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
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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
1 / 7
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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