1. Introduction to Symmetric-key and Public-key Cryptography

2. Overview
Cryptography: secure communication over insecure communication channels
Three goals
Confidentiality
Only authorized people can see the message
Integrity
If the message is tampered by an attacker, we can know
Authenticity
We can check whether a message is from a given person

3. Brief History of Crypto
2,000 years ago
Caesar Cypher: shifting each letter forward by a fixed amount
Encode and decode by hand
During World War I/II
Mechanical era: a mechanical device for encrypting messages
After World War II
Modern cryptography: rely on mathematics and electronic computers

4. Plaintext and ciphertext
Plaintext is the message before encryption
Ciphertext is the encrypted message

5. Modern Cryptography Symmetric-key cryptography
The same secret key is used by both endpoints of a communication
Public-key cryptography (or asymmetric cryptography)
Two endpoints use different keys

6. Roadmap Symmetric-key cryptography Public-key cryptography
Confidentiality
Block cipher
Stream cipher
Integrity & authenticity
HMAC
Public-key cryptography
Public-key encryption
Digital signature

7. Symmetric-key cryptography :Block Cipher
Encrypt/Decrypt messages in fixed size blocks using the same secret key
k-bit secret key
n-bit plaintext/ciphertext
n bits
n bits
E, D
Plaintext Block
Ciphertext Block
Key
k Bits

8. Examples of Block Cipher
DES - Data Encryption Standard (1977)
Works on 64 bit block with 56 bit keys
Developed by IBM (Lucifer) improved by NSA
Brute force attack feasible in 1997
AES – Advanced Encryption Standard (1997)
Block size 128 bits
Key can be 128, 192, or 256 bits

9. Modes of Operation Block ciphers encrypt fixed size blocks
eg. DES encrypts 64-bit blocks with 56-bit key
Need to en/decrypt arbitrary amounts of data
NIST SP A defines 5 modes of operation
Divide a long message into blocks and encypt each one using a block cipher

10. Stream cipher For a message with a length k
Generate a key with length k
This is often a pseudo-random bit stream generated from a master secret key
Encryption and decryption are simple
Examples
RC4 (insecure)
ChaCha20-Poly1305

11. Hash functions Properties Variable input size
Fixed output size (e.g., 512 bits)
Efficient to compute
Pseudo-random (mixes up input well)

12. Cryptographic hash functions
Cryptogtaphic hash functions add conditions
Preimage resistance
Given h, intractable to find y such that H(y)=h
Second preimage resistance
Given x, intractable to find y≠x such that H(y)=H(x)
Collision resistance
Intractable to find x, y such that y≠x and H(y)=H(x)

13. We have a cryptographic hash function crisis
Popular hash function MD5
Thoroughly broken
Government standard function SHA-1, SHA-2
Theoretical weaknesses
“New” cryptographic hash function SHA-3
Too new to fully evaluate
Maybe good enough

14. Message Integrity: Hashed Message Authentication Codes (HMACs)
Goal: provide message integrity and authenticity
ex: Protecting public binaries on disk. k k
Message m
tag
Alice
Bob
Generate tag:
tag  S(k, m)
Verify tag:
V(k, m, tag) = `yes’ ?

15. HMAC Most widely used MAC on the Internet. H: hash function.
example: SHA-256; output is 256 bits
Building a MAC out of a hash function:
Standardized method: HMAC
S( k, m ) = H( kopad , H( kipad , m ) )
opad, ipad: fixed strings

16. Public key encryption
Def: a public-key encryption system is a triple of algs. (G, E, D)
G(): randomized alg. outputs a key pair (pk, sk)
E(pk, m): randomized alg. that takes m∈M and outputs c ∈C
D(sk, c): det. alg. that takes c∈C and outputs m∈M or ⊥
Consistency: ∀(pk, sk) output by G :
∀m∈M: D(sk, E(pk, m) ) = m

17. Building Block: Trapdoor Functions (TDF)
Def: a trapdoor function over X is a triple of efficient algs. (G, F, F-1)
G(): randomized alg. outputs a key pair (pk, sk)
F(pk,⋅): deterministic alg. that defines a function X ⟼ Y
F-1(sk,⋅): defines a function Y ⟼ X that inverts F(pk,⋅)
Security: (G, F, F-1) is secure if F(pk, ⋅) is a “one-way” function:
given F(pk, x) and pk it is difficult to find x
for all x in X: F-1(sk, F(pk, x) ) = x

18. Example TDF: RSA set N  pq (3072 bits  925 digits)
alg. G(): generate two equal length primes p, q
set N  pq (3072 bits  925 digits)
set e  = 65537
set d  e-1 (mod (N))
pk = (N, e) ; sk = (N, d)
RSA(pk, x) : x  (xe mod N)
Inverting this function is believed to be as hard as factoring N
RSA-1(sk, y) : y  (yd mod N)

19. Public Key Encryption with a TDF
G(): generate pk and sk
E(pk, m):
choose random x  domain(F) and set k  H(x)
c0  F(pk, x) , c1  E(k, m) (E: symm. cipher)
send c = (c0, c1)
D(sk, c=(c0,c1) ): x  F-1(sk, c0) , k  H(x) , m  D(k, c1) c0 c1

20. Digital signature Bind message to author
An author uses its private key to create a signature
Signature is different for different message
Anyone can verify the signature with the author’s public key
It is hard for attackers to forge a signature for a message without knowing the private key

21. Digital Signatures: applications
Software distribution
Windows Update File
Microsoft’s signature on file

22. Certificates How we can know a public key belongs to the author?
Use certificate
A trusted third party issues a digital certificate saying that a public key belongs to the author
Widely used in Internet