Public-Key Encryption
Introduction to Number Theory
•A prime number is an integer that can only be divided without
remainder by positive and negative values of itself and 1. Prime
numbers play a critical role both in number theory and in
cryptography.
•Two theorems that play important roles in public-key
cryptography are Fermat's theorem and Euler's theorem.
•An important requirement in a number of cryptographic
algorithms is the ability to choose a large prime number. An area
of ongoing research is the development of efficient algorithms
for determining if a randomly chosen large integer is a prime
number.
1
Fermat's Theorem
Fermat's theorem states the following: If p is prime and a is a
positive integer not divisible by p, then
An alternative form of Fermat's theorem is also useful: If p is
prime and a is a positive integer, then
•
•
•
•
p = 5,a = 3
ap = 35 = 243 3(mod 5) = a(mod p)
p = 5, a = 10
ap = 105 = 100000 10(mod 5) = 0(mod 5) =
a(mod p)
2
Euler Totient Function ø(n)
• to compute ø(n) need to count number of
residues to be excluded
• in general need prime factorization, but
– for p (p prime)
– for p.q (p,q prime)
ø(p)
ø(pq)
= p-1
=(p-1)x(q-1)
• eg.
ø(37) = 36
ø(21) = (3–1)x(7–1) = 2x6 = 12
3
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
4
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
5
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
6
RSA
• by Rivest, Shamir & Adleman of MIT in 1977
• best known & widely used public-key scheme
• uses large integers (eg. 1024 bits)
• security due to cost of factoring large numbers
7
RSA ingredients
We are now ready to state the RSA scheme. The
:ingredients are the following
We are now ready to state the RSA scheme. The ingredients are
the following:
p,q, two prime numbers
(private, chosen)
n = pq
(public, calculated)
e, with gcd(f(n),e) = 1;1 < e <
f(n)
(public, chosen)
d e1(mod f(n))
(private, calculated)
8
RSA Algorithm
9
RSA Example - Key Setup
Select primes: p=17 & q=11
Compute n = pq =17 x 11=187
Compute ø(n)=(p–1)(q-1)=16 x 10=160
Select e: gcd(e,160)=1; choose e=7
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}
1.
2.
3.
4.
5.
10
11
Clarification
• we need to calculate C = 887 mod 187. Exploiting the
properties of modular arithmetic, we can do this as
follows:
887 mod 187 = [(884 mod 187) x (882 mod 187) x
(881 mod 187)] mod 187
881 mod 187 = 88
882 mod 187 = 7744 mod 187 = 77
884 mod 187 = 59,969,536 mod 187 = 132
887 mod 187 = (88 x 77 x 132) mod 187 =
894,432 mod 187 = 11
12
Security of RSA
Four possible approaches to attacking the RSA:
Brute force: This involves trying all possible private
keys.
Mathematical attacks: There are several approaches,
all equivalent in effort to factoring the product of two
primes.
Timing attacks: These depend on the running time of
the decryption algorithm.
Chosen cipher text attacks: This type of attack
exploits properties of the RSA algorithm.
13
Factoring Problem
• mathematical approach takes 3 forms:
– factor n=p.q, hence compute ø(n) and then d
– determine ø(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
14
Timing Attacks
• developed by Paul Kocher in mid-1990’s
• 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
15
Chosen Cipher text 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)
16
Examples on RSA
• Perform encryption and decryption using the
RSA algorithm, for the following:
1. p = 5; q = 11, e = 3; M = 9
2. p = 7; q = 11, e = 17; M = 8
3. p = 11; q = 13, e = 11; M = 7
4. p = 3; q = 11, e = 7; M = 5
17
Examples on RSA
•
•
•
•
•
•
•
•
•
•
RSA Algorithm Example
Choose p = 3 and q = 11
Compute n = p * q = 3 * 11 = 33
Compute φ(n) = (p - 1) * (q - 1) = 2 * 10 = 20
Choose e such that 1 < e < φ(n) and e and n are coprime.
Let e = 7
Compute a value for d such that (d * e) % φ(n) = 1. One
solution is d = 3 [(3 * 7) % 20 = 1]
Public key is (e, n) => (7, 33)
Private key is (d, n) => (3, 33)
The encryption of m = 2 is c = 27 % 33 = 29
The decryption of c = 29 is m = 293 % 33 = 2
18
End of Sections
Creative Minds
never gives up
Thank you
19