Unformatted text preview: is known to the public
i.e. it is computationally infeasible to derive D( ) by knowing E( )
D( b y
Instead, they were able to find a way for communication parties to establish
a shared secret via open communications only
=> This is the Diffie-Hellman Key exchange algorithm The Beginning of Public Key Cryptography in
“public” world (cont’d) Diffie and Hellman published their ideas and findings in “New
Directions in Cryptography” Nov ’76, together with the open problem of
Ron Rivest saw Diffie and Hellman’s paper and was intrigued by it.
He enlisted the help of Shamir and Adleman, all from MIT, to work on
the open problem and came up with the solution in 1977 --- this is the
Diffie, Hellman, Merkle, Rivest, Shamir, Adleman were commonly
recognized as the founders of Public Key Cryptography. RSA Algorithm
Ron Rivest, Adi Shamir, Len Adleman – found the
functions and published the results in 1978:
D[E[m]] = m = E[D[m]]
D[E[m Most widely accepted and implemented approach to
public key encryption Block cipher where m = plaintext ; and c =ciphertext
are integers, between 0 <= m , c <= n-1 for some n
This is the E[ ] Following form:
This is the D[ ]
c = me mod n
m = cd mod n
Public key is (n,e). Private key is (n,d). RSA: Choosing keys
1. Choose two large prime numbers p, q.
(e.g., 1024 bits each)
2. Compute n = pq, z = (p-1)(q-1)
3. Choose e (with e<n) that has no common factors
with z. (e, z are “relatively prime”).
4. Choose d such that ed-1 is exactly divisible by z.
(in other words: ed –1 = K * z for some integer K ,
i.e. , ed = K * z + 1,
in other words: If ed is divided by z, the remainder
is equal to 1, i.e., ed mod z = 1 ).
5. Public key is (n,e). Private key is (n,d).
priv RSA: Encryption, decryption
0. Given (n,e) and (n,d) as computed above
1. To encrypt bit pattern, m, compute e mod n (i.e., remainder when m e is divided by n)
c=m 2. To decrypt received bit pattern, c, compute d
m = c d mod n (i.e., remainder when c is divided by n)
m = (m e mod n) d mod n
c RSA example:
Bob chooses p=5, q=7. Then n=5*7 =35, z=(5-1)*(7-1) =24
e=5 (so e, z relatively prime).
d=29 (so ed-1 = 5*29 – 1 =144 which is exactly divisible by z.) encrypt: decrypt: letter m me D 4 1024 =
29 * 35 + 9 c
c c = me mod n
m = cd mod n letter
29 929= 4710128697246………
Use the following property to compute : 9 mod 35
(a * b) mod n = [(a mod n) * (b mod n)] mod n, i.e.
929 mod 35 = 910+10+9 mod 35
= [(910 mod 35) * (910 mod 35) * (99 mod 35) ] mod 35
= (16 * 16 * 29) mod 35 = 4 m = (m e mod n) d mod n RSA: Why is that Useful number theory result: If p,q prime and
n = pq, then:
y mod (p-1)(q-1)
x mod n = x
mod n e
(m mod n) d mod n = medmod n
=m ed mod (p-1)(q-1) mod n (using number theory result above) 1 = m mod n
(since we chose ed to be divisible by
(p-1)(q-1) with remainder 1 ) = m (since m < n, thus m mod n = m) RSA: Important properties
It is infeasible to determine d given e and n
K (K (m)) = m = K (K (m)) priv pub pub priv use public key
by private key use private ke...
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- Spring '13
- Cryptography, Public-key cryptography