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Public-Key Cryptography

- Dr. Ron Rymon
- Efi Arazi School of Computer Science
- IDC, Herzliya. 2010/11

Pre-Requisites Conventional Cryptography

Overview

- Public Key Cryptography
- Crossword puzzles
- Diffie-Hellman
- RSA
- Elliptic Curves
- Digital Signatures
- Key Management for Public-Key Cryptography

Public-Key Cryptography

Main sources Network Security Essential /

Stallings Applied

Cryptography / Schneier

Motivation

- Until early 70s, cryptography was mostly owned by

government and military - Key distribution is more manageable and better

funded - Symmetric cryptography not ideal for

commercialization - Enormous key distribution problem most parties

may never meet physically - Must ensure authentication, to avoid

impersonation, fabrication - Few researchers (Diffie, Hellman, Merkle), in

addition to the IBM group, started exploring

Cryptography because they realized it is critical

to the forthcoming digital world - Privacy
- Effective commercial relations
- Payment
- Voting

Public-Key Cryptography

- Idea use separate keys to encrypt and decrypt
- First proposed by Diffie and Hellman
- Independently proposed by Merkle (1976)
- Pair of keys for each user
- generated by the user himself
- Public key is advertised
- Private key is kept secret, and is

computationally infeasible to discover from the

public key and ciphertexts - Each key can decrypt messages encrypted using the

other key - Applications
- Encryption
- Authentication (Digital Signature)
- Key Exchange (to establish Session Key)

Crossword Puzzles

- Ralph Merkles Key Exchange Algorithm
- Alice generates MANY crossword puzzles and sends

to Bob - Bob chooses ONE and solves it
- The solution includes an identifier, and the key
- Bob communicates the identifier to Alice
- Alice and Bob communicate using the key
- Important observation Eve would have to solve

ALL puzzles to identify the right one and the

key. - First attempt, cumbersome, and not working, but

very revolutionary at the time - Later, Merkle suggested to use NP-Hard problems
- Hard to solve, but easy to check (e.g.,

knapsack). - Also proven inadequate later...

Diffie-Hellman Key Exchange

- First public-key algorithm, based on the

difficulty of computing discrete logarithms

modulo n - Protocol
- Use key exchange protocol to establish session

key - Use session key to encrypt actual communication
- Algorithm
- Choose a large prime n, and a primitive root g

Bob

Alice

Xgx mod n

select x

Ygy mod n

select y

Compute KYx mod n

Kgxy mod n

Compute KXy mod n

Diffie-Hellman Protocol

- DH does not offer authentication
- Trudy can use a man-in-the-middle attack
- Impersonating Alice to Bob and vice versa
- Using his own key (or different keys) with each
- Solution establish a public directory
- Each person publishes (g,n,gx) this is the

public key - Note g,n may be different from one user to

another - Make sure not to select x0/1 mod n

Two-key Public-Key Encryption

- Sender uses the public key of the receiver to

encrypt - Receiver uses her private key to decrypt

Two-Key Public-key Authentication

- The sender encrypts some message (e.g. a

certificate) with his own private key - The receiver, by decrypting, verifies key

possession

Public-Key AlgorithmsThe Requirements

- It is computationally feasible to generate a pair

of keys - It is computationally easy to encrypt using the

public key - It is computationally easy to decrypt using the

private key - It is computationally infeasible to compute the

private key from the public key - It is computationally infeasible to recover the

plaintext from the public key and ciphertext - Either of the keys can decrypt a message

encrypted using the other key

RSA

- Developed by Rivest, Shamir, and Adleman (1977)
- Most widely used public key algorithm
- Receives its security from the difficulty of

factoring large numbers - Actually discovered first by UK GCHQ (Ellis and

Cocks) in 1973 ! - Algorithm
- Works as a block cipher, where each

plaintext/ciphertext block is integer between 0

and n (for some n2k) - Each receiver chooses e, d
- The values of e, and n are made public d is kept

secret - Encryption CMe mod n
- Decryption MCd mod n Med mod n
- Requisites
- Find e, d such that MMed mod n, for all Mltn
- Make sure that d cannot be computed from n and e,

not even if a ciphertext is available

RSA Keys and Key Generation

- Select primes p and q, npq
- ?(n)(p-1)(q-1) Euler totient of n number of

integers between 1 and n that are relatively

prime to n, i.e., m gcd(m,n)1 - Select integer elt?(n) such that gcd(?(n),e)1
- Guarantees that e-1 exists
- Calculate d such that de-1 mod ?(n),
- Use Euler extended GCD algorithm
- Now, for every Mltn, we have
- Med M 1 mod ?(n) M
- Note
- The message could have been encrypted with d and

decrypted by e

Recall Math Backgrounder

- Fermats Little Theorem
- For a prime p, ?a such that 0ltaltp, a(p-1)1 mod p
- Eulers extension
- For any n, ?a such that 0ltaltn, a ?(n) mod n 1

mod n - For primes p,q, ?a such that gcd(a,pq)1,

a(p-1)(q-1) 1 mod pq - Hence, Med mod n Mk(p-1)(q-1)1 mod n 1xM M
- To generate primes, use primality test
- For a non-prime, Fermats theorem will usually

fail on a random a - Carmichael numbers are rare exception, and if

chosen decryption wont work. Can reduce the

probability by checking more as - Primes are dense enough (almost one of every k

k-bit numbers) - GCD to select e takes O(log n) time
- Calculate de-1mod ?(n) - Euler extended GCD.

O(log n) - Exponentiation (Encrypt/Decrypt) takes O(log n)

time - RSA gets its security from the difficulty of

factoring npq

RSA Example

- Key Generation
- Select p7, q17, npq119, ?(119)96
- Select e5 Calculate d77 (7753851 mod 96)

Attacks on RSA Algorithm

- If one could factor n, which is available, into p

and q, then d could be calculated (as inverse of

e), and then the message deciphered - If one could guess the value of ?(n)(p-1)(q-1),

even without factoring n, then again d could be

computed as the inverse of e

Attacks on RSA Protocol

- Chosen ciphertext attack
- Attack get sender to sign (decrypt) a chosen

message - Inputs original (unknown) ciphertext CMe
- Construct
- XRe mod n, for a random R
- YXC mod n
- Ask sender to sign Y, obtaining UYd mod n
- Compute
- TR-1 mod n
- TU mod n R-1Yd mod n R-1 Xd Cd mod n Cd mod

n M - Exploits preservation of multiplication in group
- Conclusion
- never sign a random message
- sign only hashes
- use different keys for encryption and signature

Other precautions when implementing RSA protocol

- Do not use same n for multiple users
- A third party can sometimes decipher if same

message is encrypted using both encryption

(public) keys, without needing the decryption

(private) key - Always pad messages with random numbers, making

sure that M is about same size as n - If e is small, there is an attack that uses

e(e1)/2 linearly dependent messages, and if

messages are small its easier to find linearly

dependent ones - Do not choose low values for e and d
- For e, see above, and there is also attack on

small ds

Elliptic Curves Cryptography

- ECC addresses the cost of exponentiation in DH

and RSA - Use Abelian groups w/ addition defined on cubic

equations

- E.g., y2 x3 ax b (for some a, b)
- For RPQ, find third point of intersection on

line that connects P and Q (use tangent line if

PQ). This is R, and R is its mirror. - O is a point of infinity and is defined as

OP(-P). As a result it is also the identity

since POP - Can also be defined over GF(p)
- Consider QkP mod p
- Easy to compute Q from k, P
- Difficult to determine k from P, Q (except

through brute force)

Elliptic Curves Key Exchange

- Key Generation
- Select/agree on cubic curve (p, a, b) ---

public - Select a base point G with a high order n ---

public - i.e., smallest n such that nGO
- Private key of Alice is an integer KA lt n
- Public key of Alice is KAG
- Key Exchange
- Alice and Bob send public key to each other
- Each of them multiplies the result by own private

key - Agreed Key KA KBG
- Like DH but uses addition instead of

exponentiation

Timing and Power Attacks

- Ciphertext-only attack
- No mathematical analysis
- How it works
- Measure the effort (time, power) to decrypt a

message - Correlate the effort to the probability that

certain key bits are on - Idea
- Different algorithms work more on certain

combinations of bit values - E.g., in RSA the exponentiation effort depends on

the number of bits that are 1 - Solutions
- Idle computation to randomize even out

Other Public-Key Algorithms

- Merkle-Hellman Knapsack Algorithms
- First public-key cryptography (not key exch)

algorithm (1976) - patented - Encode a message as a series of solutions to

knapsack problems (NP-Hard). Easy

(superincreasing) knapsack serves as private key,

and a hard knapsack as a public key. - Broken by Shamir and Zippel in 1980, showing a

reconstruction of superincreasing knapsacks from

the normal knapsacks - Rabin
- Based on difficulty of finding square roots

modulo n - Encryption is faster CM2 mod n (npq)
- Decryption is a bit complicated and the plaintext

has to be selected from 4 possibilities (also

makes it difficult to use it for signature) - El Gamal
- Based on difficulty of calculating discrete

logarithms in a finite field - Elliptic Curves can be used to implement El Gamal

and Diffie-Hellman faster

Digital Signatures

Main sources Network Security Essential /

Stallings Applied

Cryptography / Schneier

Public-Key Digital Signature

- Same as authentication
- The sender encrypts a message with his own

private key - The receiver, by decrypting, verifies key

possession

Digital Signatures

- It is possible to use the entire message,

encrypted with the private key, as the digital

signature - But, this is computationally expensive
- And, anyone can then decrypt the original message
- Alternatively, a digest can be used
- Should be short
- Prevent decryption of the original message
- Prevent modification of original message
- Difficult to fake signature for
- If message authentication (integrity) is needed,

we may use the hash code of the message - If only source authentication is needed, a

different message can be used (certificate)

Digital Signature Algorithm (DSA)

- Proposed in 1991 by NIST as a standard (DSS)
- Based on difficulty of computing discrete

logarithms (like Diffie-Hellman and El Gamal) - Encountered resistance because RSA was already

de-facto standard, and already drew significant

investment - DSA cannot be used for encryption or key

distribution - RSA is advantageous in most applications (exc.

smart cards) - RSA is 10x faster in signature
- DSA is faster in verification
- Concerns about NSA backdoor (table can be built

for some primes) - Key size was increased from 512 to 2048 and 3072

bits - In DSA, the key size needs to be 4 times the

security level - DSA has an Elliptic Curve version
- Faster to compute, and requires half the bits

Description of DSA

- Parameters
- p is a prime number with up to 1024 bits public

key - q is a 160-bit factor of (p-1), and itself prime

public key - gh(p-1)/q mod p (h is random) public key
- x is the private key and is smaller than q --

private key - ygx mod p is part of the public key public key

- Signature
- Given a message M, generate a random kltq -- keep

secret - Signature is a pair (r,s)
- send r(gk mod p) mod q signature
- send sk-1(H(M)xr) mod q signature
- If r0 or s0, choose a new k
- Verification
- Compute ws-1 mod q
- Compute u1H(M)w mod q u2rw mod q
- Compute v(gu1yu2 mod p) mod q
- If vr then the signature is verified verificatio

n

Key Generation in DSA

- Generate q as a SHA on an arbitrary 160-bit

string - If not prime, try another string
- Use Rabin method for primality testing
- To get (p-1)
- Concatenate additional 160 bit numbers until you

get to the right size (e.g., 1024) - Subtract the remainder after division by 2q
- q is a factor from construction
- Since p-1 is even, then 2 is also a factor
- If p is not prime, repeat the process

One-Time Signatures (Merkle)

- Key Generation
- Let t n 1 log n, where n is message size
- Select random K1, Kt (private key)
- Let ViH(Ki) for a hash function H (public key)
- Signature
- Let C be the number of 0s in message M
- Let W M C, and let A1 At be Ws bits
- Signature is (S1 Su) such that SjKl if Al is

the jth 1-bit of W - Verification
- Compute W as above
- Compute H(Si) for each bit and compare to

(properly indexed) Vj

Key Management for Public Key Cryptographic

Protocols

Main sources Network Security Essential /

Stallings Applied

Cryptography / Schneier

Certificate Authority Verifying the Public Key

- How to ensure that Charles doesnt pretend to be

Bob by publishing a public-key for Bob. Then,

using a Man-in-the-Middle attack, Charles can

read the message and reencrypt-resend to Bob

- Bob prepares certificate with his identifying

information and his public key - The Certificate Authority (CA) verifies the

details and sign Bobs certificate - Bob can publish the signed certificate

More on (Public) Key Management

- Alice may have more than one key
- e.g., personal key and work key
- Where shall Alice store her keys?
- Alice may not want to trust her work

administrator with her personal banking key - Distributed certification a la X.509
- CA certifies Agents who certify organizations who

certify others - Distributed certification a la PGP
- Alice will present her certificate with

introducers who will vouch for her (PKI

parties) - Key Escrow
- US American Escrowed Encryption Standard suggests

that private keys be broken in half and kept by

two Government agencies - Clipper for cellular phone encryption
- Capstone for computer communication

Summary

Cryptography Summary

- Cryptography (and steganography) were always

considered a strategic tool - Used mostly by governments and military

organizations - Served to keep top secrets and in wars
- Different generations were characterized by

either the cryptographers or cryptanalysts

winning the battle - Today, cryptographers seem certainly on top, with

unbreakable ciphers (but, remember Vigeneres

unbreakable cipher) - Must remember that cryptanalysis is not the only

attack - It is usually the hardest way to break a message
- May attack human weaknesses in crypto protocol
- May attack communication, hosts, etc.
- Much easier to get information using good old

3Bs bribery, burglary, and bending

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