asynchronous byzantine agreement with subquadratic ......[cks20]: shir cohen, idit keidar, and...
TRANSCRIPT
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Asynchronous Byzantine Agreement with Subquadratic Communication
Julian Loss
U. Maryland
Chen-Da Liu-Zhang
ETH Zurich
Erica
Blum
U. Maryland
TCC 2020
Jonathan
Katz
U. Maryland
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Byzantine Agreement
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Byzantine Agreement
π₯1
π₯2
π₯3
π₯6
π₯5
π₯4
![Page 4: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/4.jpg)
Byzantine Agreement
All honest parties agree on the same output
π¦
π¦
π¦
π¦
π¦
π¦
![Page 5: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/5.jpg)
Byzantine Agreement
All honest parties agree on the same output
If honest parties have the same input, they keep the same value as output
π₯
π₯
π₯
π₯
π₯
π₯
![Page 6: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/6.jpg)
Byzantine Agreement
All honest parties agree on the same output
If honest parties have the same input, they keep the same value as output
π₯
π₯
π₯
π₯
π₯
π₯
![Page 7: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/7.jpg)
Byzantine Agreement
All honest parties agree on the same output
If honest parties have the same input, they keep the same value as output
π₯
π₯
π₯
π₯
![Page 8: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/8.jpg)
![Page 9: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/9.jpg)
Is there an asynchronous BA with π(π2) communication that tolerates π(π) adaptive corruptions?
![Page 10: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/10.jpg)
Is there an asynchronous BA with π(π2) communication that tolerates π(π) adaptive corruptions?
β’ Feasibility of asynch. π(π2) BA for π < (1 β π) Ξ€π 3 using a trusted dealer(alternately, with amortized π(π2) and without setup)
![Page 11: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/11.jpg)
Is there an asynchronous BA with π(π2) communication that tolerates π(π) adaptive corruptions?
β’ Feasibility of asynch. π(π2) BA for π < (1 β π) Ξ€π 3 using a trusted dealer(alternately, with amortized π(π2) and without setup)
β’ Impossibility of asynch. π(π2) BA with π(π) corruptions without setup
![Page 12: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/12.jpg)
Related Work
Most previous subquadratic BA are synchronous or partially synchronous [KS06,KS10,M17,A+19,β¦]
Recent work by Cohen et al. [CKS20] give subquadratic asynchronous BA, but the adversary has restricted scheduling power
![Page 13: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/13.jpg)
Feasibility of asynchronous π(π2) BA for π < (1 β π) Ξ€π 3 adaptive
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Feasibility of asynchronous π(π2) BA for π < (1 β π) Ξ€π 3 adaptive
π΅π΄ πππ‘π’π
π΅π΄
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Feasibility of asynchronous π(π2) BA for π < (1 β π) Ξ€π 3 adaptive
CC: π ππππ¦ π β π
π΅π΄ πππ‘π’π
π΅π΄
Size: π ππππ¦ π
![Page 16: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/16.jpg)
Feasibility of asynchronous π(π2) BA for π < (1 β π) Ξ€π 3 adaptive
πππΆ πππ‘π’π
πππΆ
π΅π΄ πππ‘π’π
π΅π΄
CC: π ππππ¦ π β π
Size: π ππππ¦ π
![Page 17: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/17.jpg)
Feasibility of asynchronous π(π2) BA for π < (1 β π) Ξ€π 3 adaptive
πππΆ πππ‘π’π
πππΆ
π΅π΄ πππ‘π’π
π΅π΄
CC: π ππππ¦ π β π
Size: π ππππ¦ π
Initial dealer
![Page 18: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/18.jpg)
Feasibility of asynchronous π(π2) BA for π < (1 β π) Ξ€π 3 adaptive
πππΆ πππ‘π’π
πππΆ
π΅π΄ πππ‘π’π
π΅π΄
CC: π ππππ¦ π β π
Size: π ππππ¦ π
πππΆ πππ‘π’ππ΅π΄ πππ‘π’π
Initial dealer
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Feasibility of asynchronous π(π2) BA for π < (1 β π) Ξ€π 3 adaptive
πππΆ πππ‘π’π
πππΆ
π΅π΄ πππ‘π’π
π΅π΄
CC: π ππππ¦ π β π
Size: π ππππ¦ π
πππΆ πππ‘π’ππ΅π΄ πππ‘π’π
π ππππ¦ π β π
π ππππ¦ π
Initial dealer
![Page 20: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/20.jpg)
Feasibility of asynchronous π(π2) BA for π < (1 β π) Ξ€π 3 adaptive
πππΆ πππ‘π’π
πππΆ
π΅π΄ πππ‘π’π
π΅π΄
CC: π ππππ¦ π β π
Size: π ππππ¦ π
πππΆ πππ‘π’ππ΅π΄ πππ‘π’π
π ππππ¦ π β π
π ππππ¦ π
Initial dealer
πππΆπ΅π΄
β¦
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One-Time BA
πΊπΆ
πΆπππ
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πΊπΆ
πΆπππ
Graded Consensus [CR93]Input π₯π; Output (π§π , ππ)
If β honest ππ π₯π = π₯, then π§π , ππ = (π₯, 1)If β honest ππ ππ = 1, then π§π = π§π
One-Time BA
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πΊπΆ
πΆπππ
Graded Consensus [CR93]Input π₯π; Output (π§π , ππ)
If β honest ππ π₯π = π₯, then π§π , ππ = (π₯, 1)If β honest ππ ππ = 1, then π§π = π§π
Coin-FlipEach ππ obtains the same random bit ππ
One-Time BA
![Page 24: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/24.jpg)
πΊπΆ
πΆπππ
Graded Consensus [CR93]Input π₯π; Output (π§π , ππ)
If β honest ππ π₯π = π₯, then π§π , ππ = (π₯, 1)If β honest ππ ππ = 1, then π§π = π§π
Coin-FlipEach ππ obtains the same random bit ππ
β€ π(π )
If ππ = 0: π₯π = ππElse π₯π = π§π
One-Time BA
![Page 25: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/25.jpg)
π΅π΄ πππ‘π’π
πΊπΆ
πΆπππ
β€ π(π )
Graded Consensus [CR93]Input π₯π; Output (π§π , ππ)
If β honest ππ π₯π = π₯, then π§π , ππ = (π₯, 1)If β honest ππ ππ = 1, then π§π = π§π
Coin-FlipEach ππ obtains the same random bit ππ
π π π
Each party in set can prove membership
π
Each party in set has a (signed) share of ππ
π π π
One-Time BA
If ππ = 0: π₯π = ππElse π₯π = π§π
![Page 26: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/26.jpg)
π΅π΄ πππ‘π’π
πΊπΆ
πΆπππ
β€ π(π )
Graded Consensus [CR93]Input π₯π; Output (π§π , ππ)
If β honest ππ π₯π = π₯, then π§π , ππ = (π₯, 1)If β honest ππ ππ = 1, then π§π = π§π
Coin-FlipEach ππ obtains the same random bit ππ
Communication π ππππ¦ π β π
Setup size π ππππ¦ π
π π π
Each party in set can prove membership
π
Each party in set has a (signed) share of ππ
π π π
One-Time BA
If ππ = 0: π₯π = ππElse π₯π = π§π
![Page 27: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/27.jpg)
MPC
![Page 28: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/28.jpg)
MPC
Multi-Party Computation with β-output quality
π₯1
π₯2
π₯3
π₯6
π₯5
π₯4
![Page 29: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/29.jpg)
MPC
Multi-Party Computation with β-output quality
π(π₯1β² , π₯2
β² , β¦ , π₯πβ² ), where π₯π
β² = π₯π if ππ β ππ₯πβ² =β₯ otherwise
Adversary chooses π with size at least β
π₯1
π₯2
π₯3
π₯6
π₯5
π₯4
![Page 30: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/30.jpg)
MPC
Agreement on a Common Subset with β-output quality
π΄πΆπ
π₯1π₯2
π₯3π₯4
π₯π
π β₯ β with β β π honest inputs
π
β¦
![Page 31: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/31.jpg)
MPC
Agreement on a Common Subset with β-output quality
π΄πΆπ
π₯1π₯2
π₯3π₯4
β¦
π₯π
π
π β₯ β with β β π honest inputs
π΅π΄ πππ‘π’π π΅π΄ πππ‘π’πβ¦
π΅π΄ πππ‘π’π
π(β)
Communication π β β β β ππππ¦ π β π
Setup size π β β ππππ¦ π
π΄πΆπ πππ‘π’π
![Page 32: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/32.jpg)
MPCThreshold Fully Homomorphic Encryption
![Page 33: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/33.jpg)
MPCThreshold Fully Homomorphic Encryption
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π
![Page 34: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/34.jpg)
MPCThreshold Fully Homomorphic Encryption
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π π
![Page 35: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/35.jpg)
MPCThreshold Fully Homomorphic Encryption
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π π ππ ππ1, β¦ , πππ for parties in
![Page 36: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/36.jpg)
MPCThreshold Fully Homomorphic Encryption
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π [π] π ππ ππ1, β¦ , πππ for parties in
![Page 37: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/37.jpg)
MPC
π¦π
Threshold Fully Homomorphic Encryption
π₯1
π₯2
π₯3
π₯4β¦
π₯π
π
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π [π] π ππ ππ1, β¦ , πππ for parties in
![Page 38: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/38.jpg)
MPCThreshold Fully Homomorphic Encryption
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π [π] π ππ ππ1, β¦ , πππ for parties in
[π₯1]
[π₯2]
[π₯3]
[π₯4]
β¦
[π₯π]
ππ΄πΆπ
![Page 39: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/39.jpg)
MPC
ππΈπ£πππ
Threshold Fully Homomorphic Encryption
π΄πΆπ
[π₯1]
[π₯2]
[π₯3]
[π₯4]
β¦
[π₯π]
π
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π
[π₯1]
β₯
[π₯3]
β₯
[π₯π]
[π] π ππ ππ1, β¦ , πππ for parties in
β¦
![Page 40: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/40.jpg)
MPC
ππΈπ£πππ
Threshold Fully Homomorphic Encryption
π΄πΆπ
[π₯1]
[π₯2]
[π₯3]
[π₯4]
β¦
[π₯π]
π
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π
[π₯1]
β₯
[π₯3]
β₯
[π₯π]
[π] π ππ ππ1, β¦ , πππ for parties in
Decryption
ππ = π·πππβππππππ(π)
ππ
π1
ππ
β¦β¦
![Page 41: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/41.jpg)
MPC
ππΈπ£πππ
Threshold Fully Homomorphic Encryption
π΄πΆπ
[π₯1]
[π₯2]
[π₯3]
[π₯4]
β¦
[π₯π]
π
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π
[π₯1]
β₯
[π₯3]
β₯
[π₯π]
[π] π ππ ππ1, β¦ , πππ for parties in
Decryption
ππ = π·πππβππππππ(π)
ππ
π1
ππ
β¦
π¦ = π ππ({ππ})
All parties output
β¦
![Page 42: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/42.jpg)
MPC
ππΈπ£πππ
Threshold Fully Homomorphic Encryption
π΄πΆπ
[π₯1]
[π₯2]
[π₯3]
[π₯4]
β¦
[π₯π]
π
πππΆ πππ‘π’π
π΄πΆπ πππ‘π’π
[π₯1]
β₯
[π₯3]
β₯
[π₯π]
[π] π ππ ππ1, β¦ , πππ for parties in
Size: π (β + 1) β ππππ¦ π
Decryption
ππ = π·πππβππππππ(π)
ππ
π1
ππ
β¦
π¦ = π ππ({ππ})
All parties output
CC: π β + 1 β β + πͺ β ππππ¦ π β π
β¦
![Page 43: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/43.jpg)
MPC for Trusted Dealer
ππΈπ£πππ
Threshold Fully Homomorphic Encryption
πππΆ πππ‘π’π
[π] π ππ ππ1, β¦ , πππ for parties in
Size: π (β + 1) β ππππ¦ π
Decryption
ππ = π·πππβππππππ(π)
ππ
π1
ππ
β¦
π¦ = π ππ({ππ})
All parties output
CC: π β + 1 β β + πͺ β ππππ¦ π β π
![Page 44: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/44.jpg)
MPC for Trusted Dealer
ππΈπ£πππ
Threshold Fully Homomorphic Encryption
πππΆ πππ‘π’π
[π] π ππ ππ1, β¦ , πππ for parties in
Size: π ππππ¦ π
Decryption
ππ = π·πππβππππππ(π)
ππ
π1
ππ
β¦
π¦ = π ππ({ππ})
All parties output
CC: π β + 1 β β + πͺ β ππππ¦ π β π
![Page 45: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/45.jpg)
MPC for Trusted Dealer
ππΈπ£πππ
Threshold Fully Homomorphic Encryption
πππΆ πππ‘π’π
[π] π ππ ππ1, β¦ , πππ for parties in
Size: π ππππ¦ π
Decryption
ππ = π·πππβππππππ(π)
ππ
π1
ππ
β¦
π¦ = π ππ({ππ})
All parties output
CC: π ππππ¦ π β π
![Page 46: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/46.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
Other lower bounds:
[DR85, A+19] adversary can perform after-the-fact removal
[R20] similar to our lower bound, but with idealized PKI
![Page 47: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/47.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
βππ has input 1π outputs 1
ππ
πβ²
![Page 48: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/48.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
![Page 49: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/49.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 50: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/50.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 51: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/51.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 52: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/52.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 53: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/53.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 54: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/54.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 55: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/55.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 56: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/56.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 57: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/57.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 58: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/58.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 59: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/59.jpg)
Impossibility of asynch. π(π2) BA with π(π) adaptive corruptions and no setup
ππ
πβ²
βππ has input 1π outputs 1
βππ β πβ² has input 0βππ β πβ² outputs 0
π
πβ²
ππ
πβ²
π has input 1; βππ β πβ² has input 0π outputs 1; βππ β πβ² outputs 0
![Page 60: Asynchronous Byzantine Agreement with Subquadratic ......[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine agreement](https://reader035.vdocuments.mx/reader035/viewer/2022071606/61426537d9e4dc11f47f063b/html5/thumbnails/60.jpg)
References and CreditsFull version: https://eprint.iacr.org/2020/851
References:[BKLL20]: Ran Canetti and Tal Rabin. Fast asynchronous Byzantine agreement with optimal resilience. STOC 1993.[DR85]: Danny Dolev and RΓΌdiger Reischuk. Bounds on information exchange for Byzantine agreement. Journal of the
ACM 1985.[KS06]: Valerie King, Jared Saia, Vishal Sanwalani, and Erik Vee. Scalable leader election. SODA 2006.[KS10]: Valerie King and Jared Saia. Breaking the π(π2) bit barrier: scalable byzantine agreement with an adaptive
adversary. PODC 2010.[M17]: Silvio Micali. Very simple and efficient byzantine agreement. ITCS 2017.[A+19]: Ittai Abraham, T.-H. Hubert Chan, Danny Dolev, Kartik Nayak, Rafael Pass, Ling Ren, and Elaine Shi. Communication
complexity of byzantine agreement, revisited. PODC 2019.[CKS20]: Shir Cohen, Idit Keidar, and Alexander Spiegelman. Not a COINcidence: Sub-quadratic asynchronous Byzantine
agreement WHP. DISC 2020.[R20]: Matthieu Rambaud. Lower bounds for authenticated randomized Byzantine consensus under (partial)
synchrony: The limits of standalone digital signatures.
Credits:Icons: https://www.flaticon.com/