public-key cryptography cs110 fall 2002. conventional encryption
TRANSCRIPT
Public-Key Cryptography
CS110
Fall 2002
Conventional Encryption
Public Key Cryptography
Instead of a single key, there is a key pair. One of the keys is kept secret (private key). The other key is made available to anyone
(public key). If one key encrypts, then the other decrypts. If one key decrypts, then the other encrypts. “Computationally infeasible” to derive the private
key from the public key.
How Does It Work?
Each user generates a pair of keys to be used for the encryption and decryption of messages.
KUb
KRb
KUa
KRa
How Does It Work?
Each user then places their public key in a public register or other available location.
KUb
KRb
KUa
KRa
Public information
How Does It Work?
If Bob wishes to send a secret message to Alice, he encrypts the message with Alice’s public key.
KUb
KRb
KUa
KRa
Public information
How are you? Encrypt
KUa
Shy3!ks8sk&0
How Does It Work? When Alice receives a message, she decrypts it with her
private key. No one else can decrypt the message except Alice because only she knows the private key!
KUb
KRb
KUa
KRa
Public information
How are you? Encrypt
KUa
Shy3!ks8sk&0
KUa
Decrypt
KRa
How are you?
Some Things to NoticeEveryone has access to everyone else’s public keys!As long as a user protects his or her private key,
communication is secure.What if a user wants to change his or her key?
KUb
KRbKRa
Public information
How are you? Encrypt
KUa
Shy3!ks8sk&0
KUa
Decrypt
KRa
How are you?
Applications for Public-Key Crypto
The first is obvious -- confidentiality!
KUb
KRbKRa
Public information
How are you? Encrypt
KUa
Shy3!ks8sk&0
KUa
Decrypt
KRa
How are you?
Only Alicecan decrypt
Applications for Public-Key Crypto
What happens if Bob encrypts his message with his private key and sends it to Alice?
KUb
KRbKRa
Public information
How are you? Encrypt
KUa
Jly3^ks6sk%9
KRb
Decrypt How are you?
Anyone canDecrypt. Why?
KUb
Applications for Public-Key CryptoThis application is called a digital signature because
Bob signs his message with his private key. How can Alice be assured that it really came from Bob?
KUb
KRbKRa
Public information
How are you? Encrypt
KUa
Jly3^ks6sk%9
KRb
Decrypt How are you?
Anyone canDecrypt. Why?
KUb
KUb
Requirements for Public-Key Cryptography It is computationally easy to generate the
key pairs. It is computationally easy for a sender to
encrypt a message, knowing the recipient’s public key.
It is computationally easy for a receiver to decrypt a message using their own private key.
Requirements for Public-Key Cryptography (cont) It is computationally infeasible for an opponent,
knowing the public key to determine the private key.
It is computationally infeasible for an opponent, knowing the public key and ciphertext, to recover the original message.
The keys should be inverses of one another.
An Example – RSA (Generate Keys)
1 – Generate 2 very large (random)primes, p and q
2 – Calculate n = p x q
3 – Calculate Φ(n) = (p-1)(q-1)
4 – Select integer e, such that gcd(Φ(n), e)=1
5 – Calculate d, where d·e=1 mod Φ(n)
6 – The public key, KU = {e, n}
7 – The private key, KR = {d, n}
An Example – RSA
Encryption
Plaintext: M < n
Ciphertext: C=Me mod n
Decryption
Ciphertext: C
Plaintext: M=Cd mod n
RSA – Simple Example
Find the Keys
Assume we pick the random primes:
p = 3, q = 11
Generate a public/private key pair.n = p x q = 3 * 11 = 33
Φ(n) = (p-1)(q-1) = 20
Select e, such that gcd(Φ(n), e)= 1 = 7
Calculate d, where d·7=1 mod 20
RSA – Simple Example
How do we calculate d, where d·7=1 mod 20One way is to try numbers:
d = 1 1 * 7 = 7 mod 20
d = 2 2 * 7 = 14 mod 20
d = 3 3 * 7 = 21 mod 20 = 1 mod 20
So, d = 3.
RSA – Simple Example
The public key, KU = {e, n} = {7, 33}
The private key, KR = {d, n} = {3, 33}
Now, suppose we want to encrypt the message M = 24.
Ciphertext: C=Me mod n = 247 mod 33 = 18
RSA – Simple Example
How does one decrypt a message? Use the previous example. Suppose you receive the ciphertext C = 18.
Plaintext: M=Cd mod n = 183 mod 33 = 24