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MATH CAREERS AT NATIONAL SECURITY AGENCY

Jill CalhounMay 2010

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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.”

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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

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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

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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

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Application Process

• Apply at website, www.nsa.gov

• Must be U.S. citizen

• Allow 6-12 months for application process

• Onsite interview/security screening

• Math Proficiency Exam

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Benefits

• 10 Federal Holidays

• Annual Leave and Sick Leave earned per pay period

• Flexible Time

• Continuing Education Opportunities

• Internal Training Opportunities/Career Development

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

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General Idea

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internetAlice

Eve

Bob

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

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Why Use Public Key

• Encrypt Messages

• Key Exchange

• Authentication

• Digital Signatures

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RSA Public Key System

• Application of multiplication and factoring to public key cryptography

• Developed in 1977 by Rivest, Shamir, and Adelman

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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)– (xe)d = x (ed) = x mod N whenever x is not divisible by p

or q

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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

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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

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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)

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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

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D-H key exchange (1)

• Pick some group G, with generator g

• Alice picks a random number a and calculates ga (in G)

• Bob picks a random number b and calculates gb (in G)

• Alice’s private key = a• Alice’s public key = ga

• (Similarly for Bob)

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D-H Key Exchange

• Eve will see ga and gb during transmission

• She can’t calculate the shared secret key unless she know (or can guess) either a or b

• Determining a, given ga (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 1077)

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D-H Example

• G = M17, with g = 3

• Alice selects a = 12

• Bob selects b = 7

• Alice calculates 312 (mod 17) = 4 and sends it to Bob

• Bob calculates 37 (mod 17) = 11 and sends it to Alice

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QUESTIONS

Contact Information

• NSA Website: www.nsa.gov

• My information:

• Jill Calhoun

• Email: jcday3@nsa.gov

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