Information Security and Management 13. Digital Signatures and Authentication Protocols

Chih-Hung WangFall 2011

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•Dispute of message authentication▫ Message authentication protects two

parties who exchange messages from any third party. However, it does not protect the two parties against each other.

▫ Several forms of dispute between the two are possible

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Digital Signature

▫ The following disputes could arise Receiver may forge a different message and

claim that it came from sender. Sender can deny sending the message

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Problem of Authentication

Generic Model of Digital Signature Process

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•Mary may forge a different message and claim that it came from John. Mary would simply have to create a message and append an authentication code using the key that John and Mary share.

•John can deny sending the message. Because it is possible for Mary to forge a message, there is no way to prove that John did in fact send the message.

Possible Disputes Using MACs

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Simplified Depiction of Essential Elements of Digital Signature Process

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•The digital signature is analogous to the handwritten signature. It must have the following properties:▫ It must be able to verify the author and the

date and time of the signature▫ It must be able to authenticate the

contents at the time of the signature▫ The signature must be verifiable by third

parties, to resolve dispute

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Properties

• The signature must be a bit pattern that depends on the message being signed

• The signature must use some information unique to the sender, to prevent both forgery and denial

• It must be relatively easy to produce the digital signature

• It must be relatively easy to recognize and verify the digital signature

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Requirements (1/2)

• It must be computationally infeasible to forge a digital signature, either by constructing a new message for an existing digital signature or by constructing a fraudulent digital signature for a given message

• It must be practical to retain a copy of the digital signature in storage

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Requirements (2/2)

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Digital Signature Concept

Signer Receiver

Signer’s Private Key

Signed Document

Signer’s Public Key

Verify the signature

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Dispute Concept

Sender ReceiverDispute

Signer’s Digital Signature

Third Party

Verify& Judge

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RSA Digital Signature

Signer Receiver

Signer’s private key: d

Signed Document

Signer’s public key : (e,n)

Verifyh(M) ?= Sige mod n =( h(M)d)e mod n

Sig=h(M)d mod n

• The National Institute of Standards and Technology (NIST) has published Federal Information Processing Standard FIPS PUB 186, known as the Digital Signature Standard (DSS).

• The DSS makes use of the Secure Hash Algorithm (SHA)

• The DSS was originally proposed in 1991 and revised in 1993 in response to public feedback concerning the security of the scheme

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Digital Signature Standard (DSS)

•The DSS uses an algorithm that is designed to provide only the digital signature function

•Unlike RSA, it cannot be used for encryption or key exchange

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DSS Concept (1/2)

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DSS Concept (2/2)

•Discrete Logarithms (page 228-233)▫ Consider the equation

y = gx mod p Given g,x,and p, it is straightforward matter

to calculate y Given y, g, and p, it is, in general, very

difficult to calculate x Computational complexity

e((ln p)1/3ln(ln p))2/3

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DSS Algorithm

•Setup▫ p large prime: bit length of between 512

and 1024 bits in increments of 64 bits 2L-1 < p < 2L and

512 <= L <= 1024▫ q prime divisor of (p-1) and 2159 < q <

2160， i.e., the length is 160 bits▫ g = h(p-1)/q mod p, where 1<h < p-1 ， h is

an integer （ I.e., gq = 1 mod p ）

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DSS Algorithm

• Sign▫ User A’s private key:

x: random 0 < x <q▫ User A’s public key:

y = gx mod p▫ Signing

Randomly select k, 0<k<q calculate r = (gk mod p) mod q calculate s = [k-1(H(M) + xr)] mod q Signature = (r,s)

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DSS Algorithm

•Sign

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DSS Algorithm

•Verify▫ Verifying (r’, s’) & (M’)

Calculate w = (s’)-1 mod q Calculate u1 = [H(M’)w] mod q Calculate u2 = (r’)w mod q Calculate v = [(gu1yu2) mod p] mod q Verify v = r’ (?). If yes, (r’, s’) is a valid

signature on the message M’

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DSS Algorithm

•Verify

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DSS Algorithm

•DSS cannot be used for encryption or key distribution

•DSS was developed by the NSA, and there may be a trapdoor in the algorithm

•DSS is slower than RSA•RSA is the ISO 9796, the international

digital signature standard

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Criticisms of DSS (1/2)

•The DSS selection process was not public; sufficient time for analysis has not been provided

•DSS may infringe on other pattern•The key size is too small

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Criticisms of DSS (2/2)

ElGamal Digital Signatures

•In 1984, T. Elgamal announced a public-key scheme based on discrete logarithms, closely related to the Diffie-Hellman technique. ▫Use private key for encryption (signing)▫Uses public key for decryption (verification)

•The security of ElGamal is based on the difficulty of computing discrete logarithms

•Each user (eg. A) generates their key▫chooses a secret key (number): 1 < xA < q-1

▫compute their public key: yA = axA mod q

ElGamal Digital Signature•Alice signs a message M to Bob by computing

▫hash m = H(M), 0 <= m <= (q-1)▫Chose random integer K with 1 <= K <= (q-1)

and gcd(K,q-1)=1▫Compute temporary key: S1 = a

k mod q

▫Compute K-1 the inverse of K mod (q-1)▫Compute the value: S2 = K-1(m-xAS1) mod (q-1)▫Signature is:(S1,S2)

•Any user B can verify the signature by computing▫ V1 = a

m mod q

▫ V2 = yAS1 S1

S2 mod q▫ signature is valid if V1 = V2

ElGamal Signature Example •Use field GF(19) q=19 and a=10•Alice computes her key:

▫A chooses xA=16 & computes yA=1016 mod 19 = 4

•Alice signs message with hash m=14 as (3,4):▫Choosing random K=5 which has gcd(18,5)=1▫Computing S1 = 10

5 mod 19 = 3

▫Finding K-1 mod (q-1) = 5-1 mod 18 = 11▫Computing S2 = 11(14-16.3) mod 18 = 4

•any user B can verify the signature by computing▫ V1 = 10

14 mod 19 = 16

▫ V2 = 43.34 = 5184 = 16 mod 19▫ since 16 = 16 signature is valid

Schnorr Digital Signatures•Also uses exponentiation in a finite (Galois)

▫Security based on discrete logarithms, as in D-H

•Minimizes message dependent computation▫Multiplying a 2n-bit integer with an n-bit

integer•Main work can be done in idle time•Have using a prime modulus p

▫p–1 has a prime factor q of appropriate size▫Typically p 1024-bit and q 160-bit numbers

Schnorr Key Setup

•Choose suitable primes p , q•Choose a such that a

q = 1 mod p

•(a,p,q) are global parameters for all•Each user (eg. A) generates a key

▫Chooses a secret key (number): 0 < sA < q

▫Compute their public key: vA = a-sA mod q

Schnorr Signature•User signs message by

▫Choosing random r with 0<r<q and computing x = ar mod p

▫Concatenate message with x and hash result to Computing: e = H(M || x)

▫Computing: y = (r + se) mod q ▫Signature is pair (e, y)

•Any other user can verify the signature as follows: ▫Computing: x' = ayve mod p ▫Verifying that: e = H(M || x’)