Asymmetric encryption

Asymmetric encryption

• Asymmetric encryption, often called "public key" encryption, allows Alice to send Bob an encrypted message without a shared secret key

• there is a secret key, but only Bob knows what it is, and he does not share it with anyone, including Alice

Asymmetric encryption

Asymmetric encryption

• Bob creates a pair of keys, one of which he keeps secret and one of which he sends to Alice.

• Alice composes a confidential message and encrypts it using the key that Bob has sent to her.

• Alice sends the encrypted data to Bob.• Bob uses his secret key to decrypt the data and

reads the confidential message.

Asymmetric encryption

• Asymmetric algorithms include a "key generation" protocol that Bob uses to create his key pair, as shown by Figure 15-2. Following the protocol results in the creation of a pair of keys that have a mathematical relationshipthe exact detail of the protocol and the relationship between the keys is different for each algorithm.

Asymmetric encryption

Creating Asymmetric Keysusing RSA Algorithm

• each user generates a public/private key pair by: • selecting two large primes at random - p, q • computing their system modulus N=p.q

– note ø(N)=(p-1)(q-1) • selecting at random the encryption key e

• where 1<e<ø(N), gcd(e,ø(N))=1

• solve following equation to find decryption key d – e.d=1 mod ø(N) and 0≤d≤N Or d = e -1 (mod ( (n))

• publish their public encryption key: KU={e,N} • keep secret private decryption key: KR={d,p,q}

Example of RSA

Ciphertext C = Me mod nPlaintext M = Cd mod n

Example:• p = 17 , q = 11• n = 178• ø(N)= 160• e = 7• D = 23• PU = { 7 , 187 } ; PR = { 23 , 187 }• M= 88; C =88 7 mod 187 = 11• C = 1123 mod 187 = 88

Encryption

Decryption

Solve this example

• Start with p= 61 q=53