Basic Cryptography Ge Zhang What is Cryptography Cryptography - - PDF document

Basic Cryptography

Ge Zhang Karlstad University

What is Cryptography

Cryptography Cryptosystem: 5-tuple (M, C, E, D, K)

M: the set of plaintexts C: the set of ciphertexts E: M x K -> C enciphering functions D: C x K -> M deciphering functions K: the set of keys

Example: Caesar cipher

• 00000000001111111111222222
• 01234567890123456789012345
• …ABCDEFGHIJKLMNOPQRSTUVWXYZ

M={all sequences of Roman letters} K={i | i is an integer such that

0<=i<=25}

E=(m+k) mod 26 D=(c-k) mod 26

Relative Frequency of Letters in English Text

Example

Break it! WKHIDNHUDQGWKHZDONHUPHH

WQHAWZHHN

Example: Vigenère cipher

Transportation cipher

Recorder the plaintext letters Plain text: attack on tomorrow Key: 4312567 Same letter frequencies as the original

plaintext

A taxonomy of Cryptosystems

Operations

• Substitution ciphers
• Transposition ciphers

Number of Keys used

• Symmetric
• Asymmetric (public key)

The way in which the plaintext is processed

• Block cipher
• Stream cipher

Attacks on Cryptosystems

Cryptanalysis Brute-force attack: tries every possible

key

Computational secure

Time Cost

Classical Feistel Network

Block size Key size Number of rounds Subkey generation

algorithm

Round function (F)

The Data Encryption Standard (DES)

Block size: 64 bit Key size: 56 bit Subkey generation

• 56bit key->16x48bit

subkeys

Round time: 16 S-boxes: 16 X 4 Permutation rule:

Round function (F) of DES

DES-- avalanche effect

Strong avalanche effect 2 Plaintext

0000 0000 …. 0000 0000 1000 0000 …. 0000 0000

Encrypted with the same key, 34 bits

different

Input 1.5% difference Output 53% difference

Weakness of DES

Design in 1970s 56 bit key: 2 56 = 7.2 X 10 16 Brute force

1142 years, 1 decryption/us 10 hrs, 106 decryptions/us

Triple DES

Asymmetric Key Cryptography

The problems of symmetric key? Asymmetric Key Cryptography

Private/secret key Public key

RSA algorithm

Asymmetric Key Cryptography

The RSA algorithm

• each user generates a public/private key pair by:
• selecting two large primes at random - p,q
• computing n=p.q
• define 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
• ed mod ø(n)= 1 and 0≤d≤n
• publish their public encryption key: PU={e,n}
• keep secret private decryption key: PR={d,n}

The RSA algorithm

to encrypt a message M the sender:

• btains public key of recipient

PU={e,n}

computes: C = Me mod n, where 0≤M<n

to decrypt the ciphertext C the owner:

uses their private key PR={d,n} computes: M = Cd mod n

Public-Key Applications

can classify uses into 3 categories:

encryption/decryption (provide

confidentiality)

digital signatures (provide

authentication)

key exchange (of session keys)

Message Authentication

message authentication is concerned

with:

protecting the integrity of a message validating identity of originator

then two alternative functions used:

hash function message authentication code (MAC)

Hash Functions

a Hash Function produces a digest of

some file/message/data

h = H(M)

Input a variable-length message M

• utput a fixed-sized digest h

usually assume that the hash function

is public and not keyed

Usage of hash:

Requirements for Hash Functions

1.

produces fixed-length output h

2.

is easy to compute h=H(M) for any message M

3.

given h is infeasible to find x s.t. H(x)=h

• ne-way property

4.

given x is infeasible to find y s.t. H(y)=H(x)

• weak collision resistance

5.

is infeasible to find any x,y s.t. H(y)=H(x)

• strong collision resistance

Pigeonhole principle

Message Authentication Code (MAC)

• Keyed hash

depending on both message and some key like encryption though need not be reversible

• appended to message as a signature
• receiver performs same computation on

message and checks it matches the MAC

• provides assurance that message is unaltered

and comes from sender

Questions

Lab assignment

Secure communication

Blowfish

characteristics

• fast
• Less memory
• Easy to implement
• varying key size
• Allows tuning for

speed/security tradeoff

Blowfish Key Schedule

Block size either 64 bit or 128 bit uses a 32 to 448 bit key 16 rounds Subkey generation Round function

Initialize s-boxes and p-arrays

Initialize subkeys before

en(de)crypting

Update:

P-array S-boxes

Algorithm of blowfish

Blowfish Encryption

uses two primitives: addition & XOR data is divided into two 32-bit halves L0 & R0

for i = 1 to 16 do

Ri = Li-1 XOR Pi; Li = F[Ri] XOR Ri-1;

L17 = R16 XOR P18; R17 = L16 XOR i17;

where

F[a,b,c,d] = ((S1,a + S2,b) XOR S3,c) + S4,a Break 32-bit Ri into (a,b,c,d)

F function

Diffie-Hellman Algorithm

Global Public Elements

q: Prime number α: α < q and α is a primitive root of q What is a primitive root of Prime

number q?

One whose power modulo q generate

all the integers from 1 to q-1

User A Key Generation

Select private XA

XA < q

Calculate public YA

YA = α X

A

mod q

User B Key Generation

Select private XB

XB < q

Calculate public YB

YB = α X

B

mod q

Diffie-Hellman Key Exchange

Diffie-Hellman Key Exchange

XA -> YA : YA = α X

A mod q

YA -> XA : XA = d logα,q (YA)

Discrete logarithm

Notice

To simplify, we use fixed Global Public

Elements q = 353, α = 3.

The session key for blowfish should be

with 64 bit length. (Depends on your

• wn design)

Deadline: 20th Dec 2009.