1. CSCE 715: Network Systems Security
Chin-Tser Huang
University of South Carolina

2. Key Management
Asymmetric encryption helps address key distribution problems
Two aspects
distribution of public keys
use of public-key encryption to distribute secret keys
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3. Distribution of Public Keys
Four alternatives of public key distribution
Public announcement
Publicly available directory
Public-key authority
Public-key certificates
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4. Public Announcement
Users distribute public keys to recipients or broadcast to community at large
E.g. append PGP keys to messages or post to news groups or list
Major weakness is forgery
anyone can create a key claiming to be someone else’s and broadcast it
can masquerade as claimed user before forgery is discovered
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5. Publicly Available Directory
Achieve greater security by registering keys with a public directory
Directory must be trusted with properties:
contains {name, public-key} entries
participants register securely with directory
participants can replace key at any time
directory is periodically published
directory can be accessed electronically
Still vulnerable to tampering or forgery
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6. Public-Key Authority
Improve security by tightening control over distribution of keys from directory
Has properties of directory
Require users to know public key for the directory
Users can interact with directory to obtain any desired public key securely
require real-time access to directory when keys are needed
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7. Public-Key Authority
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8. Public-Key Certificates
Certificates allow key exchange without real-time access to public-key authority
A certificate binds identity to public key
usually with other info such as period of validity, authorized rights, etc
With all contents signed by a trusted Public-Key or Certificate Authority (CA)
Can be verified by anyone who knows the CA’s public key
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9. Public-Key Certificates
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10. Distribute Secret Keys Using Asymmetric Encryption
Can use previous methods to obtain public key of other party
Although public key can be used for confidentiality or authentication, asymmetric encryption algorithms are too slow
So usually want to use symmetric encryption to protect message contents
Can use asymmetric encryption to set up a session key
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11. Simple Secret Key Distribution
Proposed by Merkle in 1979
A generates a new temporary public key pair
A sends B the public key and A’s identity
B generates a session key Ks and sends encrypted Ks (using A’s public key) to A
A decrypts message to recover Ks and both use
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12. Problem with Simple Secret Key Distribution
An adversary can intercept and impersonate both parties of protocol
A generates a new temporary public key pair {KUa, KRa} and sends KUa || IDa to B
Adversary E intercepts this message and sends KUe || IDa to B
B generates a session key Ks and sends encrypted Ks (using E’s public key)
E intercepts message, recovers Ks and sends encrypted Ks (using A’s public key) to A
A decrypts message to recover Ks and both A and B unaware of existence of E
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13. Distribute Secret Keys Using Asymmetric Encryption
if A and B have securely exchanged public-keys ?
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14. Problem with Previous Scenario
Message (4) is not protected by N2
An adversary can intercept message (4) and replay an old message or insert a fabricated message
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15. Order of Encryption Matters
What can be wrong with the following protocol?
AB: N
BA: EKUa[EKRb[Ks||N]]
An adversary sitting between A and B can get a copy of secret key Ks without being caught by A and B!
Reverse the order of encryption using KRb and encryption using KUa can avoid this attack
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16. Diffie-Hellman Key Exchange
First publicly proposed public-key type scheme
By Diffie and Hellman in 1976 along with advent of public key concepts
A practical method for public exchange of secret key
Used in a number of commercial products
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17. Diffie-Hellman Key Exchange
Use to set up a secret key that can be used for symmetric encryption
cannot be used to exchange an arbitrary message
Value of key depends on the participants (and their private and public key information)
Based on exponentiation in a finite (Galois) field (modulo a prime or a polynomial) – easy
Security relies on the difficulty of computing discrete logarithms (similar to factoring) – hard
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18. Primitive Roots From Euler’s theorem: aø(n) mod n=1
Consider am mod n=1, GCD(a,n)=1
must exist for m= ø(n) but may be smaller
once powers reach m, cycle will repeat
If smallest is m= ø(n) then a is called a primitive root
if p is prime and a is a primitive root of p, then successive powers of a “generate” the group mod p
Not every integer has primitive roots
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19. Primitive Root Example: Power of Integers Modulo 19
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20. Discrete Logarithms
Inverse problem to exponentiation is to find the discrete logarithm of a number modulo p
Namely find x where ax = b mod p
Written as x=loga b mod p or x=dloga,p(b)
If a is a primitive root of p then discrete logarithm always exists, otherwise may not
3x = 4 mod 13 has no answer
2x = 3 mod 13 has an answer 4
While exponentiation is relatively easy, finding discrete logarithms is generally a hard problem
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21. Diffie-Hellman Setup All users agree on global parameters
large prime integer or polynomial q
α which is a primitive root mod q
Each user (e.g. A) generates its key
choose a private key (number): xA < q
compute its public key: yA = αxA mod q
Each user publishes its public key
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22. Diffie-Hellman Key Exchange
Shared session key for users A and B is KAB:
KAB = αxA.xB mod q
= yAxB mod q (which B can compute)
= yBxA mod q (which A can compute)
KAB is used as session key in symmetric encryption scheme between A and B
Attacker needs xA or xB, which requires solving discrete log
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23. Diffie-Hellman Example
Given Alice and Bob who wish to swap keys
Agree on prime q=353 and α=3
Select random secret keys:
A chooses xA=97, B chooses xB=233
Compute public keys:
yA=397 mod 353 = 40 (Alice)
yB=3233 mod 353 = 248 (Bob)
Compute shared session key as:
KAB= yBxA mod 353 = = 160 (Alice)
KAB= yAxB mod 353 = = 160 (Bob)
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24. Elliptic Curve Cryptography
Majority of public-key crypto (RSA, D-H) use either integer or polynomial arithmetic with very large numbers/polynomials
Imposes a significant load in storing and processing keys and messages
An alternative is to use elliptic curves
Offers same security with smaller bit sizes
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25. Real Elliptic Curves
An elliptic curve is defined by an equation in two variables x and y, with coefficients
Consider a cubic elliptic curve of form
y 2 = x 3 + ax + b
where x, y, a, b are all real numbers
also define zero point O
Have addition operation for elliptic curve
geometrically, sum of P+Q is reflection of intersection R
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26. Real Elliptic Curve Example
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27. Finite Elliptic Curves
Elliptic curve cryptography uses curves whose variables and coefficients are finite
Two families are commonly used
prime curves Ep(a,b) defined over Zp
use integers modulo a prime
best in software
binary curves E2m(a,b) defined over GF(2m)
use polynomials with binary coefficients
best in hardware
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28. Elliptic Curve Cryptography
ECC addition is analog of modulo multiply
ECC repeated addition is analog of modulo exponentiation
Need a “hard” problem equivalent to discrete logarithm
Q=kP, where Q, P belong to a prime curve
is “easy” to compute Q given k, P
but “hard” to find k given Q, P
known as the elliptic curve logarithm problem
Certicom example: E23(9,17)
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29. ECC Diffie-Hellman Can do key exchange analogous to D-H
Users select a suitable curve Ep(a,b)
Select base point G=(x1, y1) with large order n s.t. nG=O
A and B select private keys nA<n, nB<n
Compute public keys: PA=nA×G, PB=nB×G
Compute shared key: K=nA×PB, K=nB×PA
same since K=nA×nB×G
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30. ECC Encryption/Decryption
Must first encode any message M as a point on the elliptic curve Pm
Select suitable curve and point G as in D-H
Each user chooses private key nA<n and computes public key PA=nA×G
To encrypt Pm:
Cm={kG, Pm+kPB}, k random
To decrypt Cm:
Pm+kPB–nB(kG) = Pm+k(nBG)–nB(kG) = Pm
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31. ECC Security Relies on elliptic curve logarithm problem
Fastest method is “Pollard rho method”
Compared to factoring, ECC can use much smaller key sizes than with RSA
For equivalent key lengths computations are roughly equivalent
Hence for similar security ECC offers significant computational advantages
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32. Comparable Key Sizes 1
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33. Next Class Message authentication Hashing functions Message digests
Read Chapters 11 and 12
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