Post on 14-Jul-2020
Private Information Retrieval in Distributed Storage
Systems Using an Arbitrary Linear Code
Siddhartha Kumar1 Eirik Rosnes1 Alexandre Graell i Amat2
1Simula@UiB, Norway
2Chalmers University of Technology, Sweden
Institut Mittag-Leffler16th May, 2017
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Introduction
· · · · · ·
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 1 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Introduction
· · · · · ·
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 1 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Introduction
· · · · · ·
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 1 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Introduction
· · · · · ·
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 1 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Introduction
· · · · · ·
Q(1) Q(k) Q(k+1) Q(n)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 1 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Introduction
· · · · · ·
r1 rk rk+1 rn
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 1 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Timeline of PIR
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 2 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Timeline of PIR
n−se
rver
PIR
Chor et al. (1995)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 2 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Timeline of PIR
n−se
rver
PIR
PIRus
ing
Batch
Codes
Ishai et al. (2004)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 2 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Timeline of PIR
n−se
rver
PIR
PIRus
ing
Batch
Codes
PIRforDSS
s
Shah et al. (2014)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 2 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Timeline of PIR
n−se
rver
PIR
PIRus
ing
Batch
Codes
PIRforDSS
s
MDS
code
dPIR
Tajeddine and El Rouayheb (2016)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 2 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Timeline of PIR
n−se
rver
PIR
PIRus
ing
Batch
Codes
PIRforDSS
s
MDS
code
dPIR
Opt
imal
MDS
PIR
Banawan and Ulukus (2016)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 2 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
In this talk (to be presented at ISIT 2017)
Contribution
• Extend Tajeddine and El Rouayheb’s PIR protocol to an arbitrary linear codeddata of rate R > 1/2.
• Show that the protocol can be optimized using the structure of the linear code.
Outcome
• Non-MDS codes can also achieve the lower bound on the communication cost(cPoP).
Parallel Work
• [Freij-Hollanti et al., 2016] designed a protocol• Works over arbitrary linear coded data.• Multiple nodes can collude.
• Our protocol yields better cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 3 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
In this talk (to be presented at ISIT 2017)
Contribution
• Extend Tajeddine and El Rouayheb’s PIR protocol to an arbitrary linear codeddata of rate R > 1/2.
• Show that the protocol can be optimized using the structure of the linear code.
Outcome
• Non-MDS codes can also achieve the lower bound on the communication cost(cPoP).
Parallel Work
• [Freij-Hollanti et al., 2016] designed a protocol• Works over arbitrary linear coded data.• Multiple nodes can collude.
• Our protocol yields better cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 3 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
In this talk (to be presented at ISIT 2017)
Contribution
• Extend Tajeddine and El Rouayheb’s PIR protocol to an arbitrary linear codeddata of rate R > 1/2.
• Show that the protocol can be optimized using the structure of the linear code.
Outcome
• Non-MDS codes can also achieve the lower bound on the communication cost(cPoP).
Parallel Work
• [Freij-Hollanti et al., 2016] designed a protocol• Works over arbitrary linear coded data.• Multiple nodes can collude.
• Our protocol yields better cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 3 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
In this talk (to be presented at ISIT 2017)
Contribution
• Extend Tajeddine and El Rouayheb’s PIR protocol to an arbitrary linear codeddata of rate R > 1/2.
• Show that the protocol can be optimized using the structure of the linear code.
Outcome
• Non-MDS codes can also achieve the lower bound on the communication cost(cPoP).
Parallel Work
• [Freij-Hollanti et al., 2016] designed a protocol• Works over arbitrary linear coded data.• Multiple nodes can collude.
• Our protocol yields better cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 3 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
In this talk (to be presented at ISIT 2017)
Contribution
• Extend Tajeddine and El Rouayheb’s PIR protocol to an arbitrary linear codeddata of rate R > 1/2.
• Show that the protocol can be optimized using the structure of the linear code.
Outcome
• Non-MDS codes can also achieve the lower bound on the communication cost(cPoP).
Parallel Work
• [Freij-Hollanti et al., 2016] designed a protocol• Works over arbitrary linear coded data.• Multiple nodes can collude.
• Our protocol yields better cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 3 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
In this talk (to be presented at ISIT 2017)
Contribution
• Extend Tajeddine and El Rouayheb’s PIR protocol to an arbitrary linear codeddata of rate R > 1/2.
• Show that the protocol can be optimized using the structure of the linear code.
Outcome
• Non-MDS codes can also achieve the lower bound on the communication cost(cPoP).
Parallel Work
• [Freij-Hollanti et al., 2016] designed a protocol• Works over arbitrary linear coded data.• Multiple nodes can collude.
• Our protocol yields better cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 3 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
In this talk (to be presented at ISIT 2017)
Contribution
• Extend Tajeddine and El Rouayheb’s PIR protocol to an arbitrary linear codeddata of rate R > 1/2.
• Show that the protocol can be optimized using the structure of the linear code.
Outcome
• Non-MDS codes can also achieve the lower bound on the communication cost(cPoP).
Parallel Work
• [Freij-Hollanti et al., 2016] designed a protocol• Works over arbitrary linear coded data.• Multiple nodes can collude.
• Our protocol yields better cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 3 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Distributed Storage System (DSS)
• A file X(m): a β × k matrix over a GF(qℓ).
• Each row is encoded using an (n, k) linear systematic code C over GF(q).
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 4 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Distributed Storage System (DSS)
• A file X(m): a β × k matrix over a GF(qℓ).
• Each row is encoded using an (n, k) linear systematic code C over GF(q).
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 4 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Distributed Storage System (DSS)
· · ·
· · ·
· · ·
· · ·
.
.
.
.
.
.
.
.
.
• A file X(m): a β × k matrix over a GF(qℓ).
• Each row is encoded using an (n, k) linear systematic code C over GF(q).
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 4 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Distributed Storage System (DSS)
· · ·
· · ·
· · ·
· · ·
.
.
.
.
.
.
.
.
.
k
β
• A file X(m): a β × k matrix over a GF(qℓ).
• Each row is encoded using an (n, k) linear systematic code C over GF(q).
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 4 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Distributed Storage System (DSS)
· · ·
· · ·
· · ·
· · ·
.
.
.
.
.
.
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.
.
k
β
· · · · · ·
· · · · · ·
· · · · · ·
· · · · · ·
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.
.
.
.
.
.
.
.
.
.
.
.
.
.
• A file X(m): a β × k matrix over a GF(qℓ).
• Each row is encoded using an (n, k) linear systematic code C over GF(q).
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 4 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Distributed Storage System (DSS)
· · ·
· · ·
· · ·
· · ·
.
.
.
.
.
.
.
.
.
k
β
· · · · · ·
· · · · · ·
· · · · · ·
· · · · · ·
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.
.
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.
.
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.
.
n
• A file X(m): a β × k matrix over a GF(qℓ).
• Each row is encoded using an (n, k) linear systematic code C over GF(q).
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 4 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Distributed Storage System (DSS)
· · · · · ·
......
......
• A file X(m): a β × k matrix over a GF(qℓ).
• Each row is encoded using an (n, k) linear systematic code C over GF(q).
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 4 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Distributed Storage System (DSS)
· · · · · ·
......
......
k systematic nodes n − k parity nodes
fch
unks
• A file X(m): a β × k matrix over a GF(qℓ).
• Each row is encoded using an (n, k) linear systematic code C over GF(q).
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 4 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Privacy Model
• Query matrix Q(j) of size d × βf is sent to j-th node.
• Node j sends back response rj of length d.
• Let s ∈ {1, . . . , n} be a spy node in the DSS.
rj
Q(j)
Perfect information-theoretic PIR
This scheme achieves perfect information-theoretic PIR if and only if
Privacy : H(m|Q(s)) = H(m)
Recovery : H(X(m)|r1, r2, . . . , rn) = 0,
where H(·) denotes the entropy function.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 5 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Privacy Model
• Query matrix Q(j) of size d × βf is sent to j-th node.
• Node j sends back response rj of length d.
• Let s ∈ {1, . . . , n} be a spy node in the DSS.
rj
Q(j) Q(j) =
( )
Perfect information-theoretic PIR
This scheme achieves perfect information-theoretic PIR if and only if
Privacy : H(m|Q(s)) = H(m)
Recovery : H(X(m)|r1, r2, . . . , rn) = 0,
where H(·) denotes the entropy function.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 5 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Privacy Model
• Query matrix Q(j) of size d × βf is sent to j-th node.
• Node j sends back response rj of length d.
• Let s ∈ {1, . . . , n} be a spy node in the DSS.
rj
Q(j)rj =
( )
= Q(j)
Content in the node
Perfect information-theoretic PIR
This scheme achieves perfect information-theoretic PIR if and only if
Privacy : H(m|Q(s)) = H(m)
Recovery : H(X(m)|r1, r2, . . . , rn) = 0,
where H(·) denotes the entropy function.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 5 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Privacy Model
• Query matrix Q(j) of size d × βf is sent to j-th node.
• Node j sends back response rj of length d.
• Let s ∈ {1, . . . , n} be a spy node in the DSS.
rj
Q(j)rj =
( )
= Q(j)
Content in the node
Perfect information-theoretic PIR
This scheme achieves perfect information-theoretic PIR if and only if
Privacy : H(m|Q(s)) = H(m)
Recovery : H(X(m)|r1, r2, . . . , rn) = 0,
where H(·) denotes the entropy function.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 5 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Efficiency of PIR protocols
Communication Price of Privacy (cPoP)
Total amount of downloaded data per unit of retrieved data,
θ =nd
βk.
• For the considered protocol, θ ≥ 1/(1 − R) , θLB [Chan et al., 2015].
• For a general protocol (single spy node),
θ ≥ θcap , 1 +k
n+
k2
n2+ · · · +
kf−1
nf−1[Banawan and Ulukus, 2016].
• When number of files (f) is very large, θcap = θLB.
• When more than one nodes collude, an explicit lower bound is currently unknown.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 6 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Efficiency of PIR protocols
Communication Price of Privacy (cPoP)
Total amount of downloaded data per unit of retrieved data,
θ =nd
βk.
• For the considered protocol, θ ≥ 1/(1 − R) , θLB [Chan et al., 2015].
• For a general protocol (single spy node),
θ ≥ θcap , 1 +k
n+
k2
n2+ · · · +
kf−1
nf−1[Banawan and Ulukus, 2016].
• When number of files (f) is very large, θcap = θLB.
• When more than one nodes collude, an explicit lower bound is currently unknown.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 6 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Efficiency of PIR protocols
Communication Price of Privacy (cPoP)
Total amount of downloaded data per unit of retrieved data,
θ =nd
βk.
• For the considered protocol, θ ≥ 1/(1 − R) , θLB [Chan et al., 2015].
• For a general protocol (single spy node),
θ ≥ θcap , 1 +k
n+
k2
n2+ · · · +
kf−1
nf−1[Banawan and Ulukus, 2016].
• When number of files (f) is very large, θcap = θLB.
• When more than one nodes collude, an explicit lower bound is currently unknown.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 6 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Efficiency of PIR protocols
Communication Price of Privacy (cPoP)
Total amount of downloaded data per unit of retrieved data,
θ =nd
βk.
• For the considered protocol, θ ≥ 1/(1 − R) , θLB [Chan et al., 2015].
• For a general protocol (single spy node),
θ ≥ θcap , 1 +k
n+
k2
n2+ · · · +
kf−1
nf−1[Banawan and Ulukus, 2016].
• When number of files (f) is very large, θcap = θLB.
• When more than one nodes collude, an explicit lower bound is currently unknown.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 6 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Efficiency of PIR protocols
Communication Price of Privacy (cPoP)
Total amount of downloaded data per unit of retrieved data,
θ =nd
βk.
• For the considered protocol, θ ≥ 1/(1 − R) , θLB [Chan et al., 2015].
• For a general protocol (single spy node),
θ ≥ θcap , 1 +k
n+
k2
n2+ · · · +
kf−1
nf−1[Banawan and Ulukus, 2016].
• When number of files (f) is very large, θcap = θLB.
• When more than one nodes collude, an explicit lower bound is currently unknown.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 6 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
Non PIR scheme for retrieving X(1):
Queries:
q1 =(
1 0)
, q2 =(
1 0)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
Non PIR scheme for retrieving X(1):
Queries:
q1 =(
1 0)
, q2 =(
1 0)
Nodes’ responses:
r1 =(
1 0)
(
x1
y1
)
= x1, r2 =(
1 0)
(
x2
y2
)
= x2
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
PIR scheme for reterieving x1:
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
PIR scheme for reterieving x1:
Queries:
q1 =(
u1 u2
)
, q2 =(
u1 u2
)
, q3 =(
u1 u2
)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
PIR scheme for reterieving x1:
Queries:
q1 =(
u1 + 1 u2
)
, q2 =(
u1 u2
)
, q3 =(
u1 u2
)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
PIR scheme for reterieving x1:
Queries:
q1 =(
u1 + 1 u2
)
, q2 =(
u1 u2
)
, q3 =(
u1 u2
)
Nodes’ responses:
r1 =(
u1 + 1 u2
)
(
x1
y1
)
= u1x1 + u2y1 + x1 = I1 + x1
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Basic Idea
x1 x2 x1 + x2
y1 y2 y1 + y2
(1) (2) (3)
• Files X(1) = (x1, x2) and X(2) = (y1, y2).
• Data encoded using an (3, 2) SPC code.
PIR scheme for reterieving x1:
Queries:
q1 =(
u1 + 1 u2
)
, q2 =(
u1 u2
)
, q3 =(
u1 u2
)
Nodes’ responses:
r1 =(
u1 + 1 u2
)
(
x1
y1
)
= u1x1 + u2y1 + x1 = I1 + x1
r2 =(
u1 u2
)
(
x2
y2
)
= I2, r3 =(
u1 u2
)
(
x1 + x2
y1 + y2
)
= I1 + I2
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 7 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
A Brief Overview
• Queries to systematic nodes: Q(l) = U + V (l).
• Queries to parity nodes: U .
• V (l) depends upon the columns of a binary matrix E (β regular).
• Each row of E represents an erasure pattern that is correctable by a shortenedcode C̃ of C. 1 represents erasure.
• β = d̃min − 1. d̃min is the minimum distance of C̃.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 8 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
A Brief Overview
• Queries to systematic nodes: Q(l) = U + V (l).
• Queries to parity nodes: U .
• V (l) depends upon the columns of a binary matrix E (β regular).
• Each row of E represents an erasure pattern that is correctable by a shortenedcode C̃ of C. 1 represents erasure.
• β = d̃min − 1. d̃min is the minimum distance of C̃.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 8 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
A Brief Overview
• Queries to systematic nodes: Q(l) = U + V (l).
• Queries to parity nodes: U .
• V (l) depends upon the columns of a binary matrix E (β regular).
• Each row of E represents an erasure pattern that is correctable by a shortenedcode C̃ of C. 1 represents erasure.
• β = d̃min − 1. d̃min is the minimum distance of C̃.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 8 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
A Brief Overview
• Queries to systematic nodes: Q(l) = U + V (l).
• Queries to parity nodes: U .
• V (l) depends upon the columns of a binary matrix E (β regular).
• Each row of E represents an erasure pattern that is correctable by a shortenedcode C̃ of C. 1 represents erasure.
• β = d̃min − 1. d̃min is the minimum distance of C̃.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 8 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Query Construction: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1.
• Code C̃ has parity-check matrix H̃ = P . β = d̃min − 1 = 2.
U =
(
u1,1 u1,2
u2,1 u2,2
u3,1 u3,2
)
, E =
(
1 0 11 1 00 1 1
)
Q(1) =
(
u1,1 + 1 u1,2
u2,1 u2,2 + 1u3,1 u3,2
)
, Q(2) =
(
u1,1 u1,2
u2,1 + 1 u2,2
u3,1 u3,2 + 1
)
, Q(3) =
(
u1,1 u1,2 + 1u2,1 u2,2
u3,1 + 1 u3,2
)
.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 9 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Query Construction: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1.
• Code C̃ has parity-check matrix H̃ = P . β = d̃min − 1 = 2.
U =
(
u1,1 u1,2
u2,1 u2,2
u3,1 u3,2
)
, E =
(
1 0 11 1 00 1 1
)
Q(1) =
(
u1,1 + 1 u1,2
u2,1 u2,2 + 1u3,1 u3,2
)
, Q(2) =
(
u1,1 u1,2
u2,1 + 1 u2,2
u3,1 u3,2 + 1
)
, Q(3) =
(
u1,1 u1,2 + 1u2,1 u2,2
u3,1 + 1 u3,2
)
.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 9 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Query Construction: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1.
• Code C̃ has parity-check matrix H̃ = P . β = d̃min − 1 = 2.
U =
(
u1,1 u1,2
u2,1 u2,2
u3,1 u3,2
)
, E =
(
1 0 11 1 00 1 1
)
Q(1) =
(
u1,1 + 1 u1,2
u2,1 u2,2 + 1u3,1 u3,2
)
, Q(2) =
(
u1,1 u1,2
u2,1 + 1 u2,2
u3,1 u3,2 + 1
)
, Q(3) =
(
u1,1 u1,2 + 1u2,1 u2,2
u3,1 + 1 u3,2
)
.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 9 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Query Construction: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1.
• Code C̃ has parity-check matrix H̃ = P . β = d̃min − 1 = 2.
U =
(
u1,1 u1,2
u2,1 u2,2
u3,1 u3,2
)
, E =
(
1 0 11 1 00 1 1
)
Q(1) =
(
u1,1 + 1 u1,2
u2,1 u2,2 + 1u3,1 u3,2
)
, Q(2) =
(
u1,1 u1,2
u2,1 + 1 u2,2
u3,1 u3,2 + 1
)
, Q(3) =
(
u1,1 u1,2 + 1u2,1 u2,2
u3,1 + 1 u3,2
)
.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 9 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Query Construction: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1.
• Code C̃ has parity-check matrix H̃ = P . β = d̃min − 1 = 2.
U =
(
u1,1 u1,2
u2,1 u2,2
u3,1 u3,2
)
, E =
(
1 0 11 1 00 1 1
)
Q(1) =
(
u1,1 + 1 u1,2
u2,1 u2,2 + 1u3,1 u3,2
)
, Q(2) =
(
u1,1 u1,2
u2,1 + 1 u2,2
u3,1 u3,2 + 1
)
, Q(3) =
(
u1,1 u1,2 + 1u2,1 u2,2
u3,1 + 1 u3,2
)
.
Privacy : H(m|Q(s)) = H(m)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 9 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
File Retrieval: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 10 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
File Retrieval: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1, β = d̃min − 1 = 2.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 10 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
File Retrieval: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1, β = d̃min − 1 = 2.
r1 = Q(1) ( x11
x21) =
(
u11+1 u12
u21 u22+1u31 u32
)
( x11
x21) =
(
u11x11+u12x21+x11
u21x11+u22x21+x21
u31x11+u32x21
)
=(
I1+x11
I4+x21
I7
)
r2 = Q(2) ( x12
x22) =
(
u11 u12
u21+1 u22
u31 u32+1
)
( x12
x22) =
(
u11x12+u12x22
u21x12+u22x22+x12
u31x12+u32x22+x22
)
=(
I2
I5+x12
I8+x22
)
r3 = Q(3) ( x13
x23) =
(
u11 u12+1u21 u22
u31+1 u32
)
( x13
x23) =
(
u11x13+u12x23+x23
u21x13+u22x23
u31x13+u32x23+x13
)
=(
I3+x23
I6
I9+x13
)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 10 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
File Retrieval: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1, β = d̃min − 1 = 2.
r4 = Q(4)(
x11+x12
x21+x22
)
=(
u11 u12
u21 u22
u31 u32
)
(
x11+x12
x21+x22
)
=(
I1+I2
I4+I5
I7+I8
)
r5 = Q(5)(
x12+x13
x22+x23
)
=(
u11 u12
u21 u22
u31 u32
)
(
x12+x13
x22+x23
)
=(
I2+I3
I5+I6
I8+I9
)
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 10 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
File Retrieval: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1, β = d̃min − 1 = 2.
r4 = Q(4)(
x11+x12
x21+x22
)
=(
u11 u12
u21 u22
u31 u32
)
(
x11+x12
x21+x22
)
=(
I1+I2
I4+I5
I7+I8
)
r5 = Q(5)(
x12+x13
x22+x23
)
=(
u11 u12
u21 u22
u31 u32
)
(
x12+x13
x22+x23
)
=(
I2+I3
I5+I6
I8+I9
)
• Yields cPoP θ = 3·52·3
= 2.5.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 10 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
File Retrieval: (5, 3) Linear code C
C =
(
x11 x12 x13 x11 + x12 x12 + x13
x21 x22 x23 x21 + x22 x22 + x23
)
H = (P |I) =
(
1 1 0 1 00 1 1 0 1
)
Example
• We have: n = 5, k = 3, f = 1, β = d̃min − 1 = 2.
r4 = Q(4)(
x11+x12
x21+x22
)
=(
u11 u12
u21 u22
u31 u32
)
(
x11+x12
x21+x22
)
=(
I1+I2
I4+I5
I7+I8
)
r5 = Q(5)(
x12+x13
x22+x23
)
=(
u11 u12
u21 u22
u31 u32
)
(
x12+x13
x22+x23
)
=(
I2+I3
I5+I6
I8+I9
)
• Yields cPoP θ = 3·52·3
= 2.5.
Recovery : H(X(m)|r1, r2, . . . , rn) = 0
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 10 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Optimizing cPoP
β = β + 1find erasure patterns
correctable by C̃find β regular matrix, E E exists?
yes
noβ − 1
• cPoP for our protocol: θ = n/β.
• When C is an MDS code, dmin = d̃min = n − k + 1, and the PIR scheme reducesto the one in given [Tajeddine El Rouayheb].
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 11 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Optimizing cPoP
β = β + 1find erasure patterns
correctable by C̃find β regular matrix, E E exists?
yes
noβ − 1
• cPoP for our protocol: θ = n/β.
• When C is an MDS code, dmin = d̃min = n − k + 1, and the PIR scheme reducesto the one in given [Tajeddine El Rouayheb].
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 11 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Optimizing cPoP
β = β + 1find erasure patterns
correctable by C̃find β regular matrix, E E exists?
yes
noβ − 1
• cPoP for our protocol: θ = n/β.
• When C is an MDS code, dmin = d̃min = n − k + 1, and the PIR scheme reducesto the one in given [Tajeddine El Rouayheb].
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 11 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Optimizing cPoP
β = β + 1find erasure patterns
correctable by C̃find β regular matrix, E E exists?
yes
noβ − 1
• cPoP for our protocol: θ = n/β.
• When C is an MDS code, dmin = d̃min = n − k + 1, and the PIR scheme reducesto the one in given [Tajeddine El Rouayheb].
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 11 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Optimizing cPoP
β = β + 1find erasure patterns
correctable by C̃find β regular matrix, E E exists?
yes
noβ − 1
• cPoP for our protocol: θ = n/β.
• When C is an MDS code, dmin = d̃min = n − k + 1, and the PIR scheme reducesto the one in given [Tajeddine El Rouayheb].
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 11 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Optimizing cPoP
β = β + 1find erasure patterns
correctable by C̃find β regular matrix, E E exists?
yes
noβ − 1
• cPoP for our protocol: θ = n/β.
• When C is an MDS code, dmin = d̃min = n − k + 1, and the PIR scheme reducesto the one in given [Tajeddine El Rouayheb].
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 11 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Optimizing cPoP
β = β + 1find erasure patterns
correctable by C̃find β regular matrix, E E exists?
yes
noβ − 1
• cPoP for our protocol: θ = n/β.
• When C is an MDS code, dmin = d̃min = n − k + 1, and the PIR scheme reducesto the one in given [Tajeddine El Rouayheb].
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 11 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Numerical results
Code dmin d̃min θnon−opt θFH θopt θLB
C1 : (5, 3) (example) 2 3 2.5 5 2.5 2.5C2 : (11, 6) 4 4 3.6667 3.6667 2.75 2.2C3 : (12, 8) Pyramid 4 4 4 4.5 3 3C4 : (16, 10) LRC 5 5 4 4.8 3.2 2.6667C5 : (18, 12) Pyramid 5 5 4.5 4.5 3 3C6 : (154, 121) LRC 4 6 30.8 52.18 4.9677 4.6667C7 : (187, 121) LRC 7 16 12.4667 32.45 3.0656 2.8333
• θFH is the cPoP of linear coded PIR protocol of Freij-Hollanti et al.
• For all codes, θopt is close to the lower bound θLB.
• The non-MDS codes C1, C3, and C5 achieve the lower bound θLB.
• The codes C2, C4, C6, and C7 achieve the best cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 12 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Numerical results
Code dmin d̃min θnon−opt θFH θopt θLB
C1 : (5, 3) (example) 2 3 2.5 5 2.5 2.5C2 : (11, 6) 4 4 3.6667 3.6667 2.75 2.2C3 : (12, 8) Pyramid 4 4 4 4.5 3 3C4 : (16, 10) LRC 5 5 4 4.8 3.2 2.6667C5 : (18, 12) Pyramid 5 5 4.5 4.5 3 3C6 : (154, 121) LRC 4 6 30.8 52.18 4.9677 4.6667C7 : (187, 121) LRC 7 16 12.4667 32.45 3.0656 2.8333
• θFH is the cPoP of linear coded PIR protocol of Freij-Hollanti et al.
• For all codes, θopt is close to the lower bound θLB.
• The non-MDS codes C1, C3, and C5 achieve the lower bound θLB.
• The codes C2, C4, C6, and C7 achieve the best cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 12 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Numerical results
Code dmin d̃min θnon−opt θFH θopt θLB
C1 : (5, 3) (example) 2 3 2.5 5 2.5 2.5C2 : (11, 6) 4 4 3.6667 3.6667 2.75 2.2C3 : (12, 8) Pyramid 4 4 4 4.5 3 3C4 : (16, 10) LRC 5 5 4 4.8 3.2 2.6667C5 : (18, 12) Pyramid 5 5 4.5 4.5 3 3C6 : (154, 121) LRC 4 6 30.8 52.18 4.9677 4.6667C7 : (187, 121) LRC 7 16 12.4667 32.45 3.0656 2.8333
• θFH is the cPoP of linear coded PIR protocol of Freij-Hollanti et al.
• For all codes, θopt is close to the lower bound θLB.
• The non-MDS codes C1, C3, and C5 achieve the lower bound θLB.
• The codes C2, C4, C6, and C7 achieve the best cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 12 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Numerical results
Code dmin d̃min θnon−opt θFH θopt θLB
C1 : (5, 3) (example) 2 3 2.5 5 2.5 2.5C2 : (11, 6) 4 4 3.6667 3.6667 2.75 2.2C3 : (12, 8) Pyramid 4 4 4 4.5 3 3C4 : (16, 10) LRC 5 5 4 4.8 3.2 2.6667C5 : (18, 12) Pyramid 5 5 4.5 4.5 3 3C6 : (154, 121) LRC 4 6 30.8 52.18 4.9677 4.6667C7 : (187, 121) LRC 7 16 12.4667 32.45 3.0656 2.8333
• θFH is the cPoP of linear coded PIR protocol of Freij-Hollanti et al.
• For all codes, θopt is close to the lower bound θLB.
• The non-MDS codes C1, C3, and C5 achieve the lower bound θLB.
• The codes C2, C4, C6, and C7 achieve the best cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 12 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Numerical results
Code dmin d̃min θnon−opt θFH θopt θLB
C1 : (5, 3) (example) 2 3 2.5 5 2.5 2.5C2 : (11, 6) 4 4 3.6667 3.6667 2.75 2.2C3 : (12, 8) Pyramid 4 4 4 4.5 3 3C4 : (16, 10) LRC 5 5 4 4.8 3.2 2.6667C5 : (18, 12) Pyramid 5 5 4.5 4.5 3 3C6 : (154, 121) LRC 4 6 30.8 52.18 4.9677 4.6667C7 : (187, 121) LRC 7 16 12.4667 32.45 3.0656 2.8333
• θFH is the cPoP of linear coded PIR protocol of Freij-Hollanti et al.
• For all codes, θopt is close to the lower bound θLB.
• The non-MDS codes C1, C3, and C5 achieve the lower bound θLB.
• The codes C2, C4, C6, and C7 achieve the best cPoP.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 12 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Take-home messages
• Generalized the PIR protocol proposed by Tajeddine and El Rouayheb for a DSSwith a single spy node and where data is linearly coded with R > 1/2.
• Presented an algorithm to optimize the cPoP of the protocol. The optimizationleads to a cPoP close to its theoretical lower bound.
• Interestingly, for certain (non-MDS) codes, the lower bound on the cPoP can beachieved.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 13 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Take-home messages
• Generalized the PIR protocol proposed by Tajeddine and El Rouayheb for a DSSwith a single spy node and where data is linearly coded with R > 1/2.
• Presented an algorithm to optimize the cPoP of the protocol. The optimizationleads to a cPoP close to its theoretical lower bound.
• Interestingly, for certain (non-MDS) codes, the lower bound on the cPoP can beachieved.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 13 / 13
Introduction System Model Construction Optimizing cPoP Numerical Results Summary
Take-home messages
• Generalized the PIR protocol proposed by Tajeddine and El Rouayheb for a DSSwith a single spy node and where data is linearly coded with R > 1/2.
• Presented an algorithm to optimize the cPoP of the protocol. The optimizationleads to a cPoP close to its theoretical lower bound.
• Interestingly, for certain (non-MDS) codes, the lower bound on the cPoP can beachieved.
PIR in Distributed Storage Systems Using an Arbitrary Linear Code | S. Kumar, E. Rosnes and A. Graell i Amat 13 / 13