ECE-8843http://www.csc.gatech.edu/copeland/jac/8843/

Prof. John A. Copelandjohn.copeland@ece.gatech.edu

404 894-5177fax 404 894-0035

Office: GCATT Bldg 579email or call for office visit, or call Kathy Cheek, 404 894-5696

Chapter 3 - Public-Key Cryptography & Authentication

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Requirements - must be able to verify that:1. Message came from apparent source or author,2. Contents have not been altered,3. Sometimes, it was sent at a certain time or sequence.

Sometimes we would like to provide authentication without encryption (public statements do not need privacy). Still, authentication requires that the sender know something that the forger does not ( a secret key).

Conventional encryption can be used, but the sender must share the secret key with the receiver.

Authentication

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(b) Using public-keyencryption

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Secret Value is added by both parties to message before the “hash,” function is used to get the Message Integrity Check (MIC). It is removed before transmission.

MIC MIC

It is critical that a forger can not compose a different message that would produce the same MIC value.

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SHA-1

Secure Hash Algorithm 1

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HMAC Structure 9

Public-Key Cryptography(Public-Private Key)

plaintext (data file or message)

encryption by key-1 decryption by key-1

ciphertext (stored or transmitted safely)

decryption by key-2encryption by key-2

plaintext (original data or message)

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Encryption using aPublic-Key System

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Authentication using aPublic-Key System

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RSA (Rivest, Shamir, and Adleman)Key length is variable, 512 bits most common.

• The plaintext block ("m") must be less than the key

length.Key Generation

• Choose two large prime numbers, p and q (secret)

• n = pq, Ø(n) = (p-1)(q-1)

• Find a number, e, that is relatively prime to Ø(n)

• The public key is e and n (e,n)

• Find d, the multiplicative inverse to e mod Ø(n)(by “Number Theory”: d * e mod Ø(n) = 1)

The private key is d and n (d,n), public key is (e,n)

Encryption: c = m^e mod n ("m" is message)

Decryption: m = c^d mod n ("c" is ciphertext)13

Is RSA Secure?

To factor a 512-bit number (to find p and qfrom n) with the best known technique would

take 500,000 MIPs-years• In 500 years on a 1000 MIP/s CPU, an

eavesdropper can encrypt a list of all possible messages (using the Public Key), and compare the corresponding ciphertext to the transmitted ciphertext.• If the message is your password, make sure you

picked a good one (not in any dictionary).

• A defense is to add random bits to the message.

MIPs - Millions of Instructions per second.

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How Efficient are RSA Operations

Efficient techniques for doing exponentiation:

X * Y mod n = (X mod n) * (Y mod n)

• Do a "mod n" operation whenever a multiplier isbigger than n

To calculate x^10110111011000012 mod n

x^102 = (x^12 )^2

x^1002 = (x^10

2 )^2

x^(1012) = (x^1002

) * x15

c = E(m) =(m ^ e) mod n (the ciphertext)

D(c) = (c ^ d) mod n (decryption of c)

= m^(e*d) mod n

= m^(e*d mod Ø(n)) mod n (Number Theory)

= m^(1) mod n

= m (the plaintext message)

Does It Work?(Does D(E(m))=m)

16To experiment use: www.csc.gatech.edu/copeland/jac/8843/tools/RSA.xls

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Public-Key Systems

Encrypt/ Digital Key Decrypt Signature Exchange

RSA X X X

Diffie-Hellman X X

DSS X

Elliptic Curve X X X

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Diffie-Hellman TechniqueMutual Secret Keys or Public-Private Keys

Global Public Elements: q (large prime) and a (a < q)

User A‘s Keys: Select secret Xa (Xa < q)Public Key is Ya = a^Xa mod q

User B‘s Keys: Select secret Xb (Xb < q)Public Key is Yb = a^Xb mod q

Mutual Key is K = Yb ^Xa (A’s calculation)Ya ^ Xb (B’s calculation)a^(Xa*Xb) mod q (in both cases)

No one else knows either Xa or Xb, so they can not find out K

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+ a and q

+ message encrypted with “ K”

(Ya, a,q are A’s Public Key)

Diffie-Hellman as usedfor a Public-Private System

B has to send “ Yb” withmessage so A can decrypt it.

“Trudie” does not know Xa: Can not read message. 20

Certificate Authority generates the“signature” that is added to raw

“Certificate”

MIC

Hash

Raw “Certificate” has user name, public key, expiration date, ...

RawCert.

SignedCert.

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Generate hash codeof Raw Certificate

Encrypt hash code with CA’s private key to form CA’s signature

Signed CertificateRecipient can verify signature using CA’s public key.