computer security lecture 7 ch.13 digital signatures prepared by dr. lamiaa elshenawy

Computer Security

Lecture 7Ch.13

Digital SignaturesPrepared by Dr. Lamiaa Elshenawy

Digital Signatures

• Digital Signatures Properties Attacks and Forgeries Digital Signature Requirements Direct Digital Signature

• ElGamal Digital Signature Scheme• Schnorr Digital Signature Scheme• Digital Signature Standard The DSS Approach The Digital Signature Algorithm

Digital SignaturesKey Points

• Digital signature authentication mechanism that enables the creator of a message to attach a code that acts as a signature.

• Digital signature standard (DSS) NIST standard secure hash algorithm (SHA)

Encryption AlgorithmH(M)

Private Key

Digital Signature

Digital SignaturesKey Points

Most important development of public-key cryptography

Digital Signature

Digital SignaturesGeneric Model

Digital SignaturesEssential Elements

Digital SignaturesProperties

Message Authentication

Digital SignaturesProperties

Digital SignaturesAttacks and Forgeries

A C


• Key-only attack: C A’s public key• Known message attack: C set of

messages and their signatures.• Generic chosen message attack: C list of

messages independent of A’s public key• Directed chosen message attack: C list

of messages dependent of A’s public key signatures seen

know

access

choose

choose

before


• Total break: C A’s private key• Universal forgery: C efficient signing

algorithm that provides an equivalent way of constructing signatures on arbitrary messages

• Selective forgery: C signature for chosen message

• Existential forgery: C signature for at least one message. C control over the message

determine

find

forges

forges

no

Digital SignaturesRequirements

• Signature bit pattern signed message

• Signature information unique to the sender forgery and denial• Easy digital signature• Easy digital signature• Easy copy of digital signature in storage

• Infeasible computation digital signature

depends

use

prevent

produce

recognize & verify

forge

retain

Digital SignaturesDirect Digital Signature

Source Destination

Digital SignaturesSchemes

• ElGamal Digital Signature Scheme “Taher AlGamal (1985)”• Schnorr Digital Signature Scheme “Claus Peter Schnorr (1991-2008)”• Digital Signature Standard (DSS)

Digital SignaturesElGamal Scheme

Let q is prime number & α is a primitive root of q

Generate the private/ public keys

Sign the message


Verify the message

If V1=V2

No

Yes

Valid

Not valid


Let q=19; Primitive roots of q= {2, 3, 10, 13, 14, 15};

α=10


Alice wants to sign a messageLet m=4


Verify the message


Example• We consider q = 467; α = 2; XA = 127

• Now YA is calculated: YA = αXA mod p = 2127 mod 467 = 132• So the Alice’s pair of keys is (127) , (467,2,132)• We take m= 100 and K = 213 for the signature of this message• Here we notice that (213, 466)= 1• Calculate 213-1 mod 466 = 431• Having these parameters, we can start to calculate the signature of Alice on the message

M, which is represented by the pair (S1,S2)

• S1= αK mod q = 2213 mod 467 = 29

• S2= K-1 [m−(XA S1)] mod q = 431(100-127 x 29) mod 466 = 51

• Verification step:• αm mod q = (YA ) S1

(S1) S2 mod q• 2100 ≡ 189 (mod 467)• 13229 x 2951 ≡ 189 (mod 467) • The verification confirms that the signature is valid.

Digital SignaturesSchnorr Scheme


Sign the message


Verify the message


1. Choose p = 23, q = 11, where 11 is a prime factor of 22 = 23-1.2. Choose a such that a11 = 1 mod 23. Let a = 2, since 211 = 2048 = 1

mod 23.3. Choose a random integer s, 0 < s < q. Let s= 9, since 9<11.4. Generate a public key by calculating ν, where ν = 29 mod 23 =6


User’s Public key: ν = 6 User’s Private key: s = 9


Sign the message

1. Customer chooses r = 3 < 11, and computes x = 23 mod = 8 Customer sends x = 8 to merchant

2. Merchant sends e=5 to customer3. Customer calculates y = (3+9x5) mod 11 = 48 mod 11 = 4, and

returns y = 4 to the merchant. The signature (e , y)

1. Merchant calculates x’ ν e mod p = 8 x 6 5 mod 23 = 62208 mod 23 = 16

2. Merchant also calculates a y mod p = 2 4 = 16 These are the same so the merchant accepts that the customer

knows x

Verify the message

Digital SignaturesDigital Signature Standard (DSS)

• NIST Federal Information Processing Standard (FIPS 186)-DSS

• DSS Secure Hash Algorithm (SHA) new Digital Signature Algorithm (DSA)DSS (1991) (1993,1996)DSS (2000) (FIPS 186-2)DSS (2009) (FIPS 186-3)

published

use

presented

proposed revised

expand-version

updated

presented

presented

Digital SignaturesDigital Signature Standard (DSS)

random number

verify

Digital SignaturesDigital Signature Algorithm (DSA)

Let p, q, g are public to all• p= 18x71+1=1279 prime number• q= 71 (prime divisor of (p-1))• g= 318 mod 1279=1157• Let x= 15, k=10• y=115715 mod 71=851• User’s private key:{x=15}• User’s public key: {y=851}


Signing• r =(1157 10 mod 1279) mod 71 = 32• s =[10 -1 (123+15x32)] mod 71=39• m=123, Signature={32,39} Verifying• W=39-1 mod 71= 51• u1=123 x 51 mod 71=25

• u2=32 x 51 mod 71=70• ν = [(115725 85170) mod 1279] mod 71=32• Test: ν=r the signature is valid


Thank you for your attention

computer security lecture 7 ch.13 digital signatures prepared by dr. lamiaa elshenawy

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