Lecture 5 Asymmetric Cryptography

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

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

Historical Background 1976: W. Diffie and M.E. Hellman proposed the first public-key encryption algorithms -- actually an algorithm for public exchange of a secret key. 1978: L.M Adleman, R.L. Rivest and A. Shamir propose the RSA encryption method – Currently the most widely used – Basis for the spreadsheet used in the lab

Public Key History Some algorithms – Diffie-Hellman, 1976, key-exchange based on discrete logs – Merkle-Hellman, 1978, based on “knapsack problem” – McEliece, 1978, based on algebraic coding theory – RSA, 1978, based on factoring – Rabin, 1979, security can be reduced to factoring – ElGamal, 1985, based on discrete logs – Blum-Goldwasser, 1985, based on quadratic residues – Elliptic curves, 1985, discrete logs over Elliptic curves – Chor-Rivest, 1988, based on knapsack problem – NTRU, 1996, based on Lattices – XTR, 2000, based on discrete logs of a particular field

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

Why Public-Key Cryptography? Compared with symmetric-key encryption, public-key encryption requires more computation and is therefore not always appropriate for large amounts of data. However, it's possible to use public-key encryption to send a symmetric key, which can then be used to encrypt additional data. 1.Key distribution – Secret keys for conventional cryptography – Unforgeable public keys (digital certificate) 2.Message authentication

Public-Key Cryptography

Public-Key Characteristics Public-Key algorithms rely on two keys where: – it is computationally infeasible to find decryption key knowing only algorithm & encryption key – it is 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 (for some algorithms)

Public-Key Cryptosystems

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

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 uses large integers (e.g., 1024 bits) security due to cost of factoring large numbers

Mathematical background

Prime Numbers prime numbers only have divisors of 1 and self –they cannot be written as a product of other numbers –note: 1 is prime, but is generally not of interest eg. 2,3,5,7 are prime, 4,6,8,9,10 are not prime numbers are central to number theory list of prime number less than 200 is:

Prime Factorisation to factor a number n is to write it as a product of other numbers: n=a × b × c note that factoring a number is relatively hard compared to multiplying the factors together to generate the number the prime factorisation of a number n is when its written as a product of primes تحليل العدد لعناصره الاولية –eg. 91=7×13 ; 3600=2 4 ×3 2 ×5 2

Relatively Prime Numbers & GCD two numbers a, b are relatively prime if have no common divisors apart from 1 common divisors : القاسم المشترك الاعظم –eg. 8 & 15 are relatively prime since factors of 8 are 1,2,4,8 and of 15 are 1,3,5,15 and 1 is the only common factor conversely can determine the greatest common divisor by comparing their prime factorizations and using least powers –eg. 300=2 1 ×3 1 ×5 2 18=2 1 ×3 2 hence GCD(18,300)=2 1 ×3 1 ×5 0 =6

Fermat's Theorem a p-1 mod p = 1 –where p is prime and gcd(a,p)=1 also known as Fermat’s Little Theorem useful in public key and primality testing

Fermat's Theorem Example : Let a= 12, p= 7 …..gcd(12,7)= 1 So 12 (7-1) mod 7 = 1 Another example : Let a= 10, p= 3 …..gcd(10,3)= 1 So 10 (3-1) mod 3 = 1

Euler Totient Function ø(n) when doing arithmetic modulo n complete set of residues is: 0..n-1 reduced set of residues is those numbers (residues) which are relatively prime to n –eg for n=10, –complete set of residues is {0,1,2,3,4,5,6,7,8,9} –reduced set of residues is {1,3,7,9} number of elements in reduced set of residues is called the Euler Totient Function ø(n)

Euler Totient Function ø(n) to compute ø(n) need to count number of elements to be excluded in general need prime factorization, but –for p (p prime) ø(p) = p-1 –for p.q (p,q prime) ø(p.q) = (p-1)(q- 1) eg. – ø(37) = 36 – ø(21) = (3–1)×(7–1) = 2×6 = 12

Euler's Theorem a generalisation of Fermat's Theorem a ø(n) mod N = 1 –where gcd(a,N)=1 eg. – a=3;n=10; ø(10)=4; – hence 3 4 = 81 = 1 mod 10 – a=2;n=11; ø(11)=10; – hence 2 10 = 1024 = 1 mod 11

Back to RSA

RSA Key Setup  each user generates a public/private key pair by:  selecting two large primes at random : p,q  computing their system n=p.q -define 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}

RSA Example 1.Select primes: p=17 & q=11 2.Compute n = pq =17 x 11=187 3.Compute ø(n)=(p–1)(q-1)=16 x 10=160 4.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}

Another Example Select primes p=11, q=3. n = pq = 11.3 = 33 phi = (p-1)(q-1) = 10.2 = 20 Choose e=3 Check gcd(e, p-1) = gcd(3, 10) = 1 (i.e. 3 and 10 have no common factors except 1), and check gcd(e, q-1) = gcd(3, 2) = 1 therefore gcd(e, phi) = gcd(e, (p-1)(q-1)) = gcd(3, 20) = 1 Compute d such that ed ≡ 1 (mod phi) i.e. compute d = e -1 mod phi = 3 -1 mod 20 i.e. find a value for d such that phi divides (ed-1) i.e. find d such that 20 divides 3d-1. Simple testing (d = 1, 2,...) gives d = 7 Check: ed-1 = = 20, which is divisible by phi. Public key = (e, n) = (3,33) Private key = ( d, n) = (7, 33).

RSA Use in Encryption 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)

RSA Example - En/Decryption sample RSA encryption/decryption is: given message M = 88 1.Publish public key PU={7,187} 2.Keep secret private key PR={23,187} encryption: C = 88 7 mod 187 = 11 decryption: M = mod 187 = 88

RSA Key Generation users of RSA must: – determine two primes at random - p, q – select either e or d and compute the other primes p,q must not be easily derived from modulus n=p.q – means must be sufficiently large – typically guess and use probabilistic test exponents e, d are inverses, so use Inverse algorithm to compute the other

How to compute d Find d such that : 1 ≡ ( d.e mod n )  1 = Kn + d.e Using the extended Euclid Algorithm

Extended Euclid’s Algorithm Extend the algorithm to compute the integer coefficients x and y such that gcd(a, b) = ax + by Call the algorithm with the follwing parameters: Extended-Euclid (a,b) by calling Extended-Euclid (Q(n),e) The value y which is the d Extended-Euclid(a, b) 1.if b = 0 then return (a, 1, 0) 2.(d’, x’, y’ ) ← Extended-Euclid(b, a mod b) 3.(d, x, y) ← (d’, y’, x’− [a/b]y’ ) 4.return (d, x, y)

Example

Example : self Let p =11, q=3 Let e= 3 Calculate d

The results n =p.q= 33 Q(n)= (p-1)(q-1)= 20 d.e mod Q(n) = 1 d= 7

Steps to get d according to the algorithm aba/bdxy Call first time by E(20,3) which is E(Q(n),e) The result y

Assignments 1.Perform encryption and decryption using RSA algorithm, as in Figure 1, for the following: ① p = 3; q = 11, e = 7; M = 5 ② p = 5; q = 11, e = 3; M = 9 2.In a public-key system using RSA, you intercept the ciphertext C = 10 sent to a user whose public key is e = 5, n = 35. What is the plaintext M? EncryptionDecryption Plaintext 88 Ciphertext 11 Plaintext 88 7 mod187=11 23 mod187=88 KU=7,187KR=23,187 Figure1. Example of RSA Algorithm