Introduction to Number Theory

1. Public-Key Encryption • Introduction to Number Theory • A prime number is an integer that can only be divided without remainder by positive and negative values of itself and 1. Prime numbers play a critical role both in number theory and in cryptography. • Two theorems that play important roles in public-key cryptography are Fermat's theorem and Euler's theorem. • An important requirement in a number of cryptographic algorithms is the ability to choose a large prime number. An area of ongoing research is the development of efficient algorithms for determining if a randomly chosen large integer is a prime number.

2. Fermat's Theorem Fermat's theorem states the following: If p is prime and a is a positive integer not divisible by p, then • p = 5,a = 3 • ap = 35 = 243 3(mod 5) = a(mod p) • p = 5, a = 10 • ap = 105 = 100000 10(mod 5) = 0(mod 5) = a(mod p) An alternative form of Fermat's theorem is also useful: If p is prime and a is a positive integer, then

3. Euler Totient Function ø(n) • to compute ø(n) need to count number of residues to be excluded • in general need prime factorization, but • for p (p prime) ø(p) = p-1 • for p.q (p,q prime) ø(pq) =(p-1)x(q-1) • eg. ø(37) = 36 ø(21) = (3–1)x(7–1) = 2x6 = 12

4. 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

5. Why Public-Key Cryptography? • developed to address two key issues: • key distribution – how to have secure communications in general without having to trust a KDC with your key • digital signatures – how to verify a message comes intact from the claimed sender • public invention due to Whitfield Diffie & Martin Hellman at Stanford Uni in 1976 • known earlier in classified community

6. 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

7. RSA • by Rivest, Shamir & Adleman of MIT in 1977 • best known & widely used public-key scheme • uses large integers (eg. 1024 bits) • security due to cost of factoring large numbers

8. RSA ingredients We are now ready to state the RSA scheme. The ingredients are the following:

9. RSA Algorithm

10. RSA Example - Key Setup • Select primes: p=17 & q=11 • Computen = pq=17 x 11=187 • Compute ø(n)=(p–1)(q-1)=16 x 10=160 • Select e:gcd(e,160)=1; choose e=7 • Determine d:de=1 mod 160 and d < 160 Value is d=23 since 23x7=161= 10x160+1 • Publish public key PU={7,187} • Keep secret private key PR={23,187}

12. Clarification • we need to calculate C = 887 mod 187. Exploiting the properties of modular arithmetic, we can do this as follows: 887 mod 187 = [(884 mod 187) x (882 mod 187) x (881 mod 187)] mod 187 881 mod 187 = 88 882 mod 187 = 7744 mod 187 = 77 884 mod 187 = 59,969,536 mod 187 = 132 887 mod 187 = (88 x 77 x 132) mod 187 = 894,432 mod 187 = 11

13. Security of RSA Four possible approaches to attacking the RSA: Brute force: This involves trying all possible private keys. Mathematical attacks: There are several approaches, all equivalent in effort to factoring the product of two primes. Timing attacks: These depend on the running time of the decryption algorithm. Chosen cipher text attacks: This type of attack exploits properties of the RSA algorithm.

14. Factoring Problem • mathematical approach takes 3 forms: • factor n=p.q, hence compute ø(n) and then d • determine ø(n) directly and compute d • find d directly • currently believe all equivalent to factoring • have seen slow improvements over the years • as of May-05 best is 200 decimal digits (663) bit with LS • biggest improvement comes from improved algorithm • cf QS to GHFS to LS • currently assume 1024-2048 bit RSA is secure • ensure p, q of similar size and matching other constraints

15. Timing Attacks • developed by Paul Kocher in mid-1990’s • exploit timing variations in operations • eg. multiplying by small vs large number • or IF's varying which instructions executed • infer operand size based on time taken • RSA exploits time taken in exponentiation • countermeasures • use constant exponentiation time • add random delays • blind values used in calculations

16. Chosen Cipher text Attacks • RSA is vulnerable to a Chosen Ciphertext Attack (CCA) • attackers chooses ciphertexts & gets decrypted plaintext back • choose ciphertext to exploit properties of RSA to provide info to help cryptanalysis • can counter with random pad of plaintext • or use Optimal Asymmetric Encryption Padding (OASP)

17. Examples on RSA • Perform encryption and decryption using the RSA algorithm, for the following: • p = 5; q = 11, e = 3; M = 9 • p = 7; q = 11, e = 17; M = 8 • p = 11; q = 13, e = 11; M = 7 • p = 3; q = 11, e = 7; M = 5

18. Examples on RSA • RSA Algorithm Example • Choose p = 3 and q = 11 • Compute n = p * q = 3 * 11 = 33 • Compute φ(n) = (p - 1) * (q - 1) = 2 * 10 = 20 • Choose e such that 1 < e < φ(n) and e and n are coprime. Let e = 7 • Compute a value for d such that (d * e) % φ(n) = 1. One solution is d = 3 [(3 * 7) % 20 = 1] • Public key is (e, n) => (7, 33) • Private key is (d, n) => (3, 33) • The encryption of m = 2 is c = 27 % 33 = 29 • The decryption of c = 29 is m = 293 % 33 = 2

19. End of Sections Creative Minds never gives up Thank you