chapter 3 encryption algorithms & systems (part c)

Chapter 3

Encryption Algorithms & Systems (Part C)

csci5233 computer security & integrity (Chap. 3)

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Outline

NP-completeness & EncryptionNP-completeness & Encryption Symmetric (secret key) vs Asymmetric (public key) Symmetric (secret key) vs Asymmetric (public key)

EncryptionsEncryptions Popular Encryption AlgorithmsPopular Encryption Algorithms

– Merkle-Hellman KnapsacksMerkle-Hellman Knapsacks– RSA Encryption– El Gamal Algorithms– DES

Hashing Algorithms Key Escrow & Clipper


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RSA Encryption 1978: Rivest, Shamir, Adelman Public key encryption Remains secure to date Encryption key (e) and decryption key (d) are

interchangeable. The two keys, e and d, are carefully chosen such that

C = Pe mod n (encryption) and P = Cd mod n (decryption).


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Euler Totient Function (n): the number of positive integers less than n and

are relatively prime to n.

If n is prime:

(n) = n – 1

When n = p * q, where both p and q are primes and p q:

(n) = (p) * (q) = (p – 1) * (q – 1)


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RSA Encryption Public key = (e, n) Private key = (d, n)

Step 1: Choose n, p, & q

n = p * q, where both p and q are primes and p q

Example: n = 143 = p * q = 11 * 13


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RSA Encryption Step 2: Choose e.

e is relatively prime to (n). That is, e is relatively prime to (p-1)*(q-1).

Example: e = 17, which is relatively prime to 10*12.

Step 3: Compute d.

d is the inverse of e mod (p-1)*(q-1).

Use the algorithm on page 81 to compute inverses.

Note: A Java implementation of the algorithm is available at the class page.

Example: d = e-1 mod (p-1)*(q-1) = 17-1 mod 120= 113


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RSA Encryption An example (pp.94-95): P = 7

Let n = 143, p = 11, q = 13, and e = 11.

Note: e is relprime to (p-1)*(q-1).

Then d = 11

Note: d is the inverse of e mod (p-1)*(q-1).

Encryption:

C = Pe mod n = 711 mod 143 = 106

Decryption:

P = Cd mod n = 5011 mod 143 = 7


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RSA Encryption Another example: P = 7

Let n = 143, p = 11, q = 13, and e = 17.

Note: e is relprime to (p-1)*(q-1).

Then d = 113


Encryption:

C = Pe mod n = 717 mod 143 = 50

Decryption:

P = Cd mod n = 50113 mod 143 = 7


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RSA Encryption Still another example: P = 55

Let n = 285, p = 19, q = 17, and e = 37.

Note: e is relprime to (p-1)*(q-1), 288.

d = 109


Encryption:

C = Pe mod n = 5537 mod 288 = 55

Decryption:

P = Cd mod n = 55109 mod 288 = 55


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RSA Encryption The cryptographer’s job:

– Find three primes, p, q, and e, where

p * q = n and

e is relatively prime to (p-1)*(q-1).

– Compute d based on e and n.

The challenge: p, q, and e must be large enough primes.

See discussions on p.95.


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RSA Encryption The cryptanalyst’s job:

P = Cd mod n

– Available: (e, n).

– Find two primes p and q, such that p * q = n and e is relatively prime to (p-1)*(q-1).

– Compute d: d = inverse (e, (p-1)*(q-1))

Q: Where’s the secrecy?

Q: Given n and a prime e, how hard is it to find two distinct primes, p and q, such that p*q = n and (p-1)*(q-1) is relprime to e?


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El Gamal Algorithm A public key algorithm 1984 Important in the U.S. DSS (Digital Signature Standard) Digital Signatures

The sender computes the digital signature using his own private key.

DS = E (Keypriv, P)

The receiver verifies the signature using the sender’s public key.

P = D (Keypub, DS)


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El Gamal Algorithm To generate a key pair:

– Choose a prime p and two integers, a and x,

such that a < p and x < p.

– The prime p should be chosen so that (p-1) has

a large prime factor q.

– Calculate the public key: y = ax mod p.

– Private key: x

– Public key: y


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El Gamal Algorithm (The sender) To sign a message m:

– Choose a new random integer k, 0 < k < p-1 and

k is relprime to (p-1).

– Compute r = ak mod p.

– Compute s = k-1 ( m – xr ) mod (p-1)

– The message signature: r and s.

Verification: A recipient use the public key (y) to compute ( y r r s ) mod p and determine if it is equivalent to am mod p.


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Hash Algorithm A hash algorithm is a check function that protects

data against modifications. C.f., checksum in network transmission Hash functions produce a reduced form of a body of

data (called a digest or check value) such that most changes to the data will also change the reduced form.

A cryptographic hash function uses a cryptographic function as part of the hash function.

1992: Secure Hash Algorithm (SHA)


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Secure Hash Algorithm (SHA) 1992: NIST Input data < 264 bits 160-bit digest Strength: diffusion, the avalanche effect See Fig. 3-9, p.99

C.f., MD4, MD5

Both MD5 and SHA are variants of the MD4 by Rivest.

Strength: MD4 < MD5 < SHA


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Summary Public key encryption algorithms: Merkle-Hellman,

RSA, El Gamal SHA

Next: DES, Key Escrow

chapter 3 encryption algorithms & systems (part c)

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