Chapter 3:Chapter 3:Chapter 3:Chapter 3:----

Public Key Cryptography

By:- Sarthak Patel (www.sarthakpatel.in)

Outline

� Principles Of Public-Key Cryptography

� RSA Algorithm

� Key Management

� Elliptic Curve CryptographyElliptic Curve Cryptography

� Diffie-Hellman Key Exchange

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Classical Cryptography:Secret-Key or Symmetric Cryptography

� Alice and Bob agree on an encryption method and a shared key.

� Alice uses the key and the encryption method to encrypt (or encipher) a message and sends it to Bob.

� Bob uses the same key and the related decryption method to � Bob uses the same key and the related decryption method to decrypt (or decipher) the message.

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Advantages of Classical Cryptography

� There are some very fast classical encryption (and decryption) algorithms

� Since the speed of a method varies with the length of the key, faster algorithms allow one to use longer key values.

� Larger key values make it harder to guess the key value -- and � Larger key values make it harder to guess the key value -- and break the code -- by brute force.

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Disadvantages of Classical Cryptography

� Requires secure transmission of key value

� Requires a separate key for each group of people that wishes to exchange encrypted messages (readable by any group member)� For example, to have a separate key for each pair of people, 100 � For example, to have a separate key for each pair of people, 100 people would need 4950 different keys.

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Public-Key Cryptography: Asymmetric

Cryptography

� Alice generates a key value (usually a number or pair of related numbers) which she makes public.

� Alice uses her public key (and some additional information) to determine a second key (her private key).

� Alice keeps her private key (and the additional information she used to construct it) secret.

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Public-Key Cryptography (continued)

� Bob can use Alice’s public key to encrypt a message for Alice.

� Alice can use her private key to decrypt this message.

� No-one without access to Alice’s private key (or the information used to construct it) can easily decrypt the message.message.

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Private-Key Cryptography� traditional private/secret/single key cryptography uses one key

� Key is shared by both sender and receiver

� if the key is disclosed communications are compromised

� also known as symmetric, both parties are equal � hence does not protect sender from receiver forging a message & claiming is sent by sender

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Public-Key Cryptography� probably most significant advance in the 3000 year history of cryptography

� uses two keys – a public key and a private key

� asymmetric since parties are not equal

� uses clever application of number theory concepts to function

� complements rather than replaces private key cryptography

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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) signaturesmessages, and sign (create) signatures

� is asymmetric because� those who encrypt messages or verify signatures cannotdecrypt messages or create signatures

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Public-Key Cryptography

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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

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Public-Key Characteristics� Public-Key algorithms rely on two keys with the characteristics that it is:� computationally infeasible to find decryption key knowing only algorithm & encryption key

� computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known(en/decrypt) key is known

� either of the two related keys can be used for encryption, with the other used for decryption (in some schemes)

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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 � some algorithms are suitable for all uses, others are specific to one

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Security of Public Key Schemes� like private key schemes brute force exhaustive searchattack 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) between easy (en/decrypt) and hard (cryptanalyse) problems

� more generally the hard problem is known, its just made too hard to do in practise

� requires the use of very large numbers� hence is slow compared to private key schemes

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Difference between Symm/Asymm

Symmetric (Private Key) Asymmetric (Public Key)

Same key for encry/ decry Asymmetric uses both a public and private

key

Very fast encry Slower encry

Key exchange is a big problem No problem at allKey exchange is a big problem No problem at all

Mainly for Encry/ Decry, cannot be used for

digital signature

Can be used for both

More number of keys required Less than Symmetric

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RSA Algorithm� 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

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Algorithm1. Choose two large prime number P and Q.2. Calculate N=P*Q3. Select the public key(i.e. encryption key) E such that it is not a

factor of (P-1) and (Q-1)4. Select the private key(i.e. decryption key) D such that

following equation is true:following equation is true:(D*E) mod (P-1) * (Q-1)= 1

5. For encryption calculate the cipher text CT from the PT as : CT= PTE mod N

6. Send CT as the cipher text to the receiver.7. For decryption calculate the cipher text PT from the CT as :

PT= CTD mod N

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RSA Example1. Select primes: p=17 & q=11

2. Compute n = pq =17×11=187

3. Compute ø(n)=(p–1)(q-1)=16×10=160

4. Select e : gcd(e,160)=1; choose e=7

5. Determine d: d*e mod 160=1 and d < 160Value is d=23 since 23×7=161

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RSA Example cont� sample RSA encryption/decryption is:

� given message M = 88 (nb. 88<187)

� encryption:C = 887 mod 187 = 11

� decryption:� decryption:M = 1123 mod 187 = 88

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Diffie-Hellman Key Exchange1. Select two prime numbers: n and g (public)

2. Alice (Sender) chooses another random number x, and calculates A such that: A= gx mod n

3. Alice Sends it to Bob(Receiver)

4. Bob chooses another random number y and calculates B 4. Bob chooses another random number y and calculates B such that: B=gy mod n

5. Bob sends it to Alice

6. K1 = Bx mod n

7. K2 = Ay mod n

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Example for Deffie-Hellman1. n= 11, g=7

2. Let x=3. Then A=73 mod 11= 343 mod 11= 2

3. Alice sends 2 to Bob

4. Let y=6. Then B=76 mod 11= 117649 mod 11=4

Bob sends 4 to Alice5. Bob sends 4 to Alice

6. K1= 43 mod 11= 64 mod 11= 9

7. K2= 26 mod 11= 64 mod 11= 9

Note: K1=K2

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Elliptic Curve Cryptography

� The Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.fields.

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ECC� An elliptic curve is the set of solutions of an equation of the form

y2 = x3 + ax + b .

Where the co-efficeints a and b are elements of the field and

4a3 + 27b2 ≠ 0. Each value of the 'a' and 'b' gives a different elliptic

curve.

� An equation of this kind can be studied over various mathematical � An equation of this kind can be studied over various mathematical

structures, such as a ring or a field.

� One interesting property of set of solutions of elliptic curve is that it

forms a group which enables us to do Cryptography. The public key is

a point in the curve and the private key is a random number. The

public key is obtained by multiplying the private key with the

generator point G in the curve.

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ECC Diffie-Hellman Key ExchangeGlobal Public Elements

Eq(a,b) – Elliptic curve with parameter a,bG – Point on elliptic curve

User A Key GenerationSelect Private nA

Calculate Public PA PA = nA * G

User B Key Generation

Calculation of Secret Key by User AK = nA * PB

User B Key GenerationSelect Private nB

Calculate Public PB PB = nB * G

Calculation of Secret Key by User BK = nB * PA

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Applications of ECCApplications of ECCApplications of ECCApplications of ECC� Many devices are small and have limited storage and computational power

� Where can we apply ECC?� Wireless communication devices� Smart cards� Smart cards� Web servers that need to handle many encryption sessions� Any application where security is needed but lacks the power, storage and computational power that is necessary for our current cryptosystems

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Benefits of ECCBenefits of ECCBenefits of ECCBenefits of ECC�Same benefits of the other cryptosystems: confidentiality, integrity, authentication and non-repudiation but…

�Shorter key lengths� Encryption, Decryption and Signature Verification speed up

� Storage and bandwidth savings� Storage and bandwidth savings

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Key Management (Key Distribution)

Key Distribution Center (KDC)

IDA || IDB ||N1

InitiatorA

Responder B

E (Kb, [Ks || IDA])

E (Ks, N2)

E (Ks, f(N2))

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Public key Distribution� Public Announcement

� Public Available Directory

� Public Key Authority

� Public Key Certificates

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THE ENDTHE END

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