Review Overview of Cryptography Classical Symmetric Cipher Substitution Cipher Transposition Cipher Product Cipher Modern Symmetric Ciphers (DES)

Basic Terminology plaintext - the original message ciphertext - the coded message cipher - algorithm for transforming plaintext to ciphertext key - info used in cipher known only to sender/receiver encipher (encrypt) - converting plaintext to ciphertext decipher (decrypt) - recovering ciphertext from plaintext cryptography - study of encryption principles/methods cryptanalysis (codebreaking) - the study of principles/ methods of deciphering ciphertext without knowing key cryptology - the field of both cryptography and cryptanalysis Briefly review some terminology used throughout the course.

Feistel Cipher Structure Feistel cipher implements Shannon’s S-P network concept based on invertible product cipher Process through multiple rounds which partitions input block into two halves perform a substitution on left data half based on round function of right half & subkey then have permutation swapping halves Horst Feistel, working at IBM Thomas J Watson Research Labs devised a suitable invertible cipher structure in early 70's. One of Feistel's main contributions was the invention of a suitable structure which adapted Shannon's S-P network in an easily inverted structure. Essentially the same h/w or s/w is used for both encryption and decryption, with just a slight change in how the keys are used. One layer of S-boxes and the following P-box are used to form the round function.

DES (Data Encryption Standard) Published in 1977, standardized in 1979. Key: 64 bit quantity=8-bit parity+56-bit key Every 8th bit is a parity bit. 64 bit input, 64 bit output. 64 bit M 64 bit C DES Encryption 56 bits

DES Top View …... 56-bit Key 64-bit Input 48-bit K1 Generate keys Permutation Initial Permutation 48-bit K1 Round 1 48-bit K2 Round 2 …... 48-bit K16 Round 16 Swap Swap 32-bit halves Permutation Final Permutation 64-bit Output

Bit Permutation (1-to-1) 1 2 3 4 32 ……. 0 0 1 0 1 Input: 1 bit Output …….. 1 0 1 1 1 22 6 13 32 3

Per-Round Key Generation Initial Permutation of DES key C i-1 28 bits D i-1 28 bits Circular Left Shift Circular Left Shift One round Permutation with Discard Round 1,2,9,16: single shift Others: two bits 48 bits Ki C i D i 28 bits 28 bits

A DES Round One Round Encryption 32 bits Ln 32 bits Rn E 48 bits Mangler Function 48 bits Ki S-Boxes P 32 bits 32 bits Ln+1 32 bits Rn+1

Mangler Function 4 6 + S8 S1 S2 S7 S3 S4 S5 S6 Permutation The permutation produces “spread” among the chunks/S-boxes!

Bits Expansion (1-to-m) 1 2 3 4 5 32 Input: ……. 0 0 1 0 1 1 Output …….. 1 0 0 1 0 1 0 1 1 0 1 2 3 4 5 6 7 8 48

S-Box (Substitute and Shrink) 48 bits ==> 32 bits. (8*6 ==> 8*4) 2 bits used to select amongst 4 substitutions for the rest of the 4-bit quantity 2 bits row S i = 1,…8. I1 I2 I3 I4 I5 I6 O1 O2 O3 O4 4 bits column

DES Standard One round (Total 16 rounds) Cipher Iterative Action : Input: 64 bits Key: 48 bits Output: 64 bits Key Generation Box : Input: 56 bits Output: 48 bits One round (Total 16 rounds)

DES Box Summary Simple, easy to implement: Hardware/gigabits/second, software/megabits/second 56-bit key DES may be acceptable for non- critical applications but triple DES (DES3) should be secure for most applications today Supports several operation modes (ECB CBC, OFB, CFB) for different applications

Outlines Strength/weakness of DES, AES Public Key Cryptography Modular Arithmetic RSA

Avalanche Effect Key desirable property of encryption alg Where a change of one input or key bit results in changing more than half output bits DES exhibits strong avalanche

Strength of DES – Key Size 56-bit keys have 256 = 7.2 x 1016 values Brute force search looks hard Recent advances have shown is possible in 1997 on a huge cluster of computers over the Internet in a few months in 1998 on dedicated hardware called “DES cracker” by EFF in a few days ($220,000) in 1999 above combined in 22hrs! Still must be able to recognize plaintext No big flaw for DES algorithms DES finally and definitively proved insecure in July 1998, when the Electronic Frontier Foundation (EFF) announced that it had broken a DES encryption using a special-purpose "DES cracker" machine that was built for less than $250,000. The attack took less than three days. The EFF has published a detailed description of the machine, enabling others to build their own cracker [EFF98].

DES Replacement Triple-DES (3DES) Advanced Encryption Standards (AES) 168-bit key, no brute force attacks Underlying encryption algorithm the same, no effective analytic attacks Drawbacks Performance: no efficient software codes for DES/3DES Efficiency/security: bigger block size desirable Advanced Encryption Standards (AES) US NIST issued call for ciphers in 1997 Rijndael was selected as the AES in Oct-2000 The AES candidates are the latest generation of block ciphers, and now we see a significant increase in the block size - from the old standard of 64-bits up to 128-bits; and keys from 128 to 256-bits. In part this has been driven by the public demonstrations of exhaustive key searches of DES. Whilst triple-DES is regarded as secure and well understood, it is slow, especially in s/w.

AES Private key symmetric block cipher 128-bit data, 128/192/256-bit keys Stronger & faster than Triple-DES Provide full specification & design details Evaluation criteria Security: effort to practically cryptanalysis Cost: computational efficiency and memory requirement Algorithm & implementation characteristics: flexibility to apps, hardware/software suitability, simplicity

AES Shortlist After testing and evaluation, shortlist in Aug- 99: MARS (IBM) - complex, fast, high security margin RC6 (USA) - v. simple, v. fast, low security margin Rijndael (Belgium) - clean, fast, good security margin Serpent (Euro) - slow, clean, v. high security margin Twofish (USA) - complex, v. fast, high security margin Then subject to further analysis & comment The AES shortlist of 5 ciphers was as shown. Note mix of commercial (MARS, RC6, Twofish) verses academic (Rijndael, Serpent) proposals, sourced from various countries. All were thought to be good – it came down to the best balance of attributes to meet criteria, in particular the balance between speed, security & flexibility.

Outlines Strength/weakness of DES, AES Public Key Cryptography Modular Arithmetic RSA

Private-Key Cryptography Private/secret/single key cryptography uses one key Shared by both sender and receiver If this key is disclosed communications are compromised Also is symmetric, parties are equal Hence does not protect sender from receiver forging a message & claiming is sent by sender So far all the cryptosystems discussed have been private/secret/single key (symmetric) systems. All classical, and modern block and stream ciphers are of this form.

Public-Key Cryptography Probably most significant advance in the 3000 year history of cryptography Uses two keys – a public & a private key Asymmetric since parties are not equal Uses clever application of number theoretic concepts to function Complements rather than replaces private key crypto Will now discuss the radically different public key systems, in which two keys are used. Anyone knowing the public key can encrypt messages or verify signatures, but cannot decrypt messages or create signatures, counter-intuitive though this may seem. It works by the clever use of number theory problems that are easy one way but hard the other. Note that public key schemes are neither more secure than private key (security depends on the key size for both), nor do they replace private key schemes (they are too slow to do so), rather they complement them.

Public-Key Cryptography Public-key/two-key/asymmetric cryptography involves the use of two keys: a public-key, which may be known by anybody, and can be used to encrypt messages, and verify signatures a private-key, known only to the recipient, used to decrypt messages, and sign (create) signatures Asymmetric because those who encrypt messages or verify signatures cannot decrypt messages or create signatures

Public-Key Cryptography Stallings Fig 9.1

Public-Key Characteristics Public-Key algorithms rely on two keys with the characteristics that it is: computationally infeasible to find decryption key knowing only algorithm & encryption key computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known either of the two related keys can be used for encryption, with the other used for decryption (in some schemes) Analogy to delivery w/ a padlocked box Public key schemes utilise problems that are easy (P type) one way but hard (NP type) the other way, eg exponentiation vs logs, multiplication vs factoring. Consider the following analogy using padlocked boxes: traditional schemes involve the sender putting a message in a box and locking it, sending that to the receiver, and somehow securely also sending them the key to unlock the box. The radical advance in public key schemes was to turn this around, the receiver sends an unlocked box to the sender, who puts the message in the box and locks it (easy - and having locked it cannot get at the message), and sends the locked box to the receiver who can unlock it (also easy), having the key. An attacker would have to pick the lock on the box (hard).

Public-Key Cryptosystems Stallings Fig 9.4 Here see various components of public-key schemes used for both secrecy and authentication. Note that separate key pairs are used for each of these – receiver owns and creates secrecy keys, sender owns and creates authentication keys. Two major applications: encryption/decryption (provide secrecy) digital signatures (provide authentication)

Outlines Strength/weakness of DES, AES Public Key Cryptography Modular Arithmetic RSA

Modular Arithmetic Public key algorithms are based on modular arithmetic. Modular addition. Modular multiplication. Modular exponentiation.

Modular Addition Addition modulo (mod) K Poor cipher with (dk+dm) mod K, e.g., if K=10 and dk is the key. Additive inverse: addition mod K yields 0. “Decrypt” by adding inverse. + 1 2 3 4 5 6 7 8 9

Modular Multiplication Multiplication modulo K Multiplicative inverse: multiplication mod K yields 1 Only some numbers have inverse * 1 2 3 4 5 6 7 8 9

Modular Multiplication Only the numbers relatively prime to n will have mod n multiplicative inverse x, m relative prime: no other common factor than 1 Eg. 8 & 15 are relatively prime - factors of 8 are 1,2,4,8 and of 15 are 1,3,5,15 and 1 is the only common factor

Totient Function Totient function ø(n): number of integers less than n relatively prime to n if n is prime, ø(n)=n-1 if n=pq, and p, q are primes, p != q ø(n)=(p-1)(q-1) E.g., ø(37) = 36 ø(21) = (3–1)×(7–1) = 2×6 = 12 Numbers which are relative prime to n: 1, 2, … p-1 p+1, p+2, … 2p-1 2p+1, 2p+2, … 3p-1 (q-1)p +1, (q-1)p+2, … qp-1

Modular Exponentiation xy 1 2 3 4 5 6 7 8 9 Any patterns in the table? Shortcut for computing the modular exponentiation?

Modular Exponentiation xy mod n = xy mod ø(n) mod n if y = 1 mod ø(n) then xy mod n = x mod n

Outlines Strength/weakness of DES, AES Public Key Cryptography Modular Arithmetic RSA

RSA (Rivest, Shamir, Adleman) The most popular one. Support both public key encryption and digital signature. Assumption/theoretical basis: Factoring a big number is hard. Variable key length (usually 512 bits). Variable plaintext block size. Plaintext must be “smaller” than the key. Ciphertext block size is the same as the key length.

What Is RSA? To generate key pair: Pick large primes (>= 256 bits each) p and q Let n = p*q, keep your p and q to yourself! For public key, choose e that is relatively prime to ø(n) =(p-1)(q-1), let pub = <e,n> For private key, find d that is the multiplicative inverse of e mod ø(n), i.e., e*d = 1 mod ø(n), let priv = <d,n>

RSA Example Select primes: p=17 & q=11 Compute n = pq =17×11=187 Select e : gcd(e,160)=1; choose e=7 Determine d: de=1 mod 160 and d < 160 Value is d=23 since 23×7=161= 10×160+1 Publish public key KU={7,187} Keep secret private key KR={23,17,11} Here walk through example using “trivial” sized numbers. Selecting primes requires the use of primality tests. Finding d as inverse of e mod ø(n) requires use of Inverse algorithm (see Ch4)

How Does RSA Work? Given pub = <e, n> and priv = <d, n> encryption: c = me mod n, m < n decryption: m = cd mod n signature: s = md mod n, m < n verification: m = se mod n given message M = 88 (nb. 88<187) encryption: C = 887 mod 187 = 11 decryption: M = 1123 mod 187 = 88

Why Does RSA Work? Given pub = <e, n> and priv = <d, n> n =p*q, ø(n) =(p-1)(q-1) e*d = 1 mod ø(n) xed = x mod n encryption: c = me mod n decryption: m = cd mod n = med mod n = m mod n = m (since m < n) digital signature (similar)

Is RSA Secure? Factoring 512-bit number is very hard! But if you can factor big number n then given public key <e,n>, you can find d, hence the private key by: Knowing factors p, q, such that, n = p*q Then ø(n) =(p-1)(q-1) Then d such that e*d = 1 mod ø(n) Threat Moore’s law Refinement of factorizing algorithms For the near future, a key of 1024 or 2048 bits needed

Symmetric (DES) vs. Public Key (RSA) Exponentiation of RSA is expensive ! AES and DES are much faster 100 times faster in software 1,000 to 10,000 times faster in hardware RSA often used in combination in AES and DES Pass the session key with RSA