Exploring Cryptography: From Basic Concepts to Advanced Techniques

1. Cryptography 101 EECS710: Info Security and Assurance Professor Hossein Saiedian Resources: Terry Ritter’s Learning About Cryptography, Network Associates’ An Introduction to Cryptography, course textbooks

2. What is cryptography • Cryptography: transforming (enciphering) plaintext into a form where the original info is present but hidden • Plaintext: data that can be read w/o any special tool • Ciphertext: result of encryption; unreadable data • Given a plaintext, many transformations are possible; to expose the info one may have to try all (on average, half) of possible transformations

3. An elementary school approach • On a sheet of paper, write the alphabets in order in one column; write the same alphabets randomly (but uniquely) in the second column A W B J C R … … • To encipher a plaintext, substitute each letter with the associated letter from the second column

4. An elementary school example • Suppose we have the following substitution ABCDEFGHIJKLMNOPQRSTUVWXYZ QAZWSXEDCRFVTGBYHNUJMIKOLP • Plaintext message: MEET ME AT SIX • Enciphered message: TSSJ TS QJ UCO • The Caesar cipher En(x) = (x + n) mod 26 Dn(x) = (x - n) mod 26 For Caesar cipher: n = 3

5. A middle school approach • Singe (simple) substitution: the key is one particular permutation (arrangement) of the alphabet; once the sheet revealed, it is no longer good • But one can create a notebook of different permutations for the second column, each on a page; the key will be the page number • If the notebook is exposed, one must try all (or at least half) transformations

6. Weak vs. strong transformation • Simple substitution is weak: the more often a particular letter is used, the more often the ciphertext letter appears • Languages use some letters (or letter combinations) more than others, and thus possible to guess • One solution: increase the size of the cipher alphabet • Instead of single letters, use pairs of letters • For example, replace A with WK • At least 26 × 26 = 676 transformations

7. Weak vs. strong transformation [2] • How about expanding: instead of a pair of letters, select triplets, four quadruples, … • Soon a computer will be needed to do the operations • A conventional (block) cipher: A much larger alphabet • A 64-bit (eight character) block cipher: instead of using 26 letters, views each 2^64 values as a separate letter • 18,000,000,000,000,000 “letters”!

8. Keyspace for an 8-bit key • A notebook with 256 pages: 256 different keys • Decimal 256 = Binary 100000000 = 2^8 = 8 bit • Thus an “8 bit” keyspace gives 256 unique key values • If we choose one of the keys, one would have to try 256 (or probably only 128) keys to break • Thus a low design strength

9. Keyspace for longer than 8-bit keys • A 65,536 page notebook offers a “16 bit” keyspace • That is 256 times that of an “8 bit” while the key has 8 bits more • A “56 bit” keyspace: 7 × 10^16 different keys • Broken via brute force in 56 hours! • A “128 bit” (16 characters): 3.40282367 × 1038 • Strong enough

10. What cryptography can and cannot do • It can hide to facilitate confidentiality and authentication • It cannot hide contraband, a luxury lifestyle with no visible means of support, informants, or undercover spying • Keys can be lost, forgotten, stolen, or revealed for payment or under duress

11. Encryption/decryption process • Encryption: the process of disguising plaintext • Decryption: the process of reverting ciphertext to its original plaintext

12. Other related terms • Cryptanalysis: the science of analyzing and breaking secure communications • Analytical reasoning/math • Pattern matching • Patience, determination, good luck • Cryptography: the science of information security • Cryptology: cryptography + cryptanalysis

13. Strong vs. weak cryptography • Strength is measured in the time and resources required to recover a plaintext • Strong cryptography: very difficult to decipher • A billion computers doing a billion checks a second, it is not possible to decipher the result of strong cryptography in a billion year

14. How does it work • A mathematical function • Strength: (1) algorithm, (2) secrecy of the key

15. Conventional cryptography • AKA symmetric key • One key is used for encryption/decryption • Example: the Data Encryption Std (DES) used by the fed government

16. Conventional cryptography approaches • Substitution: changes (substitutes) characters in plaintext to produce ciphertext • Example: Caesar cipher where the letters are offset by 3 (or in general n) positions • SECRET  VHFUHW • Transposition: rearranges the characters in the plaintext to produce ciphertext • Example: the “rail fence” cipher where plaintext is written in two rows preceding down, then across • SECRET  SCE  SCEERT ERT

17. A longer example of transposition encipher • The number of rows is explicitly defined; pad with dummy characters to fill • An example of 3-row fence MTSPNRIE EAIMDBDX ETXUERGY • Read off/send : MTSPNRIEEAIMDBDXETXUERGY • May send in 4-char groups to avoid errors (also for better management and to confuse intruders) MTSP NRIE EAIM DBDX ETXU ERGY

18. A longer example of transposition encipher [2] • To decipherMTSP NRIE EAIM DBDX ETXU ERGY • Run the letters into a long string MTSPNRIEEAIMDBDXETXUERGY • Since there are 3 rails, divide into 3 groups of 8 MTSPNRIE EAIMDBDX ETXUERGY • Write the first letter of group 1, group 2, and group 3 followed by the second letter of group 1, etc. MEETATSIXPMUNDERBRIDGEXY MEET AT SIX PM UNDER BRIDGE XY

19. Benefits of conventional encryption • Very fast • Useful for encrypting local data that is not going anywhere • Expensive for data transmission • How to distribute the key

20. Public key encryption • Addresses key distribution • Asymmetric scheme • Uses a pair of keys • Public key: used to encrypt data • Private key: used to decrypt data • Public key is public and publically advertised • Private key is kept secret • Computationally infeasible to deduce the private key from the public key • An example: PGP

21. Public key encryption illustrated

22. Benefits of the public key approach • No need for sender and receiver to share a key • All communications involve public keys; private keys are never transmitted • Examples of public key cryptosystems • Elgamal (named for its inventor, Taher Elgamal) • RSA (named for its inventors, Ron Rivest, Adi Shamir, and Leonard Adleman) • Diffie-Hellman(named for its inventors), and • DSA, the Digital Signature Algorithm (invented by David Kravitz)

23. How PGP works • Combines the best features of conventional and public cryptography • PGP compresses the plaintext: saves modem transmission and disk space and strengthens security (complicates patterns) • PGP creates a session key: a one-time-only secret key (generated from the random movement of the mouse/keyboard strokes) • The plaintext is encrypted via a fast algorithm and the session key

24. How PGP works [2] • The session key is encrypted using the recipient's public key and transmitted

25. How PGP works [3] • Decryption works in reverse: the session key is recovered (by the recipient's private key) and is used to decrypt the ciphertext

26. The benefits of the PGP • A combination of two methods • Convenience of the public key: no key-distribution concerns • Speed of conventional encryption: about 1,000 faster than the public key encryption

27. The key issues • A value that works with encryption algorithms to produce a ciphertext • Big, big numbers: measures in bits: 1,024 bits • The bigger the key, the more secure ciphertext • Public key size and conventional cryptography secret key sizes are unrelated • A conventional 80-bit key has the same strengths of a 1,024-bit public key • The bigger the key, the more secure but the algorithms used for each is different (a comparison is like comparing apple and oranges)

28. The key issues [2] • Public and private keys are mathematically related but difficult to derive a private key from its public key • Pick large keys to be secure; small enough to be applied quickly • Large keys are good for a longer periods of time • Keys are stored in encrypted form; PGP stores on the hard-drive as keyrings • one for public and one for private uses • If the private key is lost, one will be unable to recover decrypted data

29. Digital signatures • A benefit of public key • Enable the recipient to verify the authenticity of the information’s origin, and also verify that the information is intact • Provides for authenticationand data integrity • Also provides non-repudiation: prevents the sender from claiming that he/she did not send the information

30. Digital signatures [2] • Authentication • Similar to a handwritten signature but superior in that it is nearly impossible to counterfeit • You may not care if anyone learns that you just deposited $500 in an account, but you do want to be sure it was the bank teller you were communicating with • Integrity • To verify and ensure that the information was not altered

31. How digital signature works

32. How digital signature works [2] • Problem with the above approach? SLOW • Data size to communicate too large (at least double the original) • Alternative to expedite? • Use hash functions • “A hash function is any well-defined procedure or mathematical function that converts a large, possibly variable-sized amount of data into a small datum, usually a single integer” • Create a message digest to sign the message

33. Message digests • Objective: to verify that the message received is the same as the message sent • How: hash function (checksum function) -- h: A  B -- A: a message of any length (millions of bits) -- B: A fixed length output, e.g., 160 bit -- h: ensures that if A is changed in anyway (even one bit), an entirely different output is produced • PGP calls B a message digest (used for creating signatures); one cannot alter the signature or attach to another document

34. Hash function (h: A B) properties • Easy to compute • For any y in B, infeasible to find x in A such that h(x) = y • For any x, x’ in A, x ≠ x’, infeasible to have h(x) = h(x’) • Given any x in A, infeasible to find x’ in A and x ≠ x’ and h(x’) = h(x)

35. Side note: pigeonhole principle • If there are n containers and n+1 objects, at least one container will have to hold two objects • So what? If a hash function produces 3-bit hashes and we have a set of 5-bit messages, it implies: • a^3 = 8 hashes • 2^5 = 32 messages • Thus large hash sizes are better

36. How a hash function is used

37. Digital envelopes • Creating a digital envelop (an encrypted message; no digital signature attached)

38. Digital envelopes [2] • Opening a digital envelop

39. Diffie-Hellam public key agreement • A relatively fast public key agreement • Relies on two functions, p (prime) and g (generator), and two random numbers x and y • Everything exchanged in clear text • Six step process • Works like magic!

40. Diffie-Hellam public key agreement [2] • Party X and Party Y agree on Diffie-Hellman p and g; exchange these in clear • Party X generates random number x Party Y generates random number y • Party X computes x’ = g^x mod p Party Y computes y’ = g^y mod p • The two parties exchange x’ and y’ in clear • Party X computes kx = y’^x mod p Party Y computes ky = x’^y mod p kx = y’^x mod p = g^(xy) mod p = x’^ymod p = ky • Subsequent encryption with kx or ky

41. Diffie-Hellam public key agreement [3]

42. Digital certificates • One concern with the public key approach: must ensure that you are encrypting to the correct person’s public key • Otherwise, you can only encrypt/decrypt to those key handed to you • A solution: digital certificates (or certs) • A form of credentials (like a physical passport) • Included with a person’s public key to verify that a key is valid

43. Components of a digital certificate • A digital certificate • A public key • Certificate info (identifying information such as name, ID) • One (or more) digital signatures • A stamp of approval from a trusted entity • Certificates are used when it is necessary to exchange public keys with someone (when you cannot manually exchange via a diskette or USB drive)

44. Components of a digital certificate [2]

45. Digital certificate distribution • Digital servers: a networked database that allows users to submit and receive digital certs • Example: PGP Keyserver • Public Key Infrastructures (PKIs) • Storage facilities like the certificate servers • More structured • Provide additional key management services • Issue revoke, store, and trust certificates • Certificate authority: a group of human beings authorized to issue certs (like a passport office)

46. Common certificate format • The certificate holder’s public key: the public portion of key pair and key algorithm, e.g., RSA • The certificate holder’s information: identity information about the user (e.g., name, user ID, email address, photograph, and so on) • The digital signature of the certificate owner: the signature using the corresponding private key of the public key of the certificate • The certificate’s validity period: the certificate’s start date/time and expiration date/time; The preferred symmetric encryption algorithm for the key: e.g., AES, Triple-DES, Twofish

47. Common certificate format [2]

48. Other substitution techniques • Choose a keyword, e.g., Jayhawk, drop repeated letters, thus jayhwk • The keyword defines the permutation of English letters: ABCDEFGHIJKLMNOPQRSTUVWXYZ jayhwkbcdefgilmnopqrstuvxz • Another keyword: Professional ABCDEFGHIJKLMNOPQRSTUVWXYZ profesinalbcdghjkmqtuvwxyz

49. Other substitution techniques [2] • Use every third letter (apply mod 26) adgjmpsvybehknqtwzcfilorux • Consider any possible permutation of the English letters • How many? 26! • Even applying decryption at 1 microsecond, still takes over 1,000 years • The primary issue: the knowledge of letter patterns in a text • Solution: Avoid using the same substitution for a letter

50. One-time pads (using Vigenere tableau) • Assume a set of large, non-repeating keys written on sheets of paper, glued into a pad • Assume keys are 20 characters • Assume a text that is 300 characters • Sender tears off 15 pages from the pad • Sender writes the keys one at a time above the text letters and enciphers in a prearranged chart • Receiver must have the same pad • Concerns: (1) key distribution, (2) sender/receiver must synchronize (3) need unlimited keys