Digital Signature

Digital Signature

Requirements of 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

Attack of Digital Signature Hard Easy • Key only or no message attack : Advaccess 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

Forgery of Digital Signature Hard Easy Total break : Advrecovers 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.

Construction of Digital Signature • Consists of 6 elements (M,Mh,A,K,S,V) • M : message space • Mh (or Ms) : signing space • A : signature space • K : 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)

Digital signature with appendix(I) 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

Digitalsignaturewithappendix(II) (a) signing M Mh A h sigK m m’ s*=sigK(m’) (b) verification verK(m,s*) true M x A false

Digital signature with msg recovery(I) 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.

Digital signature with msg recovery(II) (a) signing M MR A R sigK m m’ s*=sigK(m’) MS R: redundancy ft e.g., 1:1 ft MR : image of R (b) verification *This scheme can be easily changed to digital signature with appendix s.t., hashing before signing.

Comparison

Computationally secure D. S. Provably secure Unconditionally secure Probably secure ID-based Non-arbitrator Certificate-based Randomized Message-recovery Deterministic Message-appendix Randomized Arbitrator Deterministic Classification

RSA Signature (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=x1d x2d= (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

ElGamal Signature(I) 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

ElGamal Signature(II) (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

ElGamal Signature(III) • 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

DSS (I) 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)

DSS(II) p:512 bit prime, q:160 bit prime, q|p-1, gp* =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 randomk (1k  q-1,gcd(k,q)=1),sigK(m,k) = (,) where  = (k mod p) mod q, =(m+a)k-1mod q. (verification) m  p* and ,   q verK(m,, )=true  (e1e2 mod p)mod q = . e1= m-1 mod q, e2= -1 mod q.

DSS(III) (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-1mod 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 (e1e2 mod p)mod q= (17045 4567 27 mod 7879 ) mod 101 = 2518 mod 101=94 ?  =94 (valid)

Signature Scheme with additional properties

Advanced Digital Signature • 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.

Chaum’s Blind Signature(I) • 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} B A(customer) A B(Bank) (1)random number (1) select random k s.t. gcd(n,k)=1, 1<k<n-1 (2)blinding m* (3)signing (3) s*= (m*)d mod n (2) m*=mke mod n (4)unblinding s* g(SBf(m))=SB(m) f:blinding ft g:unblinding ft only A knows f(m) : blinded message (4) s=k-1 s*mod n (signature of M by B : k-1(mke)d= k-1 me ked= md)

Chaum’s Blind Signature(II) (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

Digital Signature