Cryptography and Network
Security
Chapter 9
Fourth Edition
by William Stallings
Lecture slides by Lawrie
Brown
Private-Key Cryptography
traditional private/secret/single
key cryptography uses one key
shared by both sender and receiver
if this key is disclosed
communications are compromised
also is symmetric, parties are equal
hence does not protect sender from
receiver forging a message &
claiming is sent by sender
Public-Key Cryptography
probably most significant advance in the
3000 year history of cryptography
uses two keys a public & a private key
asymmetric since parties are not equal
uses clever application of number
theoretic concepts to function
complements rather than replaces
private key crypto
Why Public-Key
Cryptography?
developed to address two key issues:
key distribution how to have secure
communications in general without
having to trust a KDC with your key
digital signatures how to verify a
message comes intact from the claimed
sender
public invention due to Whitfield
Diffie & Martin Hellman at Stanford
Uni in 1976
known earlier in classified community
Public-Key Cryptography
public-key/two-key/asymmetric
cryptography involves the use of two keys:
a public-key, which may be known by anybody,
and can be used to encrypt messages, and
verify signatures
a private-key, known only to the recipient, used
to decrypt messages, and sign (create)
signatures
is asymmetric because
those who encrypt messages or verify signatures
cannot decrypt messages or create signatures
Public-Key Cryptography
Public-Key Characteristics
Public-Key algorithms rely on two keys
where:
it is computationally infeasible to find
decryption key knowing only algorithm &
encryption key
it is computationally easy to en/decrypt
messages when the relevant (en/decrypt) key
is known
either of the two related keys can be used for
encryption, with the other used for decryption
(for some algorithms)
Public-Key Cryptosystems
Public-Key Applications
can classify uses into 3 categories:
encryption/decryption (provide
secrecy)
digital signatures (provide
authentication)
key exchange (of session keys)
some algorithms are suitable for all
uses, others are specific to one
Security of Public Key
Schemes
like private key schemes brute force
exhaustive search attack is always
theoretically possible
but keys used are too large (>512bits)
security relies on a large enough difference in
difficulty between easy (en/decrypt) and hard
(cryptanalyse) problems
more generally the hard problem is known, but
is made hard enough to be impractical to break
requires the use of very large numbers
hence is slow compared to private key
schemes
RSA
by Rivest, Shamir & Adleman of MIT in 1977
best known & widely used public-key scheme
based on exponentiation in a finite (Galois) field
over integers modulo a prime
nb. exponentiation takes O((log n)3) operations (easy)
uses large integers (eg. 1024 bits)
security due to cost of factoring large numbers
nb. factorization takes O(e
log n log log n
) operations (hard)
RSA Key Setup
each user generates a public/private key pair
by:
selecting two large primes at random - p, q
computing their system modulus n=p.q
note (n)=(p-1)(q-1)
selecting at random the encryption key e
where 1<e<(n), gcd(e,(n))=1
solve following equation to find decryption key d
e.d=1 mod (n) and 0dn
publish their public encryption key: PU={e,n}
keep secret private decryption key: PR={d,n}
RSA Use
to encrypt a message M the sender:
obtains public key of recipient PU={e,n}
computes: C = Me mod n, where 0M<n
to decrypt the ciphertext C the owner:
uses their private key PR={d,n}
computes: M = Cd mod n
note that the message M must be
smaller than the modulus n (block if
needed)
Why RSA Works
because of Euler's Theorem:
a(n)mod n = 1 where gcd(a,n)=1
in RSA have:
n=p.q
(n)=(p-1)(q-1)
carefully chose e & d to be inverses mod (n)
hence e.d=1+k.(n) for some k
hence :
Cd = Me.d = M1+k.(n) = M1.(M(n))k
= M1.(1)k = M1 = M mod n
RSA Example - Key Setup
1. Select primes: p=17 & q=11
2. Compute n = pq =17 x 11=187
3. Compute (n)=(p1)(q-1)=16 x
10=160
4. Select e: gcd(e,160)=1; choose e=7
5. Determine d: de=1 mod 160 and d < 160
Value is d=23 since 23x7=161= 10x160+1
6. Publish public key PU={7,187}
7. Keep secret private key PR={23,187}
RSA Example En/Decryption
sample RSA encryption/decryption is:
given message M = 88 (nb. 88<187)
encryption:
C = 887 mod 187 = 11
decryption:
M = 1123 mod 187 = 88
Exponentiation
can use the Square and Multiply Algorithm
a fast, efficient algorithm for exponentiation
concept is based on repeatedly squaring base
and multiplying in the ones that are needed
to compute the result
look at binary representation of exponent
only takes O(log2 n) multiples for number n
eg. 75 = 74.71 = 3.7 = 10 mod 11
eg. 3129 = 3128.31 = 5.3 = 4 mod 11
Exponentiation
c=0;f=1
fori=kdownto0
doc=2xc
f=(fxf)modn
ifbi==1 then
c=c+1
f=(fxa)modn
returnf
Efficient Encryption
encryption uses exponentiation to power
e
hence if e small, this will be faster
often choose e=65537 (216-1)
also see choices of e=3 or e=17
but if e too small (eg e=3) can attack
using Chinese remainder theorem & 3
messages with different modulii
if e fixed must ensure gcd(e,(n))=1
ie reject any p or q not relatively prime to e
Efficient Decryption
decryption uses exponentiation to
power d
this is likely large, insecure if not
can use the Chinese Remainder
Theorem (CRT) to compute mod p & q
separately. then combine to get desired
answer
approx 4 times faster than doing directly
only owner of private key who knows
values of p & q can use this technique
RSA Key Generation
users of RSA must:
determine two primes at random - p, q
select either e or d and compute the other
primes p,q must not be easily derived
from modulus n=p.q
means must be sufficiently large
typically guess and use probabilistic test
exponents e, d are inverses, so use
Inverse algorithm to compute the other
RSA Security
possible approaches to attacking RSA
are:
brute force key search (infeasible given
size of numbers)
mathematical attacks (based on difficulty
of computing (n), by factoring modulus n)
timing attacks (on running of decryption)
chosen ciphertext attacks (given
properties of RSA)
Factoring Problem
mathematical approach takes 3 forms:
factor n=p.q, hence compute (n) and then d
determine (n) directly and compute d
find d directly
currently believe all equivalent to factoring
have seen slow improvements over the years
as of May-05 best is 200 decimal digits (663) bit with LS
biggest improvement comes from improved
algorithm
cf QS to GHFS to LS
currently assume 1024-2048 bit RSA is secure
ensure p, q of similar size and matching other constraints
Timing Attacks
developed by Paul Kocher in mid-1990s
exploit timing variations in operations
eg. multiplying by small vs large number
or IF's varying which instructions executed
infer operand size based on time taken
RSA exploits time taken in exponentiation
countermeasures
use constant exponentiation time
add random delays
blind values used in calculations
Chosen Ciphertext Attacks
RSA is vulnerable to a Chosen
Ciphertext Attack (CCA)
attackers chooses ciphertexts & gets
decrypted plaintext back
choose ciphertext to exploit properties
of RSA to provide info to help
cryptanalysis
can counter with random pad of
plaintext
or use Optimal Asymmetric Encryption
Padding (OASP)
Summary
have considered:
principles of public-key cryptography
RSA algorithm, implementation, security
