods of factoring, etc. Montgomery Multiplication invented by Peter L. Montgomery in 1985 is most frequently used to implement repetitive sequence of modular multiplications in both software and hardware Montgomery Multiplication in hardware replaces division by a sequence of simple logic operations, conditional additions and right shifts
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Montgomery Modular Multiplication (1) Z = X Y mod M X Integer domain Montgomery domain X’ = X 2 n mod M Y Y’ = Y 2 n mod M Z’ = MP(X’, Y’, M) = = X’ Y’ 2 -n mod M = = (X 2 n ) (Y 2 n ) 2 -n mod M = = X Y 2 n mod M Z’ = Z 2 n mod M Z = X Y mod M X, Y, M – n-bit numbers
Image of page 4
Montgomery Modular Multiplication (2) X’ = MP(X, 2 2n mod M, M) = = X 2 2n 2 -n mod M = X 2 n mod M Z = MP(Z’, 1, M) = = (Z 2 n ) 1 2 -n mod M = Z mod M = Z X X’ Z Z’
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Montgomery Product S[0] = 0 S[i+1] = Z = S[n] S[i]+x i Y 2 S[i]+x i Y + M 2 if q i = S[i] + x i Y mod 2= 0 if q i = S[i] + x i Y mod 2= 1 for i=0 to n-1 M assumed to be odd
Image of page 6
Basic version of the Radix-2 Montgomery Multiplication Algorithm
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Classical Design by Tenca & Koc CHES 1999 Multiple Word Radix-2 Montgomery Multiplication algorithm (MWR2MM) Main ideas: Use of short precision words (w-bit each):
Image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern