UNIT IV PUBLIC KEY CRYPTOGRAPHY

Transcription

1 UNIT IV PUBLIC KEY CRYPTOGRAPHY

2 Private Key Cryptography traditional 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

3 Private Key Vs Public Key It is asymmetric, involving the use of two separate keys, in contrast to symmetric encryption, which uses only one key. Anyone knowing the public key can encrypt messages or verify signatures, but cannot decrypt messages or create signatures It works by the clever use of number theory problems that are easy one way but hard the other. public key schemes are neither more nor less secure than private key,nor do they replace private key schemes, rather they complement them. Both also have issues with key distribution, requiring the use of some suitable protocol.

4 Why Public Key Cryptography? evolved from two of the most difficult problems associated with symmetric encryption: key distribution and digital signatures. first practical scheme, which was for key distribution only, was published in 1977 by Diffie & Hellman.

5 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 is asymmetric because those who encrypt messages or verify signatures cannot decrypt messages or create signatures

6 Public Key Cryptography

7 Public Key Characteristics Asymmetric algorithms rely on one key for encryption and a different but related key for decryption. It is computationally infeasible to determine the decryption key given only knowledge of the cryptographic algorithm and the encryption key. utilise problems that are easy (P type) one way but hard (NP type) the other way eg exponentiation vs logs, multiplication vs factoring.

8 Public Key Cryptosystems

9 Public Key Cryptosystems public key schemes can be used for either secrecy or authentication, or both In this case, separate key pairs are used for each of these purposes. The receiver owns and creates secrecy keys, sender owns and creates authentication keys.

10 Public Key Applications can classify uses into 3 categories: encryption/decryption (provide secrecy) digital signatures (provide authentication) key exchange (of session keys) some algorithms are suitable for all uses, others are specific to one

11 RSA by Rivest, Shamir & Adleman of MIT in 1977 best known & widely used public key scheme based on exponentiation in a finite (Galois) field over integers modulo a prime exponentiation takes O((log n) 3 ) operations (easy) uses large integers (eg bits) security due to cost of factoring large numbers factorization takes O(e log n log log n ) operations (hard)

12 RSA Key Setup each user generates a public/private key pair by: selecting two large primes at random p, q computing their system modulus n=p.q note ø(n)=(p-1)(q-1) selecting at random the encryption key e where 1<e<ø(n), gcd(e,ø(n))=1 solve following equation to find decryption key d e.d=1 mod ø(n) and 0 d n publish their public encryption key: PU={e,n} keep secret private decryption key: PR={d,n}

13 RSA Use to encrypt a message M the sender: obtains public key of recipient PU={e,n} computes: C = M e mod n, where 0 M<n to decrypt the ciphertext C the owner: uses their private key PR={d,n} computes: M = C d mod n note that the message M must be smaller than the modulus n (block if needed)

14 Why RSA Works because of Euler's Theorem: a ø(n) mod n = 1 where gcd(a,n)=1 in RSA have: n=p.q ø(n)=(p-1)(q-1) carefully chose e & d to be inverses mod ø(n) hence e.d=1+k.ø(n) for some k hence : C d = M e.d = M 1+k.ø(n) = M 1.(M ø(n) ) k = M 1.(1) k = M 1 = M mod n

15 RSA Example Key Setup 1. Select primes: p=17 & q=11 2. Compute n = pq =17 x 11= Compute ø(n)=(p 1)(q-1)=16 x 10= Select e: gcd(e,160)=1; choose e=7 5. Determine d: de=1 mod 160 and d < 160 Value is d=23 since 23x7=161= 10x Publish public key PU={7,187} 7. Keep secret private key PR={23,187}

16 RSA Security possible approaches to attacking RSA are: brute force key search (infeasible given size of numbers) mathematical attacks (based on difficulty of computing ø(n), by factoring modulus n) timing attacks (on running of decryption) chosen ciphertext attacks (given properties of RSA)

17 Key Management public key encryption helps address key distribution problems have two aspects of this: distribution of public keys use of public key encryption to distribute secret keys

18 Distribution of Public Keys Techniques proposed for the distribution of public keys, public announcement publicly available directory public key authority public key certificates

19 Public Announcement Any participant can send his or her public key to any other participant, or broadcast the key to the community at large. Its major weakness is forgery, anyone can create a key claiming to be someone else and broadcast it, and until the forgery is discovered they can masquerade as the claimed user.

20 Publicly Available Directory can obtain greater security by registering keys with a public directory directory must be trusted with properties: contains {name,public key} entries participants register securely with directory participants can replace key at any time directory is periodically published directory can be accessed electronically still vulnerable to tampering or forgery

21 Public Key Authority improve security by tightening control over distribution of keys from directory has properties of directory and requires users to know public key for the directory then users interact with directory to obtain any desired public key securely does require real time access to directory when keys are needed

22 Public Key Authority

23 Public Key Certificates certificates allow key exchange without realtime access to public key authority a certificate binds identity to public key usually with other info such as period of validity, rights of use etc with all contents signed by a trusted Public Key or Certificate Authority (CA) can be verified by anyone who knows the public key authorities public key

24 Public Key Certificates

25 Public Key Distribution of Secret Keys use previous methods to obtain public key can use for secrecy or authentication but public key algorithms are slow so usually want to use private key encryption to protect message contents hence need a session key have several alternatives for negotiating a suitable session

26 Simple Secret Key Distribution proposed by Merkle in 1979 A generates a new temporary public key pair A sends B the public key and their identity B generates a session key K sends it to A encrypted using the supplied public key A decrypts the session key and both use problem is that an opponent can intercept and impersonate both halves of protocol

27 Simple Secret Key Distribution

28 Hybrid Key Distribution A hybrid approach in use on IBM mainframes. Retains the use of a key distribution center (KDC) that shares a secret master key with each user and distributes secret session keys encrypted with the master key. A public key scheme is used to distribute the master keys. The addition of a public key layer provides a secure, efficient means of distributing master keys. This is an advantage in a configuration in which a single KDC serves a widely distributed set of users.

29 Diffie Hellman Key Exchange Enable two users to securely exchange a key that can then be used for subsequent encryption of messages. The Diffie-Hellman algorithm uses exponentiation in a finite (Galois) field (modulo prime or a polynomial), Its effectiveness lies on the difficulty of computing discrete logarithms.

30 Diffie Hellman Setup all users agree on global parameters: large prime integer or polynomial q a being a primitive root mod q each user (eg. A) generates their key chooses a secret key (number): x A < q compute their public key: y A = a xa mod q Similarly user B generates y B Each side keeps the X value private and makes the Y value available publicly to the other side

31 Diffie Hellman Key Exchange x User A computes the key as K = y A B mod q x User B computes the key as K = y B A mod q These two calculations produce identical results x K= y A B mod q = (a x B mod q) xa mod q = (a xb ) xa mod q = a x B xa mod q = (a xa ) xb mod q = (a xa mod q) xb mod q = y xb mod q

32 Example Find Y A and Y B and compute the common secret key q=23, a= 7, x A = 3, x B = 6 q=353, a= 3, x A = 97, x B = 233

33 Man in Middle Attack Suppose Alice and Bob wish to exchange keys, and Darth is the adversary The attack proceeds as follows: 1. Darth prepares for the attack by generating two random private keys X D1 and X D2 and then computing the corresponding public keys Y D1 and Y D2. 2. Alice transmits Y A to Bob. 3. Darth intercepts Y A and transmits Y D1 to Bob. Darth also calculates K 2 =(Y A ) X D2 mod q. 4. Bob receives Y D1 and calculates K 1 =(Y D1 ) X E mod q 5. Bob transmits X A to Alice. 6. Darth intercepts X A and transmitsy D2 to Alice. Darth calculates K 1 =(Y B ) X D1 mod q. 7. Alice receives Y D2 and calculates K 2 =(Y D2 ) X A mod q. 8. At this point, Bob and Alice think that they shareasecret

34 ELLIPTIC CURVE CRYPTOGRAPHY majority of public key crypto (RSA, D H) use either integer or polynomial arithmetic with very large numbers/polynomials imposes a significant load in storing and processing keys and messages an alternative is to use elliptic curves offers same security with smaller bit sizes newer, but not as well analysed

35 Real Elliptic Curves an elliptic curve is defined by an equation in two variables x & y, with coefficients consider a cubic elliptic curve of form y 2 = x 3 + ax + b where x, y, a, b are all real numbers Zero point O If three points on an elliptic curve lie on a straight line, their sum is O

36 Example

37 Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables & coefficients are finite have two families commonly used: prime curves E p (a,b) defined over Z p use integers modulo a prime best in software binary curves E 2 m(a,b) defined over GF(2 n ) use polynomials with binary coefficients best in hardware

38 ECC Diffie Hellman can do key exchange analogous to D H users select a suitable curve E p (a,b) select base point G=(x 1,y 1 ) with large order n s.t. ng=o A & B select private keys n A <n, n B <n compute public keys: P A =n A G, P B =n B G compute shared key: K=n A P B, K=n B P A same since K=n A n B G

39 ECC Encryption/Decryption several alternatives, will consider simplest must first encode any message M as a point on the elliptic curve P m select suitable curve & point G as in D H each user chooses private key n A <n and computes public key P A =n A G to encrypt P m : C m ={kg, P m +kp b }, k random decrypt C m compute: P m +kp b n B (kg) = P m +k(n B G) n B (kg) = P m