07-Digital Signature Standards

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Chapter 7

Digital Signature Standards


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Some Well-Known Digital Signature

Standards

The U.S.A. national standard : DSS.

The Russia national standard : GOST.

Japan national standard : ESIGN.

De factor standard : RSA. ISO/ICE 9796 : RSA

The U.S.A national standard X9.30-199 : RSA,

ElGamal.


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ElGamal

Schnorr

DSA

GOST 28147 (, 89 )

GOST 34.11 (, 94 )

GOST 34.10(, 94 )


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ElGamal Signature Algorithm


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(1/3)

Some mathematics : Suppose two integers a and n such that GCD(a, n)=1. Then

ax(mod n) ax(mod(n)) mod n.

Some proof :

Let for some k.ynkxnxy )())((mod

)(mod)(mod1)(mod

1)(mod1)(mod

)(mod)(mod

)(

)()(

)(

naanana

nana

nana

nkxxx

nkn

ynkx

)(mod)(mod))((mod

nananxy


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Key Generation :1. Select a large prime numberp and the primitive root g in

Zp*, then publishp and g.

2. Select a numberx in Zp-1.

3. Calculateygx (mod p).

4. The public key isy and the private key isx.


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For plaintext M and signature S : Sign :

1. The signer randomly select a number kZp-1.

2. Calculate r=gk(mod p).

3. Calculate s=k-1(M-xr) (mod p-1)

4. The signature S = (r, s).

Verify :

1. Is gM=yrrs (mod p) true ?

2. If true, the signature is right.3. Otherwise, the signature is wrong.


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Schnorr Signature Algorithm


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(1/3)

Key Generation :

1. Select a large prime number p2512.

2. Select a prime q such that q|p-1 and q 2160.

3. Select a ordered q primitive root g in Zp* and g1.

4. Publish p, q, and g.

5. Select a numberx in Zq.




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1. The signer randomly select a number kZq.

2. Calculate r=gk

(mod p).3. Calculate e=h(r, M), where h is a public one-way hash

function.

4. Calculate s=k-xe (mod q)

5. The signature S = (e, s).


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(3/3)

For signature S : Verify :

1. Calculate r=gsye (mod p).

2. Is e=h(r, M) true ? If true, the signature is right.

Otherwise, the signature is wrong.


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DSA

Digital Signature Algorithm

Nation Institute of Standard and Technology(NIST), 1991.


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DSA Signature Algorithm (1/3)

Key Generation :1.Select a 512 ~ 1024-bit prime number p.

2.Select a 160-bit prime q such that q|p-1.

3.Select an ordered q primitive root g in Zp* and g1.4.Publish p, q, and g.

5.Select a numberx in Zq.

6.Calculateygx (mod p).7.The public key isy and the private key isx.


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1.The signer randomly select a number kZq.

2.Calculate r=gk(mod p) (mod q).

3.Calculate s=k-1(M+xr) (mod q)

4.The signature S = (r, s).


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For signature S : Verify :

1.Check 0rq and 0sq. If one condition is not

true, the signature is wrong.

2.Calculate t=Ms-1 (mod q).

3.Calculate u=rs-1 (mod q).

4.Is r=gtyu (mod p) (mod q) true ?If true, the signature is right.

Otherwise, the signature is wrong.


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Verification

r=gtyu (mod p) (mod q)?

gtyu =gM/s gx(r/s) =g(M+xr)/s =gk=r (mod q)


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DSS and DSA

DSA (Digital Signature Algorithm) was published byNation Institute of Standards and Technology (NIST)

in 1991.

DSS (Digital Signature Standard) is a signature

standard based on DSA.


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Some Positive Comments on DSS

The length of signature is shorter. The key generation is faster.

The processing time cost is less if signer generates

many r and store them first. Its been authenticated by the U.S.A government.


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Some Negative Comments on DSS

DSS and RSA are not compatible. If s is 0, the verification will fail.

The verification process is 100 times slower than

RSA.


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GOST Family

GOST 34.10 is the digital signature standard ofRussia.

Some other standards:

GOST34.28147a block cipher standard.

GOST34.11a one-way hash function.


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GOST34.11 (1/3)

Key Generation :

1. Select a prime number p such that 2509p 2512 or

21020p21024.

2. Select a prime q such that q|p-1 and 2254p2256.

3. Select a ordered q primitive root g in Zp* and g1.

4. Publish p, q, and g.

5. Select a numberx in Zq.




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1. Calculate h(M), where h is a one-way hash

function.2. The signer randomly select a number kZq.

3. Calculate r=gk(mod p) (mod q).

4. Calculate s=k-1

(M+xr) (mod q)5. The signature S = (r, s).

GOST34.11 (2/3)


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For signature S :

Verify :

1. Check 0

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