1. [[ NET-CENTRIC CAPABILITIES TURBULENCE TECHNICAL OVERVIEW : AUGUST 2007 ]] MATH CAREERS AT NATIONAL SECURITY AGENCY Jill Calhoun May 2010

2. 2 NSA HISTORY Founded in 1952 as part of Department of Defense National Intelligence Directorate Mission to Secure Nation’s Communication while Exploiting Foreign Signals Intelligence Located at Ft Meade, Maryland, halfway between Baltimore and Washington DC Largest Employer of Mathematicians in the United States “The ability to understand the secret communications of our foreign adversaries while protecting our own communications gives our nation a unique advantage.”

3. 3 NSA’S MISSION AREAS Signals Intelligence (SIGINT) – Produce foreign signal intelligence information – Communications and data processing using high technology – Foreign language analysis and research – Cryptanalysis is decoding encrypted transmissions = codebreaking Information Assurance (IA) – Protect U.S. information systems by safeguarding classified/sensitive information stored on or sent by U.S. government equipment – Cryptography is developing codes and ciphers = codemaking

4. 4 Mathematicians at NSA Work on math projects involving signals analysis, data mining, information retrieval, speech processing, data compression, supercomputing, biometrics, and more Use analysis, abstract algebra, number theory, graph theory, coding theory, probability, statistics Design systems, develop programs to protect sensitive U.S. information on long- term basis

5. 5 Employment Opportunities Mathematics Summer Programs – Directors Summer Program (DSP) – Mathematics Summer Employment Program (MSEP) – Graduate Mathematics Program (GMP) Full-time Mathematics Positions – 3-year development program – 4-6 short-term assignments in different offices – Internal training curriculum Mathematics Sciences Program – Grants and Sabbaticals

6. 6 Application Process Apply at website, www.nsa.govwww.nsa.gov Must be U.S. citizen Allow 6-12 months for application process Onsite interview/security screening Math Proficiency Exam

7. 7 Benefits 10 Federal Holidays Annual Leave and Sick Leave earned per pay period Flexible Time Continuing Education Opportunities Internal Training Opportunities/Career Development

8. PUBLIC KEY CRYPTOGRAPHY Users who wish to communicate via secure means must share a cryptovariable (a.k.a., a key) Physical meeting or courier exchange keys ==inconvenient Need a secure way to transmit over a public line 8

9. General Idea 9 internet Alice Eve Bob

10. General Idea Alice and Bob agree on a public key (PK) system Bob sends Alice his public key Alice encrypts her message with Bob’s public key, and sends it to him Bob uses his private key to decrypt and read Alice’s message 10

11. Why Use Public Key Encrypt Messages Key Exchange Authentication Digital Signatures 11

12. RSA Public Key System Application of multiplication and factoring to public key cryptography Developed in 1977 by Rivest, Shamir, and Adelman 12

13. RSA Public Key System Select two large prime numbers, p and q Compute n = pq ( n is the modulus) Choose e such that e < n and e is relatively prime to (p-1) (q-1) Compute d, the inverse of e –i.e. ed = 1 mod (p-1)(q-1) –(x e ) d = x ( ed ) = x mod N whenever x is not divisible by p or q 13

14. RSA (cont’d) e = public exponent d = private exponent Public key is the pair (n, e) Private key = d Factors p and q are secret 14

15. How Secure is RSA? Need to be able to factor n into p and q to recover d, the private key But factoring products of large prime numbers is hard, and requires a lot of computational power 15

16. RSA Examples Let p = 61 and q = 53 Then, n = pq = 61*53 = 3233 Also, (p-1)(q-1) = (61-1)(53-1) = 3120 Now, choose e = 17 Notice that de = 1 mod 3120, so d = 2753 Public key = (n = 3233, e = 17) Private key = (n = 3233, d = 2753) 16

17. Diffie-Hellman public key system Application of exponentiation and logarithms to public key cryptography Exponentiation done over a large finite group, not over real numbers Developed in 1975 by W. Diffie and M. Hellman Invented by Malcolm Williamson at GCHQ before Diffie-Hellman 17

18. D-H key exchange (1) Pick some group G, with generator g Alice picks a random number a and calculates g a (in G ) Bob picks a random number b and calculates g b (in G ) Alice’s private key = a Alice’s public key = g a (Similarly for Bob) 18

19. D-H Key Exchange Eve will see g a and g b during transmission She can’t calculate the shared secret key unless she know (or can guess) either a or b Determining a, given g a (in G ) is called the discrete logarithm problem This is hard to solve for a sufficiently large group G Real world prime moduli can be very big – 256 to 2048 bits (256 bits is about 10 77 ) 19

20. D-H Example G = M 17, with g = 3 Alice selects a = 12 Bob selects b = 7 Alice calculates 3 12 (mod 17) = 4 and sends it to Bob Bob calculates 3 7 (mod 17) = 11 and sends it to Alice 20

21. 21 QUESTIONS

22. Contact Information NSA Website: www.nsa.gov My information: Jill Calhoun Email: jcday3@nsa.gov 22