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Cramer & Shoup Encryption. Cramer and Shoup: A practical public key crypto system provably secure against adaptive chosen ciphertext attack. Crypto 1998 These slides are partially based on Jonathan Katz’s lecture notes. Benny Applebaum. Generate (PK,SK) PK D SK (c 1 ) D SK (c p )

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## Cramer & Shoup Encryption

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**Cramer & Shoup Encryption**Cramer and Shoup: A practical public key crypto system provably secure against adaptive chosen ciphertext attack. Crypto 1998 These slides are partially based on Jonathan Katz’s lecture notes. Benny Applebaum**Generate (PK,SK)**PK DSK(c1) DSK(cp) b{0,1} C=EPK(mb) CCA1 Security A c1 cp (m0,m1) b’ A wins if b=b’. The scheme is CCA1 secure if any efficient A wins with probability <1/2+neg**DDH Assumption**• Let G be a cyclic group of (prime) order q • DH tuple: (g,ga,gb,gab) • Rand tuple (g,ga,gb,gc) • where g is a random generator and a,b,cZq • DDH Assumption: Hard to distinguish Rand from DDH • |Pr[A(DH)=1]-Pr[A(Rand)=1]|<negl, for any poly-time A**Cramer & Shoup Lite**• PK= (g1,g2,h=g1xg2y, c= g1ag2b) • g1,g2 are random generators and x,y,a,bZq • SK = (x,y,a,b) • EPK(m): choose r Zq; set C=(g1r,g2r, hr m, cr) • DSK(u,v,w,e): • If euavb then output • Else, output w/(uxvy)**Cramer & Shoup Lite**• PK= (g1,g2,h=g1xg2y, c= g1ag2b) • g1,g2 are random generators and x,y,a,bZq • SK = (x,y,a,b) • EPK(m): choose r Zq; set C=(g1r,g2r, hr m, cr) • DSK(u,v,w,e): • If euavb then output • Else, output w/(uxvy) • Correctness: Easy…**x,y,a,bZq; SK=(x,y,a,b)**PK= (g1,g2,h=g1xg2y, c= g1ag2b) DSK(c1) DSK(cp) b{0,1} C=(g3 ,g4, g3xg4y mb, g3ag4b) CSL is CCA1 secure • Assume that A breaks CSL via CCA1 • Construct A’ that breaks DDH A’ (g1,g2,g3,g4) A c1 cp (m0,m1) b’ If b=b’ then output “DDH” otherwise output “Rand”**CSL is CCA1 secure**Thm. Under the DDH, CSL is CCA1 secure. Proof: • |Pr[A’(DH)=1]-Pr[A’(Rand)=1]|<negl follows from DDH Assum. and since A’ is poly-time • Claim: Pr[A’=1|DH]=Pr[A CCA1 breaks CSL] • Claim: |Pr[A’=1|Rand]| ½ + negl Hence: Pr[A CCA1 breaks CSL] =Pr[A’=1|DH] |Pr[A’=1|Rand]|+negl 1/2+negl**CSL is CCA1 secure**Claim 3: |Pr[A’=1|Rand]| ½ + negl Proof: • Show that (except w/neg prob) A attacks a perfect cipher. • I.e, g3xg4y is random (according to A’s view). • Let (g1,g2 = g1,g3 = g1r ,g4 = g1 r’) • Except w/neg prob 0,rr’ • From PK, A knows h=g1xg2y;that is, logg1 h=x+y (*) • We saw: if A knows only (*) then g3xg4y is random (from A’s view). Lemma: in phase 2 (except w/neg prob) A doesn’t learn info regarding (x,y). Proof: • A query (u,v,w,e) is bad if logg1 u logg2 v and DSK(u,v,w,e) Claim 4: (except w/neg prob) A’s queries are all good Claim 5: If A’s queries are all good then A does not learn additional info regarding (x,y) in phase 2**CSL is CCA1 secure**• Is CSL CCA2 secure? • Why the argument fail to prove CCA2 security?**Generate (PK,SK)**PK DSK(c1) DSK(cp) b{0,1} C*=EPK(mb) DSK(c1) DSK(cp) CCA2 Security A c1 cp (m0,m1) c’1c* c’p c* b’ A wins if b=b’. The scheme is CCA2 secure if any efficient A wins with probability <1/2+neg**The Cramer & Shoup Cryptosystem**• PK= (g1,g2,h=g1xg2y, c= g1ag2b , d= g1a’g2b’,H) g1,g2 are random generators, x,y,a,b,a’,b’Zq and H is a hash function • SK = (x,y,a,b,a’,b’) • EPK(m): choose r Zq; set C=(g1r,g2r, hr m, (cd)r), where =H(g1r,g2r, hr m) • DSK(u,v,w,e): • If eua + a’vb+ b’ (where =H(g1r,g2r, hr m)) then output • Else, output w/(uxvy) • Correctness: Easy…**x,y,a,b,a’,b’Zq; SK=(x,y,a,b,a’,b’)**PK= (g1,g2,h=g1xg2y, c= g1ag2b, d= g1a’g2b’,H) DSK(c1) DSK(cp) b{0,1} C=(g3 ,g4, g3xg4y mb, g3a+ a’g4b + b’) where =H(g3 ,g4, g3xg4y mb) CS is CCA2 secure • Assume that A breaks CS via CCA2 • Construct A’ that breaks DDH A’ (g1,g2,g3,g4) A c1 cp (m0,m1) c’1 c’p b’ If b=b’ then output “DDH” otherwise output “Rand”**CS is CCA2 secure**Thm. Under the DDH, CS is CCA2 secure. Proof: • |Pr[A’(DH)=1]-Pr[A’(Rand)=1]|<negl follows from DDH Assum. and since A’ is poly-time • Claim: Pr[A’=1|DH]=Pr[A CCA2 breaks CS] • Claim: |Pr[A’=1|Rand]| ½ + negl Hence: Pr[A CCA2 breaks CS] =Pr[A’=1|DH] |Pr[A’=1|Rand]|+negl 1/2+negl**CS is CCA2 secure**Claim 3: |Pr[A’=1|Rand]| ½ + negl Proof: • Show g3xg4y is random (according to A’s view). • Let (g1,g2 = g1,g3 = g1r ,g4 = g1 r’) • Except w/neg prob 0,rr’ • From PK, A knows h=g1xg2y;that is, logg1 h=x+y (*) • We saw: • if A knows only (*) then g3xg4y is random (from A’s view). • in phase 2 (except w/neg prob) A doesn’t learn info regarding (x,y). Lemma: in phase 3 (except w/neg prob) A doesn’t learn info regarding (x,y). Proof: • A query (u,v,w,e) is bad if logg1 u logg2 v and DSK(u,v,w,e) Claim 4: (except w/neg prob) A’s queries are all good Claim 5: If A’s queries are all good then A does not learn additional info regarding (x,y) in phase 3

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