Other Topics in Cryptography. Truong Tuan Anh

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1 Other Topics in Cryptography Truong Tuan Anh

2 2 Outline Public-key cryptosystem Cryptographic hash functions Signature schemes

3 Public-Key Cryptography Truong Tuan Anh CSE-HCMUT

4 4 Outline Public-key cryptosystem

5 Classification 5

6 Cryptography: Classical Model 6

7 Cryptography: Classical Model Secret, common key K e k and d k for each key K: d k is either the same as e k or easily derived from e k Disclose d k or e k will make the system insecure Symmetric-key Cryptosystem Require prior communication of the key K (using a secure channel) Difficult to achieve in practice Public-key cryptosystem 7

8 8 Public-key Cryptosystem Was put forward by Diffie and Hellman in 1976 The most important cryptosystems: RSA and ElGamal Computationally infeasible to determine d k given e k e k is public Alice sends to Bob an encrypted message using e k of Bob Bob is the only one who can decrypt the message using his d k (private key)

9 Example: a Scenario 9

10 10 Public-key Cryptosystem (cont.) Encryption function is easy to compute The inverse function (i.e., the decryption function) should be hard to compute (except for Bob) one-way function Example: suppose n is the product of two large primes p and q; b is a positive integer

11 11 Trapdoor One-way Functions From Bob s view, he does not want e k to be oneway provide Bob a trapdoor: which consists of secret information for the inversion of e k Trapdoor one-way function: a one-way function but it is easy to invert with the knowledge of a certain trapdoor

12 12 Trapdoor One-way Functions (cont.) Usually, we need to specify a family of trapdoor one-way functions F A function f is chosen from F randomly and used as the public encryption function Its inverse function is the private decryption function

13 13 Exercise Implement a very simple public key cryptosystem as follows: P = C = Z 256 K = 0,1,2,, 255 Where public key + private key = 256 Encryption rule: for each public key k E k (x) = (ASCII_value_of_x + k) mod 256 Decryption rule: for each private key k D k (y) = (ASCII_value_of_y + k) mod 256

14 Cryptographic Hash Functions Truong Tuan Anh CSE-HCMUT

15 15 Outline Data integrity Hash functions

16 16 Data Integrity Data usually transferred over insecure channel Many factors affecting the data Example: Alice sends to Bob a message. Bob needs to know if the message has been changed over the channel? Need to assure data integrity

17 17 Hash Functions Provide assurance of data integrity Compute a short fingerprint of some data If the data is changed, the fingerprint will no longer valid Check the data integrity by re-computing the fingerprint and verify that the fingerprint has not changed The fingerprint = message digest

18 18 Hash Functions Let x be some data, its message digest is h(x) where h is a hash function The message digest should be short, commonly 160 bits Keyed and unkeyed hash functions Keyed hash functions are often used as message authentication codes (MACs)

19 19 Hash Function: Key and Unkeyed Example: Alice wants to send to Bob message x Unkeyed hash function: Alice computes y = h(x), then sends x over the channel and stores y in a secure channel Bob receives x and reads y and verifies if y = h(x), if yes, Bob is confident that x is originated from Alice Keyed hash function: Alice and Bob share a secret key K which determines function h K : Compute y = h K (x), then send (x, y) over the channel Bob receives (x, y) and verifies if y = h K (x), if yes, Bob is confident that neither x or y was modified by attackers

20 Hash Family 20

21 21 Hash Family Unkeyed hash function: Also a hash family in which there is only one possible key All we want is to prevent the construction of certain types of valid pairs by an adversary

22 22 Security of Hash Function It is desirable that the only way to produce a valid pair (x, y) is to first choose x and then compute y = h(x) by applying the function h to x Other security requirement depending on the particular applications Usually, a hash function is considered to be secured if the following problems are difficult to solve

23 Hash Function: Security Requirement 23

24 Hash Function: Security Requirement 24

25 Hash Function: Security Requirement 25

26 26 Exercises 1. Suppose a.equals(b). What can you conclude about a.hashcode() and b.hashcode()? (Assume hashcode() is properly implemented) 2. Suppose a.hashcode() == b.hashcode(). What can you conclude about a.equals(b)? (Assume equals() is properly implemented)

27 27 Exercises: a Simple Hash Function 3. Implement a hash function int HashMe(string orgstring, int tablesize) { //return the hash value of orgstring in tablesize } Note: Hash value of a string in a hash table is: calculate the sum of the ASCII values of letters in the string, then modulo the sum to the size of the table

28 Truong Tuan Anh CSE-HCMUT Signature Schemes

29 29 Outline Signature schemes

30 30 Context A conventional handwritten signature Attached to a document Specify the person responsible for the document Used in everyday situations: writing withdrawing money, signing a contract, a letter, Electronic document? Digital Signatures = Signature Schemes A method of signing a message stored in electronic form and can be transmitted over a computer network

31 31 Digital vs. Conventional Signatures Signing a document Conventional Signatures: a part of the physical document being signed Digital Signatures: not attached physically to the message being signed; There is an algorithm to bind the signature to the message Verification Conventional Signatures: compare to authentic signatures; Not a very secure method Digital Signatures: can be verified by a public verification algorithm; Prevent the possibility of forgeries

32 32 Signature Schemes A signature Scheme: consists of two components A signing algorithm A verification algorithm Alice signs a message using the private signing algorithm Bob verifies the signature using Alice s corresponding public verification algorithm

33 Signature Schemes 33

34 Signature Schemes (cont.) 34

35 35 Signature Schemes: Notes The functions sig K and ver K should be polynomial-time functions sig K is the private function and ver K is the public function It should be computationally infeasible for anyone than Alice to compute a signature y such that ver K (x,y) = true If Oscar can compute y such that ver K (x,y) = true, then y is a forgery

36 36 Signature Schemes: An Example RSA cryptosystem can be used to provide digital signatures RSA signature scheme Alice signs a message using the RSA decryption rule d K = sig K Anyone can verify the signature using the RSA encryption rule e K = ver K Anyone can forge Alice s RSA signature by randomly choosing y and computing if x = e K (y)

37 Signature Schemes: An Example 37

38 38 RSA Signature Scheme: How it works Alice wishes to send an encrypted, signed message to Bob (given the plaintext is x) 1. Alice computes her signature y = sig Alice (x) 2. Alice encrypts both x and y using e Bob, then z = e Bob (x, y) 3. Alice sends z to Bob 4. Bob receives z 5. Bob decrypts z using d Bob to get x, y 6. Bob uses Alice s public verification function to check that ver Alice (x, y) = true

39 39 RSA Signature Scheme: How it works Alice wishes to send an encrypted, signed message to Bob (given the plaintext is x) 1. Alice encrypts x using e Bob, then z = e Bob (x) 2. Alice computes her signature y = sig Alice (z) 3. Alice sends (z, y) to Bob 4. Bob receives (z, y) 5. Bob decrypts z using d Bob to get x 6. Bob uses Alice s public verification function to check that ver Alice (z, y) = true What is the problem?

40 Signature Schemes: In Reality 40

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