CIT 380: Securing Computer Systems. Symmetric Cryptography

Transcription

1 CIT 380: Securing Computer Systems Symmetric Cryptography

2 Topics 1. Modular Arithmetic 2. What is Cryptography? 3. Transposition Ciphers 4. Substitution Ciphers 1. Cæsar cipher 2. Vigènere cipher 5. Cryptanalysis: frequency analysis 6. Block Ciphers 7. AES and DES 8. Stream Ciphers

3 Congruence Modular Arithmetic a = b (mod N) iff a = b + kn ex: 37=27 mod 10 b is the residue of a, modulo N Integers 0..N-1 are the set of residues mod N Modulo 12 number system

4 What is Cryptography? Cryptography: The art and science of keeping messages secure. Cryptanalysis: the art and science of decrypting messages. Cryptology: cryptography + cryptanalysis

5 Terminology Plaintext: message P to be encrypted. Also called cleartext. Plaintext Encryption: altering a message to keep its contents secret. Encryption Procedure Ciphertext: encrypted message C. Ciphertext

6 Cæsar cipher Plaintext is HELLO WORLD Change each letter to the third letter following it (X goes to A, Y to B, Z to C) Key is 3, usually written as letter D Ciphertext is KHOOR ZRUOG

7 ROT 13 Cæsar cipher with key of chosen since encryption and decryption are same operation Used to hide spoilers, punchlines, and offensive material online.

8 Kerckhoff s Principle Security of cryptosystem should only depend on 1. Quality of shared encryption algorithm E 2. Secrecy of key K Security through obscurity tends to fail ex: DVD Content Scrambling System

9 Cryptanalysis Goals 1. Decrypt a given message. 2. Recover encryption key. Threat models vary based on 1. Type of information available to adversary 2. Interaction with cryptosystem.

10 Cryptanalysis Threat Models ciphertext only: adversary has only ciphertext; goal is to find plaintext, possibly key. known plaintext: adversary has ciphertext, corresponding plaintext; goal is to find key. chosen plaintext: adversary may supply plaintexts and obtain corresponding ciphertext; goal is to find key.

11 Brute Force Attack Exhaustive search of keyspace by decrypting ciphertext C with all possible keys K. Must determine if D K (C) is a likely plaintext Requires some knowledge of format (language, doc type) For N possible keys, Worst case is N decryptions. Mean case is N/2 decryptions. Example: DES has 56-bit keys Average time to find key is 2 55 decryptions.

12 Is 128 bits enough? 128-bit keyspace permits keys 340,282,366,920,938,463,463,374,607,431,768,211,456 or 3.4 x keys Cracking 1 trillion (10 12 ) keys per second requires 3.4 x seconds or 1.08 x years Cracking 1 trillion keys per second on 1 billion CPUs requires 1.08 x years = 10.8 billion years

13 Classical Cryptography Sender and receiver share common key Keys may be the same, or be trivial to derive from one another. Sometimes called symmetric cryptography. P encrypt C decrypt P K K

14 Brute Force vs. Cæsar Cipher Brute Force attack Only 26 possible keys. PC can try all in <1s. Decryption key (26-K) Candidate plaintext 0 exxegoexsrgi 1 dwwdfndwrqfh 2 cvvcemcvqpeg 3 buubdlbupodf 4 attackatonce 5 zsszbjzsnmbd 6 yrryaiyrmlac haahjrhavujl 24 gzzgiqgzutik 25 fyyfhpfytshj

15 General Simple Substitution Cipher Keys: All permutations of alphabet (26! keys) Encryption: Replace each plaintext letter x with K(x) Decryption: Replace each ciphertext letter y with K -1 (y) Example: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z K= F U B A R D H G J I L K N M P O S Q Z W X Y V T C E CRYPTO BQCOWP

16 General Simple Substitution Security Exhaustive search impossible Key space size is 26! =~ 4 x Historically thought to be unbreakable. However, languages have different frequencies of letters digraphs (groups of 2 letters) trigraphs (groups of 3 letters) etc. Simple substitution ciphers preserve letter frequencies.

17 English Letter Frequencies

18 Additional Frequency Features Digraph frequencies Common digraphs: EN, RE, ER, NT Trigraph frequencies Common trigraphs: THE, AND, ING Digraph and trigraph tables can be found at The letter Q is followed only by U.

19 Countering Frequency Analysis Primary weakness of simple substitution: Each ciphertext letter corresponds to only one letter of plaintext. Solution: polyalphabetic substitution Use multiple cipher alphabets. Switch between cipher alphabets from character to character in the plaintext.

20 Letter Frequency Distributions

21 Reproduction of CSA Cipher Disk Vigènere Cipher Use phrase instead of letter as key. Example Message THE BOY HAS THE BALL Key VIG Encipher using Cæsar cipher for each letter: key VIGVIGVIGVIGVIGV plain THEBOYHASTHEBALL cipher OPKWWECIYOPKWIRG

22 Rotor Machines (1920s-1970s) Observation: If Vigènere key is very long, frequency analysis won t work. Implement: multiple rounds of Vigènere substitution. Machine contains multiple cylinders. Each cylinder has 26 states (ciphers.) Cylinders rotate to change states on different schedules. m-cylinder machine has 26 m substitution ciphers.

23 One-Time Pad A Vigenère cipher with a random key at least as long as the message. Provably unbreakable. Example ciphertext: DXQR. Equally likely to correspond to plaintext DOIT (key AJIY) plaintext DONT (key AJDY) and any other 4 letters.

24 Binary One Time Pad Encrypt a message M with pad P to produce ciphertext C = M P where is the exclusive OR operator. Decrypt a ciphertext C with the same pad P M = C P

25 One Time Pad Problems 1. The one-time pad must be random. Software pseudo-random number generators are not random. Pad needs hardware randomness. 2. Transmission of long pads is difficult. The pad is just as long as all the messages you ll ever send with it, so you ve just moved the problem of transmitting secret messages to transmitting a secret pad. 3. Pad must always be kept secret. If pad is ever discovered, then attacker can decrypt old messages. Pads must be securely destroyed at end of use.

26 Block Ciphers Encrypt groups (blocks) of chars at once. Improvement over single char substitution Cryptanalysis must use digraph frequencies for two-char blocks. Longer blocks are more difficult to analyze. Modern ciphers are block ciphers. Example: Playfair Cipher, 1854

27 SP-Networks Combine Substitution+Permutation (transposition) Confusion: adding unknown key values will confuse attacker about value of plaintext symbol. Diffusion: Spread plaintext data throughout ciphertext. Designing for Security Block Size Number of rounds Each input bit is XOR of several output bits from previous round. Substitution algorithm

28 Substitution Boxes Substitution can be done using a matrix, which acts as a lookup table for substituting one set of bits with another. Such tables are called substitution boxes, or S- boxes.

29 Overview of the DES Block cipher (64 bit blocks) 64-bit key is actually a 56-bit key + 8 parity bits Product cipher substitution + transposition 16 rounds (iterations) of encryption round key generated from user key

30 Feistel Function (F)

31 Differential Cryptanalysis A chosen ciphertext attack Biham and Shamir rediscovered in late 1980s Examines pairs of plaintext with particular differences. Requires 2 47 plaintext, ciphertext pairs. Only 2 14 pairs required with 8 round DES. Revealed several properties S-box designed to resist differential cryptanalysis. IBM revealed knowledge of technique at design time. Linear cryptanalysis improves result Linear approximation of DES. Requires 2 43 plaintext, ciphertext pairs. DES not designed to resist this technique.

32 Electronic Code Book Mode Encrypt each block independently. E(block) = C block each time block appears Therefore attacker can build dictionary of blocks. ECB encryption of bitmap hides colors but image is still discernible.

33 Cipher Block Chaining Mode XOR each block with previous ciphertext block. Random initialization vector (IV) used for 1 st. CBC encryption of bitmap looks random.

34 Cipher Block Chaining Mode Formula for CBC encryption (i=1 is 1 st block) Formula for CBC decryption

35 Triple DES Encrypt-Decrypt-Encrypt Mode (3 keys: k, k, k ) c = DES k (DES k 1 (DES k (m))) Middle decrypt allows backward compatibility if all keys are equal: k = k = k Double-encryption vulnerable to meet-in-middle attack, reducing difficulty from to 2 57.

36 DES is Insecure Brute force attacks can be completed in <1 day. Distributed computing attacks. RIVYERA FPGA-based parallel computer breaks DES in <1 day for a hardware cost of <$10,000. Linear cryptanalysis faster than brute force Need 2 41 known plaintexts

37 Advanced Encryption Standard (AES) Winner of open NIST competition ( ) Rijndael, designed by Joan Daemen and Vincent Rijmen. Published as FIPS 197 in November bit block cipher 128-, 192-, or 256-bit keys. 10, 12, or 14 rounds, depending on key size. Replacement for DES DES vulnerable to brute force attacks due to 56-bit keys. Triple DES is very slow.

38 AES Round Structure Round keys derived from user key using AES key schedule. Each round transforms 128-bit state, X i in 4 steps: 1. SubBytes: S-box substitution. 2. ShiftRows: permutation. 3. MixColumns: matrix multiplication. 4. AddRoundKey: XOR with round key for this round.

39 AES Round Steps

40 AES Cryptanalysis Biclique attack (2011) Faster than brute force by a factor of 4 So can break AES-128 with operations. Related key attacks (2009) Requires operations to break AES-256 Requires operations to break AES-192 Due to weak key scheduling for AES-256 AES-128 is more secure than AES-256!

41 Modern Block Ciphers Blowfish 64-bit block cipher designed in 1993 Variable key length: 32 to 448 bit Weak key attacks exist Twofish (AES finalist) 128-bit block cipher with up to 256 bit keys Designed by Blowfish team, no known breaks Serpent (AES finalist) 32-round 128-bit block cipher with up to 256 bit keys Known attacks against reduced round versions

42 Stream Ciphers Combine plaintext with cipher bitstream Cipher generates stream of pseudo-random bits Loosely inspired by one time pad. RC4 is most widely used stream cipher Variable length key: 40 to 128 bits Aircrack breaks 104-bit RC4 used in WEP in 1min No known practical attacks against 128-bit, but Much speculation that NSA can crack 128-bit RC4. Block cipher to stream cipher Using block cipher in Counter Mode. XOR counter mode data with plaintext.

43 Key Points Types of ciphers Substitution (monoalphabetic and polyalphabetic) Transposition (permutation) Product (Substitution + Permutation) Cryptanalysis Kerchoff s principle Brute force attack Find key in at most n tries, for n possible keys. Find key in an average of n/2 tries. A cryptosystem with x-bit keys has 2 x possible keys. Frequency analysis. One-time pad is provably secure Block ciphers ECB mode insecure; need to use CBC for block ciphers. DES obsolete due to small 56-bit keys. 3DES=112 bit key. AES current standard, best symmetric cipher is AES-128.

44 References 1. Ross Anderson, Security Engineering, 2 nd edition, Wiley, Matt Bishop, Introduction to Computer Security, Addison-Wesley, Neil Daswani et. al., Foundations of Security, Apress, Goodrich and Tammasia, Introduction to Computer Security, Pearson, David Kahn, The Codebreakers, MacMillan, Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, Handbook of Applied Cryptography, CRC Press, NIST, FIPS Publication 46-3: Data Encryption Standard (DES), 1999, 8. Bruce Schneier, Applied Cryptography, 2 nd edition, Wiley, US Government Dept of the Army, FM FIELD MANUAL, 1990,