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Digital Signature Digital Signature

Efficiency Unforgeability : only signer can generate Not reusable : not to use for other message Unalterable : No modification of signed message Authentication of a signer Non-repudiation : not denying the act of signing

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Key only or no message attack : Adv access only to public parameters and public keys

Message attack : Adv has access to pairs of message texts and corresponding signatures. Depending on Adv’s power of selecting messages signed by S.◦ Known-messages : Adv doesn’t choose message signed by S.◦ Generic chosen-messages : Adv chooses a set of messages to be

signed before knowing the actual S targeted for attack.◦ Directed chosen-message : Adv chooses a set of messages to be signed

after selecting a specific S but the actual attack. ◦ Adaptive chosen-message : Adv chooses message for signing

dynamically after inspecting signatures he obtained for previous messages.

Adv: Adversary, S: legitimate signer

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Hard

Easy

Total break : Adv recovers the secret key of S under attack.

Universal forgery : Adv doesn’t obtain the secret key of S, but gains the ability to generate valid signatures for any message.

Selective forgery : Adv doesn’t obtain the secret key of S, but gains the ability to generate valid signatures for any set of preselected messages.

Existential forgery : Adv can create at least one new message and signature pair without knowing the secret key. The messages are only arbitrary bit strings and Adv doesn’t have any power over their composition.

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Hard

Easy

Consists of 6 elements (M,Mh,A,K,S,V) M : message spaceMh (or Ms) : signing space A : signature spaceK : key space For K K, signing algorithm sigK S and

its corresponding verification algorithm verK V.

Each sigK : M A

verK : M x A {t, f} are functions s.t., verK(x,y)= t if y = sigK(x) or

verK(x,y)=f if y sigK(x)

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Signature generation (a) get private key, Ks

(b) m’=h(m) : hash algorithm and s*=sigKs(m’) (c) m, s* : signature

Signature verification (a) obtain public key, Kp

(b) compute m’=h(m) and u=verKp(m’,s*)

(c) accept signature iff u=true.

(Ex.) DSA, ElGamal, Schnorr

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(a) signing

Mm

Mh

m’A

s*=sigK(m’)

h sigK

(b) verification

M x Atrue

false

verK(m,s*)

Signature generation (a) get private key, Ks

(b) m’=R(m) : redundancy function and s*=sigKs(m’) (c) s* : signature

Signature verification (a) obtain public key KP

(b) compute m’= verKp(s*) (c) verify that m’ MR ( if m’ MR, then reject) (d) recover m from m’ by computing R-1(m’)

(Ex.) RSA, Rabin, Nyberg-Rueppel * R() and R-1() are easy to compute.

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(a) signing

M

m

MR

m’

A

s*=sigK(m’)

R sigK

MS

*This scheme can be easily changed to digital signature with appendix s.t., hashing before signing.

(b) verification R: redundancy ft e.g., 1:1 ftMR : image of R

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Classification Analog Digital

Size of Signature fixed variable

Digital Copy difficult easy

Operation simple mathematical

Legality possible possible

Forgeability un-countermeasured countermeasured

Generation pen computer

Auxiliary tool nothing necessary (hash ft.)

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D. S. Computationally secure

Unconditionally secureProvably secure

Probably secure

ID-based

Certificate-based

Message-recovery

Message-appendix

Non-arbitrator

Arbitrator

Randomized

Deterministic

Randomized

Deterministic

(preparation) n=pq, M=A=n K ={(n,p,q,e,d) : n=pq, ed=1 mod (n)} public key : {e,n}, private key : {d,p,q} (signing) K=(n,p,q,e,d),sigK(x)=xd mod n (verification)verK(x,y)=true x=ye mod n, x,y n

(Problem) If an adversary know signature of x1 and x2 to be s1 and s2, he can create signature of x3=x1x2, i.e., s3{=s1s2=x1

d x2

d= (x1x2)d} without knowing private key,d. To prevent this, hashing before signing or other means must be used.

Note that E(m1)E(m2)=E(m1m2) in RSA

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p : prime, p*: primitive element,

M =p*, A = p

* x p-1*

K ={(p,,a,) : = a mod p} public: (p,,), private: a

(signing) K=(p,,a,), secret random k p-1*

sigK(m,k) = (,) where = k mod p, =(m-a)k-1 mod (p-1)

(verification) m, p* and p-1

*

verK(m,, )=true =m mod p * =a k = m mod p

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(Preparation) p=467, =2, a=127 =a mod p = 2127mod467=132 message m=100 random k=213 s.t., gcd(213,466)=1. 213-1mod466 =431 signing = k = 2213 mod 467= 29 =(m-a)k-1 mod (p-1)=(100 - 127 x 29) 431 mod

466=51 Verification on m, and =m mod p ? =13229 2951=189 mod 467 m =2100 =189 mod 467

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Security : without knowing a, forgery of x’s signature is reducible to DLP of finding () chosen ().

Note◦ Keep k to be secret◦ Not to use k two times.

Generalization from p* to any finite Abelian group

is possible

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After 1991 August for 3-year public debate, NIST announced DSS (Digital Signature Standard) documented FIPS-186 in 1994 December.

RSA was not selected since its patent Introduce efficient operation under

subgroup in ElGamal signature scheme Used with DHA (Digital Hash Algorithm)

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p:512 bit prime, q:160 bit prime, q|p-1, gp*

=g(p-1)/qmod p (q-th root of 1 mod p), M =p

*, A= q x q, K={(p,q,,a,):=a mod p} public: (p,q,,), private: a. (signing) K=(p,q,,a,), secret random k (1k q-

1,gcd(k,q)=1), sigK(m,k) = (,) where = (k mod p) mod q, =(m+a)k-1 mod q.

(verification) m p* and , q

verK(m,, )=true (e1e2 mod p)mod q = .

e1= m-1 mod q, e2= -1 mod q.

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(Ex.) q=101, p=78q+1=7879, g=3, = 378 mod 7879 =170, a=75, = a mod 7879 = 4567

(signing) message m=1234, random k=50, k-1 mod 101=99. = (k mod p) mod q=(17050mod 7879) mod101=2518 mod101=94. =(m+a)k-1 mod q=(1234 +75x94) 99 mod101 = 97.

(verification) sigK(m,k) = (,) =(94,97), m=1234

-1=97-1 mod 101=25, e1= m-1 mod q =1234 x 25 mod 101 =45,

e2= -1 mod q = 94 x 25 mod 101 = 27

(e1e2 mod p)mod q= (17045 4567 27 mod 7879 ) mod 101 = 2518 mod 101=94 ? =94 (valid)

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Signature Scheme with additional properties

Signature Scheme with additional properties

Blind signature One-time signature

◦ Lamport scheme or Bos-Chaum scheme Undeniable signature

◦ Chaum-van Antwerpen scheme Fail-stop signature

◦ van Heyst-Peterson scheme Proxy signature Group (Ring) signature: group member can generate signature

if dispute occurs, identify member. etc.

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Without B seeing the content of message M, A can get a signature of M from B.

RSA scheme, B’s public key :{n,e}, private key:{d}

(1) select random ks.t. gcd(n,k)=1, 1<k<n-1

A(customer) B(Bank)

(2) m*=mke

mod n

m*

(3) s*= (m*)d mod ns*

(4) s=k-1 s*mod n(signature of M by B : k-1(mke)d= k-1 me ked = md)

(1)randomnumber

(2)blinding

(3)signing

(4)unblinding

A B

g(SBf(m))=SB(m)f:blinding ftg:unblinding ft only A knowsf(m) : blinded message

(Preparation) p=11, q=3, n=33,(n)= 10 x 2=20 gcd(d, (n))=1 => d=3, ed =1 mod (n) => 3 d = 1 mod 20 => e=7 B: public key :{n,e}={33,7}, private key ={d}={3}

(1) A’s blinding of m=5 select k s.t. gcd(k,n)=1. gcd(k,33)=1 => k=2 m* = m ke mod n= 5 27 mod 33 = 640 mod 33 = 13 mod 33(2) B’s signing without knowing the original m s*= (m*)d mod n = 133 mod 33 =2197 mod 33 =19 mod 33(3) A’s unblinding s=k-1 s* mod n (2 k-1=1 mod 33 => k=17) = 17 19 mod 33 =323=26 mod 33

* Original Signature : md mod n = 53 mod 33 =125 =26 mod 33

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