Prepared by Dr. Lamiaa Elshenawy Computer Security Lecture 7 Ch.13 Digital Signatures Prepared by Dr. Lamiaa Elshenawy

Digital Signatures ElGamal Digital Signature Scheme 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 Signatures Key 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 Algorithm H(M) Private Key Digital Signature

Digital Signatures Key Points Most important development of public-key cryptography Digital Signature

Digital Signatures Generic Model

Digital Signatures Essential Elements

Digital Signatures Properties Message Authentication

Digital Signatures Properties

Digital Signatures Attacks and Forgeries

Digital Signatures Attacks and Forgeries 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

Digital Signatures Attacks and Forgeries 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 Signatures Requirements Signature bit pattern signed message Signature information unique to the sender forgery and denial Easy digital signature Easy copy of digital signature in storage Infeasible computation digital signature depends use prevent produce recognize & verify retain forge

Digital Signatures Direct Digital Signature Source Destination

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

Digital Signatures ElGamal Scheme Let q is prime number & α is a primitive root of q Generate the private/ public keys Sign the message

Digital Signatures ElGamal Scheme Verify the message If V1=V2 No Not valid Yes Valid

Digital Signatures ElGamal Scheme Let q=19; Primitive roots of q= {2, 3, 10, 13, 14, 15}; α=10 Generate the private/ public keys Alice wants to sign a message Let m=4

Digital Signatures ElGamal Scheme Verify the message

Digital Signatures ElGamal Scheme 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 Signatures Schnorr Scheme Generate the private/ public keys Sign the message

Digital Signatures Schnorr Scheme Verify the message

Digital Signatures Schnorr Scheme Generate the private/ public keys Choose p = 23, q = 11, where 11 is a prime factor of 22 = 23-1. Choose a such that a11 = 1 mod 23. Let a = 2, since 211 = 2048 = 1 mod 23. Choose a random integer s, 0 < s < q. Let s= 9, since 9<11. 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

Digital Signatures Schnorr Scheme Sign the message Customer chooses r = 3 < 11, and computes x = 23 mod = 8 Customer sends x = 8 to merchant Merchant sends e=5 to customer Customer calculates y = (3+9x5) mod 11 = 48 mod 11 = 4, and returns y = 4 to the merchant. The signature (e , y) Verify the message Merchant calculates x’ ν e mod p = 8 x 6 5 mod 23 = 62208 mod 23 = 16 Merchant also calculates a y mod p = 2 4 = 16 These are the same so the merchant accepts that the customer knows x

Digital Signatures Digital 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 presented updated presented

Digital Signatures Digital Signature Standard (DSS) verify random number

Digital Signatures Digital Signature Algorithm (DSA)

Digital Signatures Digital Signature Algorithm (DSA)

Digital Signatures Digital 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}

Digital Signatures Digital Signature Algorithm (DSA) 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

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