Private Information Retrieval Over Gaussian MAC Ori Shmuel Joint - PowerPoint PPT Presentation

Private Information Retrieval PIR for Gaussian Multiple Access Channels Private Information Retrieval Over Gaussian MAC Ori Shmuel Joint Work with Asaf Cohen Ben Gurion University of the Negev shmuelor@bgu.ac.il IEEE International Symposium on Information Theory June 21, 2020 1 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - Problem Description 1 2 𝑂 ◦ N non-communicating and W W W " " " non-colluding databases (servers). W # W # W # … ◦ N noiseless orthogonal channels. … … … ◦ All servers are identical with M messages of length L . W $ W $ W $ ◦ The user wants W i privately. ◦ What is the most efficient way to retrieve W i ? User ◦ Non-efficient way: Download all messages. ◦ First introduced by Chor et al. 1 2 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - Problem Description 1 2 𝑂 ◦ N non-communicating and W W W " " " non-colluding databases (servers). W # W # W # … ◦ N noiseless orthogonal channels. … … … ◦ All servers are identical with M messages of length L . W $ W $ W $ ◦ The user wants W i privately. ◦ What is the most efficient way to retrieve W i ? User ◦ Non-efficient way: Download all messages. ◦ First introduced by Chor et al. 1 2 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - Problem Description 1 2 𝑂 ◦ N non-communicating and W W W " " " non-colluding databases (servers). W # W # W # … ◦ N noiseless orthogonal channels. … … … ◦ All servers are identical with M messages of length L . W $ W $ W $ ◦ The user wants W i privately. ◦ What is the most efficient way to retrieve W i ? User ◦ Non-efficient way: Download all messages. ◦ First introduced by Chor et al. 1 2 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - Problem Description 1 2 𝑂 ◦ N non-communicating and W W W " " " non-colluding databases (servers). W # W # W # … ◦ N noiseless orthogonal channels. … … … ◦ All servers are identical with M messages of length L . W $ W $ W $ ◦ The user wants W i privately. ◦ What is the most efficient way to retrieve W i ? User ◦ Non-efficient way: Download all messages. ◦ First introduced by Chor et al. 1 2 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - Problem Description 1 2 𝑂 ◦ N non-communicating and W W W " " " non-colluding databases (servers). W # W # W # … ◦ N noiseless orthogonal channels. … … … ◦ All servers are identical with M messages of length L . W $ W $ W $ ◦ The user wants W i privately. ◦ What is the most efficient way to retrieve W i ? User ◦ Non-efficient way: Download all messages. ◦ First introduced by Chor et al. 1 1 B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan, ”Private information retrieval,” in Proceedings of IEEE 36th Annual Foundations of Computer Science. IEEE, 1995, pp. 41-50. 2 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - General Course of Action 1 2 𝑂 ◦ Queries: Q 1 ( i ) , Q 2 ( i ) , ..., Q N ( i ) W W W " " " ◦ Answers: A 1 ( i ) , A 2 ( i ) , ..., A N ( i ) W # W # W # … ◦ Example: N=2 … … … − User: generates a uniform binary vector b ∈ F M 2 . W $ W $ W $ − Q 1 ( i ) = b 𝑅 # (𝑗) − Q 2 ( i ) = b ⊕ e i 𝑅 * (𝑗) 𝑅 " (𝑗) M User � A 1 ( i ) = b m W m mod 2 , m =1 M � A 2 ( i ) = ( b m + δ { m = i } ) W m mod 2 , m =1 3 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - General Course of Action 1 2 𝑂 ◦ Queries: Q 1 ( i ) , Q 2 ( i ) , ..., Q N ( i ) W W W " " " ◦ Answers: A 1 ( i ) , A 2 ( i ) , ..., A N ( i ) W # W # W # … ◦ Example: N=2 … … … − User: generates a uniform binary vector b ∈ F M 2 . W $ W $ W $ − Q 1 ( i ) = b 𝐵 # (𝑗) − Q 2 ( i ) = b ⊕ e i 𝐵 * (𝑗) 𝐵 " (𝑗) M User � A 1 ( i ) = b m W m mod 2 , m =1 M � A 2 ( i ) = ( b m + δ { m = i } ) W m mod 2 , m =1 3 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - General Course of Action 1 2 ◦ Queries: Q 1 ( i ) , Q 2 ( i ) , ..., Q N ( i ) W W " " W # W # ◦ Answers: A 1 ( i ) , A 2 ( i ) , ..., A N ( i ) ◦ Example: N=2 … … − User: generates a uniform W $ W $ binary vector b ∈ F M 2 . − Q 1 ( i ) = b − Q 2 ( i ) = b ⊕ e i User M � A 1 ( i ) = b m W m mod 2 , m =1 M � A 2 ( i ) = ( b m + δ { m = i } ) W m mod 2 , m =1 3 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels PIR - General Course of Action 1 2 ◦ Queries: Q 1 ( i ) , Q 2 ( i ) , ..., Q N ( i ) W W " " W # W # ◦ Answers: A 1 ( i ) , A 2 ( i ) , ..., A N ( i ) ◦ Example: N=2 … … − User: generates a uniform W $ W $ binary vector b ∈ F M 2 . − Q 1 ( i ) = b − Q 2 ( i ) = b ⊕ e i User M � A 1 ( i ) = b m W m mod 2 , m =1 M A 1 ( i ) + A 2 ( i ) = W i � A 2 ( i ) = ( b m + δ { m = i } ) W m mod 2 , m =1 3 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results 1 2 ◦ We define the PIR rate as, W W " " W # W # R � L D, … … ◦ D is the total bits downloaded. W $ W $ ◦ In the example: R � L 2 L = 1 2 User A 1 ( i ) + A 2 ( i ) = W i 4 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results 1 2 ◦ We define the PIR rate as, W W " " W # W # R � L D, … … ◦ D is the total bits downloaded. W $ W $ ◦ In the example: R � L 2 L = 1 2 User A 1 ( i ) + A 2 ( i ) = W i 4 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results 1 2 ◦ We define the PIR rate as, W W " " W # W # R � L D, … … ◦ D is the total bits downloaded. W $ W $ ◦ In the example: R � L 2 L = 1 2 User A 1 ( i ) + A 2 ( i ) = W i 4 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results 1 2 ◦ We define the PIR rate as, W W " " W # W # R � L D, … … ◦ D is the total bits downloaded. W $ W $ ◦ In the example: R � L 2 L = 1 2 User ◦ Can we do better than that? A 1 ( i ) + A 2 ( i ) = W i 4 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results 1 2 𝑂 ◦ The PIR capacity is 1 , W W W " " " W # W # W # � � 1 − 1 … N C P IR = � � M � � 1 … … … 1 − N W $ W $ W $ ◦ L = N M ◦ Example: N = 2 , M = 2 ◦ → L = 4 , C P IR = 2 3 . User 1 H. Sun and S. A. Jafar, ”The capacity of private information retrieval,” IEEE Transactions on Information Theory, vol. 63, no. 7, pp. 4075-4088, 2017 5 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results ◦ The PIR capacity is 1 , � � 1 − 1 N C P IR = � � M � � 1 1 − N ◦ L = N M ◦ Example: N = 2 , M = 2 ◦ → L = 4 , C P IR = 2 3 . 5 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results ◦ The PIR capacity is 1 , ◦ The user generates a random private permutations of L = 4 � � 1 − 1 indices. N C P IR = � � M � � 1 − W 1 : [ a 1 , a 2 , a 3 , a 4 ] 1 − N − W 2 : [ b 1 , b 2 , b 3 , b 4 ] ◦ L = N M ◦ Assume the desired message is W 1 ◦ Example: N = 2 , M = 2 ◦ → L = 4 , C P IR = 2 3 . 5 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results ◦ The PIR capacity is 1 , ◦ The user generates a random private permutations of L = 4 � � 1 − 1 indices. N C P IR = � � M � � 1 − W 1 : [ a 1 , a 2 , a 3 , a 4 ] 1 − N − W 2 : [ b 1 , b 2 , b 3 , b 4 ] ◦ L = N M ◦ Assume the desired message is W 1 ◦ Example: N = 2 , M = 2 Server 1 Server 2 ◦ → L = 4 , C P IR = 2 3 . 5 / 18

Private Information Retrieval General background PIR for Gaussian Multiple Access Channels Performance Metric and Known Results ◦ The PIR capacity is 1 , ◦ The user generates a random private permutations of L = 4 � � 1 − 1 indices. N C P IR = � � M � � 1 − W 1 : [ a 1 , a 2 , a 3 , a 4 ] 1 − N − W 2 : [ b 1 , b 2 , b 3 , b 4 ] ◦ L = N M ◦ Assume the desired message is W 1 ◦ Example: N = 2 , M = 2 Server 1 Server 2 ◦ → L = 4 , C P IR = 2 3 . 𝑏 " 5 / 18