Foundations of Cryptography: Basics and Beyond

1. Chapter 1: Foundations Dulal C. Kar Based on Applied Cryptography by Schneier

2. Terminology • Sender and Receiver • Messages and Encryption • Encryption and decryption • Plaintext and ciphertext • Cryptography and cryptographers • Cryptanalysis and cryptanalysts • Cryptology and cryptologists

3. Encryption and Decryption • Notations • Message, M • Plaintext, P • Ciphertext, C • Encryption function, E • E(M) = C • Decryption function, D • D(C) = M • Identity:D(E(M)) = M Plaintext Original Plaintext Ciphertext Encryption Decryption

4. Authentication, Integrity, and Nonrepudiation • In addition to providing confidentiality (privacy), other cryptographic services are: • Authentication • Ascertaining a message’s true origin • Integrity • Verifying that a message has not been modified in transit • Nonrepudiation • A sender should not be able to falsely deny later that he sent a message

5. Algorithms and Keys • Cryptographic algorithm • Also called cipher • Mathematical function for encryption and decryption • Restricted algorithm • Secret algorithm, popular for low security applications, no standard • Modern cryptography • Key, K • Keyspace • Range of possible values of the key

6. Key-Based Cryptography • Key-based encryption and decryption • Ek(M) = C • Dk(C) = M • Identity • Dk(Ek(M)) = M Key Key Plaintext Original Plaintext Ciphertext Encryption Decryption

7. Different Keys for Encryption and Decryption • Encryption key, K1 • Decryption key, K2 • Ek1(M) = C • Dk2(C) = M • Dk2(Ek1(M)) = M Decryption Key Encryption Key Plaintext Original Plaintext Ciphertext Encryption Decryption

8. Cryptosystem • An algorithm, plus all possible plaintexts, ciphertexts, and keys

9. Symmetric Algorithms • Same key for encryption and decryption • Also called, secret-key algorithms, one-key algorithms,single-key algorithms • Ek(M) = C • Dk(C) = M • Two categories • Stream algorithms or stream ciphers (operate on a single bit or sometimes bytes) • Block algorithms or block ciphers (a typical block size is 64 bits for modern computer algorithms)

10. Public-Key Algorithms • Key used for encryption is different from key used for decryption • Encryption key is often called the public key • Decryption key is often called the private key • Decryption key cannot be calculated from encryption key in any reasonable amount of time • Also called asymmetric algorithms • Ek1(M) = C • Dk2(C) = M • For digital signatures, messages are encrypted with the private key and decrypted with the public key

11. Cryptanalysis • Science of recovering plaintext of a message without access to the key • An attempted cryptanalysis is called an attack • Kerckhoffs’ assumptions in cryptanalysis • Cryptographic algorithm and implementation known • Secrecy resides only in the key

12. Cryptanalytic Attacks • Ciphertext-only attack • Given ciphertext of several messages, recover plaintext, key(s), or algorithm • Formally Given:C1 = Ek(P1), C2 = Ek(P2), . . ., Ci = Ek(Pi) Deduce:Either P1, P2, . . . Pi; k; or an algorithm to infer Pi+1 from Ci+1 = Ek(Pi+1) • Known-plaintext attack • Given ciphertext and corresponding plaintext of several messages, deduce the key(s) or algorithm • Formally Given:(P1, C1), (P2, C2), . . ., (Pi, Ci) where Ci = Ek(Pi) Deduce:Either k, or an algorithm to infer Pi+1 from Ci+1 = Ek(Pi+1)

13. Cryptanalytic Attacks (cont’d) • Chosen-plaintext attack Given: (P1, C1), (P2, C2), . . ., (Pi, Ci) where the cryptanalyst gets to choose P1, P2, . . ., Pi Deduce:Either k, or an algorithm to infer Pi+1 from Ci+1 = Ek(Pi+1) • Adaptive-chosen-plaintext attack • A special case of chosen-plaintext attack • Cryptanalyst modifies his or her choice of plaintext based on the results of previous encryption • Chosen-ciphertext attack • Given:C1, P1 = Dk(C1), C2, P2 = Dk(C2), . . ., Ci, Pi = Dk(Ci) • Deduce:k • Primarily applicable to public-key algorithms • Sometimes effective against a symmetric algorithm as well

14. Cryptanalytic Attacks (cont’d) • Chosen-key attack • Cryptanalyst has some knowledge about the relationship between different keys • Not very practical • Rubber-hose cryptanalysis • Cryptanalyst threatens, blackmails, or tortures someone to get the key • Purchase-key attack • Bribe someone to get the key • Author’s Comments • The best cryptographic algorithms are the ones that have been made public, have been attacked by the world’s best cryptographers for years, and are still unbreakable • Those who claim to have an unbreakable cipher simply because they cannot break are either geniuses or fools • Good cryptographers rely on peer review to separate the good algorithms from the bad

15. Security of Algorithms • Different algorithms offer different degrees of security • Cost of breaking must be greater than the value of encrypted data • Value of most data decreases over time • An algorithm is unconditionally secure if, no matter how much ciphertext a cryptanalyst has, there is not enough information to recover the plaintext (ex: one-time pad) • Cryptography is concerned with cryptosystems that are computationally infeasible to break • An algorithm is considered computationally secure (also called strong) if it cannot be broken with available resources, either current or future.

16. Complexity Measures of An Attack • Data complexity • Amount of data needed as input to the attack • Processing complexity (also called work factor) • Time needed to perform the attack • Storage requirements • Amount of memory needed to do the attack • As a rule of thumb, the complexity of an attack is taken to be the minimum of the three factors • Some attacks involve trading off the three complexities

17. Steganography • Existence of a secret message is concealed by hiding it in other messages • Example • Hiding secret messages in graphic images

18. Substitution Ciphers and Transposition Ciphers • Substitution Ciphers • Classical cryptography • Each character in the plaintext is substituted for another character in the ciphertext. Ex: Caesar Cipher • Transposition Ciphers • Plaintext remains the same, but the order of characters is shuffled around. Ex: columnar transposition cipher • Rotor Machines • Enigma machine • Used by the Germans during World War II • A team of Polish cryptographers broke the first German Enigma

19. Simple XOR • A symmetric algorithm • Plaintext is XORed with a keyword to generate ciphertext • Not secured at all; trivial to break • How to break • Discover the length of the key by a procedure known as counting coincidences • Shift the ciphertext by that length and XOR it with itself

20. One-Time Pads • Perfect encryption scheme, primarily used for ultra-secure low bandwidth channels • Invented by Major Joseph Mauborgne and AT&T’s Gilbert Vernam in 1917 • Start by representing the message as a sequence of 0s and1s • Key is a random sequence of 0s and 1s of the same length as the message • Add the key to the message mod 2, bit by bit (Actually XOR operation) • Discard the key, once used and never use again (plaintext) 00101001 (key) + 10101100 ------------------------------------ (ciphertext) 10000101

21. A Variation of One-Time Pads • Consider plaintext as a sequence of letters • Key is a random sequence of shifts, each one between 0 and 25 • Decryption uses the same key, but subtracts instead of adding the shifts • This encryption is completely unbreakable for a ciphertext only attack • Example: Assume A  1, B  2, . . ., Z  0 Plaintext:ONETIMEPADkey:TBFRGFARFM Ciphertext:IPKLPSFHGQ How?O+T mod 26 = I N + B mod 26 = P etc. • Caveats: • key letters have to be generated randomly. Using pseudo-random number generator is not useful. • No authenticity.

22. Computer Algorithms • Thee most common cryptographic algorithms • DES (Data Encryption Standard) • Most popular symmetric key algorithm • 3DES • AES (Advanced Encryption Standard) • RSA (Rivest, Shamir, and Adleman) • Most popular public-key algorithm • Can be used for both encryption and digital signatures • DSA (Digital Signature Algorithm) • A public-key algorithm; cannot be used for encryption, but only for digital signatures