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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
• 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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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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Contact Information
• NSA Website: www.nsa.gov
• My information:
• Jill Calhoun
• Email: [email protected]
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