QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic...
Transcript of QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic...
![Page 1: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/1.jpg)
QUANTUM HOMOMORPHIC ENCRYPTION
Christian Schaffner
(joint work with Yfke Dulek and Florian Speelman)http://arxiv.org/abs/1603.09717
Centrum Wiskunde&Informa3ca
Ins3tuteforLogic,LanguageandComputa3on(ILLC)UniversityofAmsterdam
ResearchCenterforQuantumSoCware
TrustworthyQuantumInforma1on2016,Shanghai,China,Wednesday29June2016
![Page 2: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/2.jpg)
EXAMPLE: IMAGE TAGGING
![Page 3: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/3.jpg)
EXAMPLE: IMAGE TAGGING
![Page 4: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/4.jpg)
EXAMPLE: IMAGE TAGGING
![Page 5: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/5.jpg)
EXAMPLE: IMAGE TAGGING
SKYLINE JED
![Page 6: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/6.jpg)
EXAMPLE: IMAGE TAGGING
![Page 7: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/7.jpg)
EXAMPLE: IMAGE TAGGING
![Page 8: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/8.jpg)
EXAMPLE: IMAGE TAGGING
![Page 9: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/9.jpg)
EXAMPLE: IMAGE TAGGING
![Page 10: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/10.jpg)
EXAMPLE: IMAGE TAGGING
SKYLINE JED
![Page 11: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/11.jpg)
EXAMPLE: IMAGE TAGGING
SKYLINE JED
![Page 12: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/12.jpg)
EXAMPLE: IMAGE TAGGING
SKYLINE JED
![Page 13: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/13.jpg)
1. HOMOMORPHIC ENCRYPTION
2. PREVIOUS RESULTS
3. NEW RESULT
![Page 14: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/14.jpg)
HOMOMORPHIC ENCRYPTION
![Page 15: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/15.jpg)
HOMOMORPHIC ENCRYPTION
KEY GENERATION
![Page 16: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/16.jpg)
HOMOMORPHIC ENCRYPTION
public keyKEY GENERATION
![Page 17: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/17.jpg)
HOMOMORPHIC ENCRYPTION
public keysecret key
KEY GENERATION
![Page 18: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/18.jpg)
HOMOMORPHIC ENCRYPTION
public keysecret keyevaluation key
KEY GENERATION
![Page 19: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/19.jpg)
HOMOMORPHIC ENCRYPTION
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION
![Page 20: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/20.jpg)
HOMOMORPHIC ENCRYPTION
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION + ↦
![Page 21: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/21.jpg)
HOMOMORPHIC ENCRYPTION
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION +(secure)
↦
![Page 22: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/22.jpg)
HOMOMORPHIC ENCRYPTION
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION +(secure)
↦
![Page 23: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/23.jpg)
HOMOMORPHIC ENCRYPTION
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION +(secure)
↦
![Page 24: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/24.jpg)
HOMOMORPHIC ENCRYPTION
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION
EVALUATION
+(secure)
↦
![Page 25: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/25.jpg)
HOMOMORPHIC ENCRYPTION
JED ↦
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION
EVALUATION
+
+
(secure)↦
![Page 26: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/26.jpg)
HOMOMORPHIC ENCRYPTION
JED ↦
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION
EVALUATION
DECRYPTION
+
+
(secure)↦
![Page 27: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/27.jpg)
HOMOMORPHIC ENCRYPTION
JED ↦
JED JED↦
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION
EVALUATION
DECRYPTION
+
+
+
(secure)↦
![Page 28: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/28.jpg)
HOMOMORPHIC ENCRYPTION
↦
↦
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION
EVALUATION
DECRYPTION
+
+
+
(secure)↦x x
x f(x)
f(x) f(x)
![Page 29: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/29.jpg)
HOMOMORPHIC ENCRYPTION
↦
↦
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION
EVALUATION
DECRYPTION
+
+
+
(secure)↦|ψ⟩ |ψ⟩
|ψ⟩ U|ψ⟩
U|ψ⟩ U|ψ⟩
![Page 30: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/30.jpg)
HOMOMORPHIC ENCRYPTION
↦
↦
public keysecret keyevaluation key
KEY GENERATION
ENCRYPTION
EVALUATION
DECRYPTION
+
+
+
(secure)↦|ψ⟩ |ψ⟩
|ψ⟩ U|ψ⟩
U|ψ⟩ U|ψ⟩
(quantum)
![Page 31: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/31.jpg)
1. HOMOMORPHIC ENCRYPTION
2. PREVIOUS RESULTS
3. NEW RESULT
![Page 32: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/32.jpg)
PREVIOUS RESULTS: OVERVIEW
C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015Y.Ouyang,S-H.Tan,J.Fitzsimons.Quantumhomomorphicencryp3onfromquantumcodes.arxiv:1508.00938
![Page 33: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/33.jpg)
PREVIOUS RESULTS: OVERVIEW
Classical homomorphic encryption: solved! [Gentry 2009]
C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015Y.Ouyang,S-H.Tan,J.Fitzsimons.Quantumhomomorphicencryp3onfromquantumcodes.arxiv:1508.00938
![Page 34: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/34.jpg)
PREVIOUS RESULTS: OVERVIEW
Classical homomorphic encryption: solved! [Gentry 2009]
Quantum homomorphic encryption: only partial results
Clifford scheme allowing evaluation of {P, H, CNOT}
schemes for {P, H, CNOT} + limited # of T gates
C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015Y.Ouyang,S-H.Tan,J.Fitzsimons.Quantumhomomorphicencryp3onfromquantumcodes.arxiv:1508.00938
![Page 35: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/35.jpg)
PREVIOUS RESULTS: OVERVIEW
Classical homomorphic encryption: solved! [Gentry 2009]
Quantum homomorphic encryption: only partial results
Clifford scheme allowing evaluation of {P, H, CNOT}
schemes for {P, H, CNOT} + limited # of T gates
C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015Y.Ouyang,S-H.Tan,J.Fitzsimons.Quantumhomomorphicencryp3onfromquantumcodes.arxiv:1508.00938
![Page 36: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/36.jpg)
SCHEME FOR {P, H, CNOT}
[AMTW00]A.Ambainis,M.Mosca,A.Tapp,andR.DeWolf.Privatequantumchannels.FOCS’00[Gentry09]C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09
![Page 37: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/37.jpg)
SCHEME FOR {P, H, CNOT}
Ingredient 1: quantum encryption (one-time pad)
[AMTW00]A.Ambainis,M.Mosca,A.Tapp,andR.DeWolf.Privatequantumchannels.FOCS’00[Gentry09]C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09
![Page 38: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/38.jpg)
SCHEME FOR {P, H, CNOT}
Ingredient 1: quantum encryption (one-time pad)
encryption:
[AMTW00]A.Ambainis,M.Mosca,A.Tapp,andR.DeWolf.Privatequantumchannels.FOCS’00[Gentry09]C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09
![Page 39: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/39.jpg)
SCHEME FOR {P, H, CNOT}
Ingredient 1: quantum encryption (one-time pad)
a,bencryption: pick a,b ∈R {0,1}
[AMTW00]A.Ambainis,M.Mosca,A.Tapp,andR.DeWolf.Privatequantumchannels.FOCS’00[Gentry09]C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09
![Page 40: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/40.jpg)
SCHEME FOR {P, H, CNOT}
Ingredient 1: quantum encryption (one-time pad)
|ψ⟩ a,b
a,bencryption: pick a,b ∈R {0,1}
|ψ⟩ ↦ XaZb|ψ⟩ =
[AMTW00]A.Ambainis,M.Mosca,A.Tapp,andR.DeWolf.Privatequantumchannels.FOCS’00[Gentry09]C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09
![Page 41: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/41.jpg)
SCHEME FOR {P, H, CNOT}
Ingredient 1: quantum encryption (one-time pad)
|ψ⟩ a,b
a,bencryption: pick a,b ∈R {0,1}
|ψ⟩ ↦ XaZb|ψ⟩
decryption:
=
[AMTW00]A.Ambainis,M.Mosca,A.Tapp,andR.DeWolf.Privatequantumchannels.FOCS’00[Gentry09]C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09
![Page 42: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/42.jpg)
SCHEME FOR {P, H, CNOT}
Ingredient 1: quantum encryption (one-time pad)
|ψ⟩ a,b
a,bencryption: pick a,b ∈R {0,1}
|ψ⟩ ↦ XaZb|ψ⟩
decryption: XaZb|ψ⟩ ↦ |ψ⟩
=
[AMTW00]A.Ambainis,M.Mosca,A.Tapp,andR.DeWolf.Privatequantumchannels.FOCS’00[Gentry09]C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09
![Page 43: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/43.jpg)
SCHEME FOR {P, H, CNOT}
Ingredient 2: classical homomorphic encryption
Ingredient 1: quantum encryption (one-time pad)
|ψ⟩ a,b
a,bencryption: pick a,b ∈R {0,1}
|ψ⟩ ↦ XaZb|ψ⟩
decryption: XaZb|ψ⟩ ↦ |ψ⟩
=
[AMTW00]A.Ambainis,M.Mosca,A.Tapp,andR.DeWolf.Privatequantumchannels.FOCS’00[Gentry09]C.Gentry:Fullyhomomorphicencryp3onusingideallaJces.STOC’09
![Page 44: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/44.jpg)
SCHEME FOR {P, H, CNOT}
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 45: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/45.jpg)
SCHEME FOR {P, H, CNOT}
|ψ⟩
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 46: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/46.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
a,b
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 47: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/47.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
a,b
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 48: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/48.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,a
H|ψ⟩a,b H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 49: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/49.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 50: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/50.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b H
H ( ) a,b|ψ⟩
=HXaZb|ψ⟩
=XbZaH|ψ⟩
=
b,aH|ψ⟩
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 51: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/51.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 52: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/52.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b
H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 53: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/53.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b
H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 54: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/54.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b
b,a
UPDATEFUNCTION(x,y) ↦ (y,x) H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 55: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/55.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b
b,a
UPDATEFUNCTION(x,y) ↦ (y,x) H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 56: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/56.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b
b,a
UPDATEFUNCTION(x,y) ↦ (y,x) H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 57: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/57.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
b,aH|ψ⟩
a,b
b,a
UPDATEFUNCTION(x,y) ↦ (y,x) H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 58: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/58.jpg)
a,b
SCHEME FOR {P, H, CNOT}
|ψ⟩
H|ψ⟩
a,b
b,a
UPDATEFUNCTION(x,y) ↦ (y,x) H
Folklore,lastformalizedby[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015
![Page 59: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/59.jpg)
THE CHALLENGE: T GATE
![Page 60: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/60.jpg)
THE CHALLENGE: T GATE
H
![Page 61: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/61.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
H
![Page 62: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/62.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
b,aH|ψ⟩
H
![Page 63: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/63.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
b,aH|ψ⟩
H T
![Page 64: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/64.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
b,aH|ψ⟩
0,b|ψ⟩
H T
![Page 65: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/65.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
b,aH|ψ⟩
0,b|ψ⟩
H
T|ψ⟩ 0,b
T
![Page 66: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/66.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
b,aH|ψ⟩
0,b|ψ⟩
H
T|ψ⟩ 0,b
1,b|ψ⟩
T T
![Page 67: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/67.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
b,aH|ψ⟩
0,b|ψ⟩
H
T|ψ⟩ 0,b P ( ) T|ψ⟩ 1,b
1,b|ψ⟩
T T
![Page 68: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/68.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
b,aH|ψ⟩
0,b|ψ⟩
H
T|ψ⟩ 0,b P ( ) T|ψ⟩ 1,b
1,b|ψ⟩
T Terror!
![Page 69: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/69.jpg)
THE CHALLENGE: T GATE
a,b|ψ⟩
b,aH|ψ⟩
0,b|ψ⟩
H
T|ψ⟩ 0,b P ( ) T|ψ⟩ 1,b
1,b|ψ⟩
T T
how to apply correction P-1 iff a = 1?
error!
![Page 70: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/70.jpg)
PREVIOUS RESULTS: OVERVIEW
(comparisonbasedonStaceyJeffery’sslides)[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015[OTF15]Y.Ouyang,S-H.Tan,J.Fitzsimons.Quantumhomomorphicencryp3onfromquantumcodes.arxiv:1508.00938
![Page 71: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/71.jpg)
PREVIOUS RESULTS: OVERVIEWhomomorphic for compactness security
Not encrypting Quantum circuits yes no
append evaluation description Quantum circuits
Complexity of Dec prop to (# gates) yes
Quantum OTP no yes inf theoretic
Clifford Scheme Clifford circuits yes computational
(comparisonbasedonStaceyJeffery’sslides)[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015[OTF15]Y.Ouyang,S-H.Tan,J.Fitzsimons.Quantumhomomorphicencryp3onfromquantumcodes.arxiv:1508.00938
![Page 72: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/72.jpg)
PREVIOUS RESULTS: OVERVIEWhomomorphic for compactness security
Not encrypting Quantum circuits yes no
append evaluation description Quantum circuits
Complexity of Dec prop to (# gates) yes
Quantum OTP no yes inf theoretic
Clifford Scheme Clifford circuits yes computational
[BJ15]: AUX QCircuits with constant T-depth
yes computational
[BJ15]: EPR Quantum circuits Comp of Dec is prop to (#T-gates)^2
computational
[OTF15] QCircuits with constant #T-gates
yes inf theoretic
(comparisonbasedonStaceyJeffery’sslides)[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015[OTF15]Y.Ouyang,S-H.Tan,J.Fitzsimons.Quantumhomomorphicencryp3onfromquantumcodes.arxiv:1508.00938
![Page 73: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/73.jpg)
PREVIOUS RESULTS: OVERVIEWhomomorphic for compactness security
Not encrypting Quantum circuits yes no
append evaluation description Quantum circuits
Complexity of Dec prop to (# gates) yes
Quantum OTP no yes inf theoretic
Clifford Scheme Clifford circuits yes computational
[BJ15]: AUX QCircuits with constant T-depth
yes computational
[BJ15]: EPR Quantum circuits Comp of Dec is prop to (#T-gates)^2
computational
[OTF15] QCircuits with constant #T-gates
yes inf theoretic
Our resultQCircuits of
polynomial size (levelled FHE)
yes computational
(comparisonbasedonStaceyJeffery’sslides)[BJ15]A.Broadbent,S.Jeffery.QuantumHomomorphicEncryp3onforCircuitsofLowT-gateComplexity.CRYPTO2015[OTF15]Y.Ouyang,S-H.Tan,J.Fitzsimons.Quantumhomomorphicencryp3onfromquantumcodes.arxiv:1508.00938
![Page 74: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/74.jpg)
1. HOMOMORPHIC ENCRYPTION
2. PREVIOUS RESULTS
3. NEW RESULT
![Page 75: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/75.jpg)
ERROR-CORRECTION “GADGET”
![Page 76: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/76.jpg)
A quantum state that:
can be efficiently constructed and used
ERROR-CORRECTION “GADGET”
GADGET
![Page 77: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/77.jpg)
A quantum state that:
can be efficiently constructed and used
applies correction iff error was present (iff a = 1)
ERROR-CORRECTION “GADGET”
GADGET
![Page 78: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/78.jpg)
A quantum state that:
can be efficiently constructed and used
applies correction iff error was present (iff a = 1)
ERROR-CORRECTION “GADGET”
P ( ) T|ψ⟩ 1,b
GADGET
![Page 79: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/79.jpg)
A quantum state that:
can be efficiently constructed and used
applies correction iff error was present (iff a = 1)
ERROR-CORRECTION “GADGET”
T|ψ⟩ 1,b
GADGET
![Page 80: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/80.jpg)
A quantum state that:
can be efficiently constructed and used
applies correction iff error was present (iff a = 1)
ERROR-CORRECTION “GADGET”
T|ψ⟩ 0,b
GADGET
![Page 81: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/81.jpg)
A quantum state that:
can be efficiently constructed and used
applies correction iff error was present (iff a = 1)
ERROR-CORRECTION “GADGET”
T|ψ⟩ 0,b
GADGET
![Page 82: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/82.jpg)
A quantum state that:
can be efficiently constructed and used
applies correction iff error was present (iff a = 1)
is destroyed after a single use
ERROR-CORRECTION “GADGET”
GADGET
![Page 83: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/83.jpg)
A quantum state that:
can be efficiently constructed and used
applies correction iff error was present (iff a = 1)
is destroyed after a single use
ERROR-CORRECTION “GADGET”
![Page 84: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/84.jpg)
EXCURSIONTheoretical Computer Science
![Page 85: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/85.jpg)
PERMUTATION BRANCHING PROGRAM
![Page 86: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/86.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)
![Page 87: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/87.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
![Page 88: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/88.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
0: π
1: σ’
1: σ’’
![Page 89: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/89.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
0: π
1: σ’
1: σ’’
![Page 90: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/90.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
0: π
1: σ’
1: σ’’
permutations of {1,2, …, k}
∈ Sk∈ Sk
∈ Sk∈ Sk
∈ Sk∈ Sk
![Page 91: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/91.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
output: … ° σ’’ ° σ’ ° π0: π
1: σ’
1: σ’’
permutations of {1,2, …, k}
∈ Sk∈ Sk
∈ Sk∈ Sk
∈ Sk∈ Sk
![Page 92: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/92.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
output: … ° σ’’ ° σ’ ° πid
0: π
1: σ’
1: σ’’
permutations of {1,2, …, k}
∈ Sk∈ Sk
∈ Sk∈ Sk
∈ Sk∈ Sk
![Page 93: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/93.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
output: … ° σ’’ ° σ’ ° πid(fixed) cycle
0: π
1: σ’
1: σ’’
permutations of {1,2, …, k}
∈ Sk∈ Sk
∈ Sk∈ Sk
∈ Sk∈ Sk
![Page 94: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/94.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
output: … ° σ’’ ° σ’ ° πid(fixed) cycle
0: π
1: σ’
1: σ’’
⇒ f(x,y) = 0
permutations of {1,2, …, k}
∈ Sk∈ Sk
∈ Sk∈ Sk
∈ Sk∈ Sk
![Page 95: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/95.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
output: … ° σ’’ ° σ’ ° πid(fixed) cycle
0: π
1: σ’
1: σ’’
⇒ f(x,y) = 0⇒ f(x,y) = 1
permutations of {1,2, …, k}
∈ Sk∈ Sk
∈ Sk∈ Sk
∈ Sk∈ Sk
![Page 96: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/96.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
output: … ° σ’’ ° σ’ ° πid(fixed) cycle
0: π
1: σ’
1: σ’’
⇒ f(x,y) = 0⇒ f(x,y) = 1
length: # of instructions
permutations of {1,2, …, k}
∈ Sk∈ Sk
∈ Sk∈ Sk
∈ Sk∈ Sk
![Page 97: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/97.jpg)
PERMUTATION BRANCHING PROGRAM
computes some Boolean function f(x,y)list of instructions:
xi 1: σ
yj0: π’
xk0: π’’
…
output: … ° σ’’ ° σ’ ° πid(fixed) cycle
0: π
1: σ’
1: σ’’
⇒ f(x,y) = 0⇒ f(x,y) = 1
length: # of instructionswidth: k
permutations of {1,2, …, k}
∈ Sk∈ Sk
∈ Sk∈ Sk
∈ Sk∈ Sk
![Page 98: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/98.jpg)
EXAMPLE PBP (OR)
length 4, width 5:
![Page 99: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/99.jpg)
EXAMPLE PBP (OR)
x1
y1
x1
y1
1: id0: (12453)
0: (54321)
0: (12345)
1: id
1: id0: (15243)1: (14235)
length 4, width 5:
![Page 100: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/100.jpg)
EXAMPLE PBP (OR)
x1
y1
x1
y1
OR(0,0)
output: id0
1: id0: (12453)
0: (54321)
0: (12345)
1: id
1: id0: (15243)1: (14235)
length 4, width 5:
![Page 101: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/101.jpg)
EXAMPLE PBP (OR)
x1
y1
x1
y1
OR(0,0) OR(0,1)
output: id0
(14235)1
1: id0: (12453)
0: (54321)
0: (12345)
1: id
1: id0: (15243)1: (14235)
length 4, width 5:
![Page 102: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/102.jpg)
EXAMPLE PBP (OR)
x1
y1
x1
y1
OR(0,0) OR(0,1) OR(1,0) OR(1,1)
output: id0
(14235)1
(14235)1
1: id0: (12453)
0: (54321)
0: (12345)
1: id
1: id0: (15243)1: (14235)
length 4, width 5:
![Page 103: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/103.jpg)
EXAMPLE PBP (OR)
x1
y1
x1
y1
OR(0,0) OR(0,1) OR(1,0) OR(1,1)
output: id0
(14235)1
(14235)1
(14235)1
1: id0: (12453)
0: (54321)
0: (12345)
1: id
1: id0: (15243)1: (14235)
length 4, width 5:
![Page 104: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/104.jpg)
BARRINGTON’S THEOREMTheorem (variation): if f : {0,1}n x {0,1}m → {0,1} is in NC1, then there exists a permutation branching program for f with:
[Barrington89]Bounded-WidthPolynomial-SizeBranchingProgramsRecognizeExactlyThoseLanguagesinNC1,J.Comput.Syst.Sci.38(1):150–164,1989[BV11]Z.Brakerski,V.Vaikuntanathan.Efficientfullyhomomorphicencryp3onfrom(standard)LWE.FOCS2011
![Page 105: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/105.jpg)
BARRINGTON’S THEOREMTheorem (variation): if f : {0,1}n x {0,1}m → {0,1} is in NC1, then there exists a permutation branching program for f with:
width 5
[Barrington89]Bounded-WidthPolynomial-SizeBranchingProgramsRecognizeExactlyThoseLanguagesinNC1,J.Comput.Syst.Sci.38(1):150–164,1989[BV11]Z.Brakerski,V.Vaikuntanathan.Efficientfullyhomomorphicencryp3onfrom(standard)LWE.FOCS2011
![Page 106: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/106.jpg)
BARRINGTON’S THEOREMTheorem (variation): if f : {0,1}n x {0,1}m → {0,1} is in NC1, then there exists a permutation branching program for f with:
width 5length polynomial in (n+m)
[Barrington89]Bounded-WidthPolynomial-SizeBranchingProgramsRecognizeExactlyThoseLanguagesinNC1,J.Comput.Syst.Sci.38(1):150–164,1989[BV11]Z.Brakerski,V.Vaikuntanathan.Efficientfullyhomomorphicencryp3onfrom(standard)LWE.FOCS2011
![Page 107: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/107.jpg)
BARRINGTON’S THEOREMTheorem (variation): if f : {0,1}n x {0,1}m → {0,1} is in NC1, then there exists a permutation branching program for f with:
width 5length polynomial in (n+m)
[Barrington89]Bounded-WidthPolynomial-SizeBranchingProgramsRecognizeExactlyThoseLanguagesinNC1,J.Comput.Syst.Sci.38(1):150–164,1989[BV11]Z.Brakerski,V.Vaikuntanathan.Efficientfullyhomomorphicencryp3onfrom(standard)LWE.FOCS2011
P
NC1
L
NP
no proof that NP≠NC1
![Page 108: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/108.jpg)
BARRINGTON’S THEOREMTheorem (variation): if f : {0,1}n x {0,1}m → {0,1} is in NC1, then there exists a permutation branching program for f with:
width 5length polynomial in (n+m)
Classical homomorphic decryption functionshappen to be in NC1… [BV11]
[Barrington89]Bounded-WidthPolynomial-SizeBranchingProgramsRecognizeExactlyThoseLanguagesinNC1,J.Comput.Syst.Sci.38(1):150–164,1989[BV11]Z.Brakerski,V.Vaikuntanathan.Efficientfullyhomomorphicencryp3onfrom(standard)LWE.FOCS2011
P
NC1
L
NP
no proof that NP≠NC1
![Page 109: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/109.jpg)
ERROR CORRECTION GADGET
![Page 110: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/110.jpg)
ERROR CORRECTION GADGET
GADGET
![Page 111: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/111.jpg)
ERROR CORRECTION GADGET
GADGET
![Page 112: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/112.jpg)
ERROR CORRECTION GADGET
GADGET
PBP fordecrypt( , )a {
![Page 113: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/113.jpg)
ERROR CORRECTION GADGET
GADGET
P-1 P-1 P-1 P-1
PBP fordecrypt( , )a {P-1 iff permutation ≠ id {
![Page 114: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/114.jpg)
ERROR CORRECTION GADGET
GADGET
P-1 P-1 P-1 P-1
PBP fordecrypt( , )a {P-1 iff permutation ≠ id {
reverse PBP fordecrypt( , )a {
![Page 115: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/115.jpg)
ERROR CORRECTION GADGET
GADGET
P-1 P-1 P-1 P-1
PBP fordecrypt( , )a {P-1 iff permutation ≠ id {
reverse PBP fordecrypt( , )a {
![Page 116: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/116.jpg)
ERROR CORRECTION GADGET
![Page 117: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/117.jpg)
ERROR CORRECTION GADGET
![Page 118: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/118.jpg)
ERROR CORRECTION GADGET
1: σ0: π
i
1: σ’’0: π’’
k
1: σ’0: π’
a j
1: σ’’’0: π’’’
a l
……
![Page 119: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/119.jpg)
ERROR CORRECTION GADGET
1: σ0: π
i
1: σ’’0: π’’
k
1: σ’0: π’
a j
1: σ’’’0: π’’’
a l
……
![Page 120: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/120.jpg)
ERROR CORRECTION GADGET
1: σ0: π
i
1: σ’’0: π’’
k
1: σ’0: π’
a j
1: σ’’’0: π’’’
a l
………
![Page 121: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/121.jpg)
ERROR CORRECTION GADGET
1: σ0: π
i
1: σ’’0: π’’
k
1: σ’0: π’
a j
1: σ’’’0: π’’’
a l
………
![Page 122: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/122.jpg)
ERROR CORRECTION GADGET
1: σ0: π
i
1: σ’’0: π’’
k
1: σ’0: π’
a j
1: σ’’’0: π’’’
a l
EPR pairs
EPR pairs
………
![Page 123: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/123.jpg)
ERROR CORRECTION GADGET
1: σ0: π
i
1: σ’’0: π’’
k
1: σ’0: π’
a j
1: σ’’’0: π’’’
a l
EPR pairs
EPR pairs
Bellmeasurements
Bellmeasurements
………
![Page 124: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/124.jpg)
ERROR CORRECTION GADGET
GADGET
P-1 P-1 P-1 P-1
PBP fordecrypt( , )a {P-1 iff permutation ≠ id {
reverse PBP fordecrypt( , )a {
![Page 125: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/125.jpg)
ERROR CORRECTION GADGET
GADGET
P-1 P-1 P-1 P-1
PBP fordecrypt( , )a {P-1 iff permutation ≠ id {
reverse PBP fordecrypt( , )a {
![Page 126: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/126.jpg)
NEW SCHEME: OVERVIEW
![Page 127: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/127.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATION
![Page 128: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/128.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys
![Page 129: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/129.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
![Page 130: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/130.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTION|ψ⟩
![Page 131: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/131.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTIONapply quantum one-time pad a,b|ψ⟩ a,b
![Page 132: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/132.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTIONapply quantum one-time pad classically encrypt pad keys a,b|ψ⟩ a,b
![Page 133: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/133.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTIONapply quantum one-time pad classically encrypt pad keys a,b|ψ⟩ a,b
EVALUATION
![Page 134: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/134.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTIONapply quantum one-time pad classically encrypt pad keys a,b|ψ⟩ a,b
EVALUATIONafter / / : classically update keysH P CNOT
![Page 135: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/135.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTIONapply quantum one-time pad classically encrypt pad keys a,b|ψ⟩ a,b
EVALUATIONafter / / : classically update keysafter : use
H P CNOT
T
![Page 136: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/136.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTIONapply quantum one-time pad classically encrypt pad keys a,b|ψ⟩ a,b
EVALUATIONafter / / : classically update keysafter : use
DECRYPTION c,dU|ψ⟩ c,d
H P CNOT
T
![Page 137: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/137.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTIONapply quantum one-time pad classically encrypt pad keys a,b|ψ⟩ a,b
EVALUATIONafter / / : classically update keysafter : use
DECRYPTIONclassically decrypt pad keys c,dU|ψ⟩ c,d
H P CNOT
T
![Page 138: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/138.jpg)
NEW SCHEME: OVERVIEW
KEY GENERATIONclassical keys gadgets
ENCRYPTIONapply quantum one-time pad classically encrypt pad keys a,b|ψ⟩ a,b
EVALUATIONafter / / : classically update keysafter : use
DECRYPTIONclassically decrypt pad keys remove quantum one-time pad U|ψ⟩
c,d
H P CNOT
T
![Page 139: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/139.jpg)
FUTURE WORK
![Page 140: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/140.jpg)
FUTURE WORK
non-leveled QFHE?
![Page 141: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/141.jpg)
FUTURE WORK
non-leveled QFHE?
verifiable delegated quantum computation
![Page 142: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/142.jpg)
FUTURE WORK
non-leveled QFHE?
verifiable delegated quantum computation
quantum obfuscation?
![Page 143: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/143.jpg)
FUTURE WORK
non-leveled QFHE?
verifiable delegated quantum computation
quantum obfuscation?
…
![Page 144: QUANTUM HOMOMORPHIC ENCRYPTIONschaffne/mypresentations/... · 2016. 6. 29. · Quantum homomorphic encryption: only partial results Clifford scheme allowing evaluation of {P, H, CNOT}](https://reader033.fdocuments.net/reader033/viewer/2022051806/5ffa9e5f358fa561e75f2e68/html5/thumbnails/144.jpg)
THANK YOU!
is hiring two principle investigators: http://tinyurl.com/qusoft-job
Application deadline: 1 September 2016