1. Introduction to Public Key Cryptography
Lecture 4
CPSC415 Biometrics and Cryptography

2. CPSC415 Biometrics and Cryptography
Outline
Public Key Encryption
Public Key Cryptographic System
Public Key vs. Symmetric Key
Digital Signatures
Digital Envelope
CPSC415 Biometrics and Cryptography

3. Insufficiencies with Symmetric Encryption
Symmetric encryption is not enough to address two key issues
key distribution – how to have secure communications in general without having to trust a KDC with your key?
digital signatures – how to verify that a received message really comes from the claimed sender?
CPSC415 Biometrics and Cryptography

4. Advent of Asymmetric Encryption
Probably most significant advance in the 3000 year history of cryptography
Use two keys: a public key and a private key
Asymmetric since parties are not equal
Clever application of number theory concepts instead of merely substitution and permutation
CPSC415 Biometrics and Cryptography

5. How Asymmetric Encryption Works
Asymmetric encryption uses two keys that are related to each other
a public key, which may be known to anybody, is used to encrypt messages, and verify signatures
a private key, known only to the owner, is used to decrypt messages encrypted by the matching public key, and create signatures
the key used to encrypt messages or verify signatures cannot decrypt messages or create signatures
CPSC415 Biometrics and Cryptography

6. CPSC415 Biometrics and Cryptography
Public key Encryption
Alice has a key pair: public and private
publish the public key such that the key is publicly known
Alice keeps the private key secret
Other people use Alice’s public key to encrypt messages for Alice
Alice uses her private key to decrypt
Only Alice can decrypt since only Alice has the private key
Public key
Message
Encrypt
rfwekfs
Private key
Message
Decrypt
rfwekfs
Trick: To compute the private key from the public key is a difficult problem.
CPSC415 Biometrics and Cryptography

7. Asymmetric Encryption for Confidentiality
Bob
Alice
CPSC415 Biometrics and Cryptography

8. Asymmetric Encryption for Authentication
Bob
Alice
CPSC415 Biometrics and Cryptography

9. Applications for Asymmetric Encryption
Three categories
Encryption/decryption: sender encrypts a message with receiver’s public key
Digital signature: sender “signs” a message with its private key
Key exchange: two sides exchange a session key
CPSC415 Biometrics and Cryptography

10. Security of Asymmetric Encryption
Like symmetric schemes brute-force exhaustive search attack is always theoretically possible, but keys used are too large (>512bits)
Not more secure than symmetric encryption, dependent on size of key
Security relies on a large enough difference in difficulty between easy (en/decrypt) and hard (cryptanalyse) problems
Generally the hard problem is known, just made too hard to do in practice
Require using very large numbers, so is slow compared to symmetric schemes
CPSC415 Biometrics and Cryptography

11. Public key Cryptographic System
cryptanalysis M
private key
Eve M M C
encryption
decryption
Alice
Alice’s public key
private key
Public key directory
C = EPK(M)
M = DSK(C) = DSK(EPK(M))
Public keys are published.
Each private key is known to the receiver only.
Difficult for Eve to find out SK from PK.
CPSC415 Biometrics and Cryptography

12. Public key vs. Symmetric key
Two parties MUST trust each other
Two parties DO NOT need to trust each other
Both share same key
Two separate keys: a public and a private key
(or one key is computable from the other)
Typically faster
Typically slower
Examples:
DES, IDEA, RC5, CAST, AES, …
Examples:
RSA, ElGamal Encryption, ECC…
CPSC415 Biometrics and Cryptography

13. CPSC415 Biometrics and Cryptography
Digital signatures
Is there a functional equivalent to a handwritten signature?
Easy for legitimate user to sign
But hard for anyone else to forge
Easy for anyone to verify
Dependent on message & signer (key)
Public key!
Sign: “invert” function using private key
Verify: compute function using public key
CPSC415 Biometrics and Cryptography

14. CPSC415 Biometrics and Cryptography
Digital signatures
Private key
Sign
Message
rfwekfs
(fixed-length signature)
Public key
Message
Verify
Valid/Invalid
rfwekfs
Only the signer (who has a private key) can generate a valid signature
Everyone (since the corresponding public key is published) can verify if a signature with respect to a message is valid
CPSC415 Biometrics and Cryptography

15. Digital Envelopes -- Symmetric + Asymmetric
Generate a secret key (session key) at random.
Encrypt the message using the session key and symmetric algorithm.
Encrypt the session key with the recipient’s public key. This becomes the “digital envelope”.
Send the encrypted message and the digital envelope to the recipient.
Figure …
CPSC415 Biometrics and Cryptography

16. CPSC415 Biometrics and Cryptography
Digital Envelopes
Session Key
Session Key
Cipher
Plain
Cipher
Plain
Digital Envelope
Digital Envelope
Session Key
Recipient’s
Public key
Session Key
CPSC415 Biometrics and Cryptography

17. CPSC415 Biometrics and Cryptography
RSA
CPSC415 Biometrics and Cryptography

18. CPSC415 Biometrics and Cryptography
Motivation
Revision
One problem with symmetric key algorithms is that the sender needs a secure method of telling the receiver the key.
Plus, you need a separate key for everyone you might communicate with.
Public key algorithms use a public-key and private-key pair to tackle key management problem.
Each receiver has a public key pair.
The public key is publicly known (published).
A sender uses the receiver’s public key to encrypt a message.
Only the receiver can decrypt it with the corresponding private key.
CPSC415 Biometrics and Cryptography

19. CPSC415 Biometrics and Cryptography
RSA
Invented by Rivest, Shamir & Adleman of MIT in 1977
Best known and widely used public-key scheme
Based on exponentiation in a finite (Galois) field over integers modulo a prime
exponentiation takes O((log n)3) operations (easy)
Use large integers (e.g bits)
Security due to cost of factoring large numbers
factorization takes O(e log n log log n) operations (hard)
CPSC415 Biometrics and Cryptography

20. CPSC415 Biometrics and Cryptography
RSA Key Setup
Each user generates a public/private key pair by
select two large primes at random: p, q
compute their system modulus n=p·q
note ø(n)=(p-1)(q-1)
select 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
publish their public encryption key: KU= {e,n}
keep secret private decryption key: KR= {d,n}
CPSC415 Biometrics and Cryptography

21. CPSC415 Biometrics and Cryptography
RSA Usage
To encrypt a message M:
sender obtains public key of receiver KU={e,n}
computes: C=Me mod n, where 0≤M<n
To decrypt the ciphertext C:
receiver uses its private key KR={d,n}
computes: M=Cd mod n
Message M must be smaller than the modulus n (cut into blocks if needed)
CPSC415 Biometrics and Cryptography

22. RSA Example: Computing Keys
Select primes: p=17, q=11
Compute n=pq=17×11=187
Compute ø(n)=(p–1)(q-1)=16×10=160
Select e: gcd(e,160)=1 and e<160
choose e=7
Determine d: de=1 mod 160 and d<160
d=23 since 23×7=161=10×160+1
Publish public key KU={7,187}
Keep secret private key KR={23,187}
CPSC415 Biometrics and Cryptography

23. RSA Example: Encryption and Decryption
Given message M = 88 (88<187)
Encryption KU={7,187} :
C = 887 mod 187 = 11
Decryption KR={23,187} :
M = 1123 mod 187 = 88
CPSC415 Biometrics and Cryptography