Modern Cryptography

1
CIT 380 Securing Computer Systems

• Modern Cryptography

2
Overview

• Cryptographic Checksums
• Hash Functions
• HMAC
• Number Theory Review
• Public Key Cryptography
• One-Way Trapdoor Functions
• Diffie-Helman
• RSA
• Modern Steganography

3
Hash Functions

• Checksum to verify data integrity.
• Hash Function h A?B
• Input A variable length
• Output B fixed length fingerprint of input
• Many inputs produce same output.
• Example Hash Function
• Sum 32-bit words of message mod 232.

4
Hash Function ASCII Parity

• ASCII parity bit
• ASCII has 7 bits 8th bit is for parity
• Even parity even number of 1 bits
• Odd parity odd number of 1 bits
• Bob receives 10111101 as bits.
• Sender is using even parity 6 1 bits, so
  character was received correctly
• Note could be garbled, but 2 bits would need to
  have been changed to preserve parity
• Sender is using odd parity even number of 1
  bits, so character was not received correctly

5
Cryptographic Checksums

• Hash with authentication/integrity protection
• Cannot obtain original message from hash.
• Cannot find another message with same hash.
• Additional Names
• Message Authentication Code
• Message Digest

6
One-Way Function

• Function f easy to compute, hard to reverse
• Given x, easy to calculate f(x).
• Given f(x), hard to compute x.
• Whats easy and whats hard?
• easy polynomial time
• hard exponential time
• Are there any one-way functions?

7
Cryptographic Checksum Definition

• A function h A?B such that
• For any x IN A, h(x) is easy to compute.
• For any y IN B, it is computationally infeasible
  to find x IN A such that h(x) y.
• It is computationally infeasible to find x, x IN
  A such that x ? x and h(x) h(x).

8
Collisions

• If x ? x and h(x) h(x), x and x collide.
• Pigeonhole principle if there are n containers
  for n1 objects, then at least one container will
  have 2 objects in it.
• Application suppose n 5 and k 3. Then there
  are 32 elements of A and 8 elements of B, so at
  least one element of B has at least 4
  corresponding elements of A.

9
Hash Function Examples

• Input
• Cryptography
• Output (base64 encoded)
• http//www.xml-dev.com/blog/sha1.php
• MD5 (128-bit)
• 64ef07ce3e4b420c334227eecb3b3f4c
• SHA1 (160-bit)
• b804ec5a0d83d19d8db908572f51196505d09f98

10
MD5/ SHA-1 Hash Web site

• http//www.fileformat.info/tool/hash.htm

11
Keyed Hash Function

• Hash function secret key
• Why?
• Authentication
• How?
• HMAC algorithm
• Incorporate key into a keyless hash algorithm.

12
HMAC

• HMAC Hash Function Key
• Inputs
• h keyless cryptographic checksum function that
  takes data in blocks of b bytes and outputs
  blocks of l bytes.
• k cryptographic key.
• k k modified to be of length b.
• If short, pad with 0 bytes.
• If long, hash to length b.

13
HMAC

• HMAC-h(k, m) h(k ? opad h(k ? ipad m))
• ? exclusive or
• concatenation
• ipad is 00110110 repeated b times.
• opad is 01011100 repeated b times.
• Security depends on security of hash function h.

14
Current State of Hash Functions

• MD4, MD5, SHA-0 Collisions (2004)
• SHA-1 Collisions (2005)
• Effort required is 269 instead of 280.
• No effective pre-image attacks discovered yet.
• Whats the impact?
• Attacker could create two documents.
• Document A requires payment of 500.
• Document B requires payment of 50,000.
• Digital signatures sign MAC, not document.
• Both documents have same MAC.
• Use SHA-256 for now.