Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

RSA Cryptosystem: A Method for Digital Signatures and Public-Key Cryptography Systems, Study notes of Computer Science

An overview of the rsa (rivest-shamir-adleman) cryptosystem, a valuable tool for online commerce and private communication. The authors, r.l. Rivest, a. Shamir, and l. Adlemen, present the algorithm and explain how to compute the public and private keys. The document also discusses related work, such as the needham-schroeder paper, diffie-hellman cryptosystem, and pkcs standards.

Typology: Study notes

Pre 2010

Uploaded on 09/24/2009

koofers-user-y0t
koofers-user-y0t 🇺🇸

10 documents

1 / 7

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
A M e t h o d f o r O b t a i n i n g D i g i t a l S i g n a t u r e s
a n d P u b l i c - K e y C r y p t o s y s t e m s
Authors: R.L. Rivest, A. Shamir, and L.
Adlemen
Presented by Justin Fidler
Published 1977
pf3
pf4
pf5

Partial preview of the text

Download RSA Cryptosystem: A Method for Digital Signatures and Public-Key Cryptography Systems and more Study notes Computer Science in PDF only on Docsity!

A M e t h o d f o r O b t a i n i n g D i g i t a l S i g n a t u r e s

a n d P u b l i c - K e y C r y p t o s y s t e m s

● Authors: R.L. Rivest, A. Shamir, and L. Adlemen ● Presented by Justin Fidler ● Published 1977

C o n t r i b u t i o n

– The RSA (Rivest Shamir Adleman)

cryptosystem is still used today

Valuable tool for online commerce and useful

for any other kind of private communication

(SSL, SSH, PGP, etc...)

– Believed to be secure (with some modern

enhancements) given a sufficiently long key

R S A A l g o r i t h m

Uses public key /private key as described

in last lecture

C ≡ E(M) ≡ M

e

(mod n)

  • (^) To encrypt a message M with public encryption key (e,n) ●

M ≡ D(C) ≡ C

d

(mod n)

  • (^) To decrypt a ciphertext C with private decryption key (d,n)

H o w t o c o m p u t e?

n?

– n must be the product of two primes p and q

  • (^) Very large, “random” primes

– n will be public, but it will be enormously

difficult to factor n

d?

– d must be a large, random integer relatively

prime to (p-1)*(q-1)

  • (^) Relatively prime - the prime factorization of d and (p-1)(q-1) share no factors ●

e?

– e*d ≡ 1 ≡ (mod(p-1)(q-1))

  • (^) e is the multiplicative inverse of d mod (p-1)(q-1)

P r o o f P a r t 2

D(E(M)) ≡ (E(M))

d

≡ (M

e

d

(mod n)

E(D(M)) ≡ (D(M))

e

≡ (M

d

e

(mod n)

  • (^) Just applying the algorithm definitions ●

M

e*d

≡ M

k* φ(n)+

(mod n) (for some k)

  • (^) ed ≡ 1 (mod φ(n); by definition of modulo ed = k* φ(n)+ ●

M

p-

≡ 1 (mod p)

  • (^) given Euler + Fermat Identity applied to a prime p ●

M

k* φ(n)+

= M (mod p)

  • (^) Mk^ φ(n)+1^ ≡ MMk^ φ(n) )^ ≡ MMk*(p-1)(q-1)^ (mod p)
  • (^) ≡ M(M(p-1))k(q-1)^ ≡ M(1)k(q-1)^ ≡ M (mod p)
  • (^) similarly for q ●

M

e*d

≡ M

k*φ(n)+

= M (mod n=p*q)

We started with M

ed

(mod n), we end up with M

(mod n). In other words, it works