Formulation of Privacy What information can be published? Average - PowerPoint PPT Presentation

Zhenjie Zhang Advanced Digital Sciences Center, Singapore (Thanks to Xiaokui Xiao for contributing slides)

Formulation of Privacy  What information can be published?  Average height of US people  Height of an individual  Intuition:  If something is insensitive to the change of any individual tuple, then it should not be considered private  Example:  Assume that we arbitrarily change the height of an individual in the US  The average height of US people would remain roughly the same  i.e., The average height reveals little information about the exact height of any particular individual

𝜻 -Differential Privacy  Definition:  Neighboring datasets: Two datasets 𝑬 and 𝑬′ , such that 𝑬′ can be obtained by changing one single tuple in 𝑬  A randomized algorithm 𝑩 satisfies 𝛇 -differential privacy, iff for any two neighboring datasets 𝑬 and 𝑬′ and for any output 𝑷 of 𝑩 , Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷 Name Gender Age Diabetes Name Gender Age Diabetes Alice F 28 Y Alice F 28 Y Bob M 19 Y Bob M 19 Y Chris M 25 N Chris M 23 Y Doug M 30 N Doug M 30 N

𝜻 -Differential Privacy Probabilities ≤ exp (𝜻) ratio Pr 𝑩 𝑬 = 𝑷 Pr 𝑩 𝑬′ = 𝑷  Intuition:  It is OK to publish information that is insensitive to changes of any particular tuple # of diabetes patients  Definition:  Neighboring datasets: Two datasets 𝑬 and 𝑬′ , such that 𝑬′ can be obtained by changing one single tuple in 𝑬  A randomized algorithm 𝑩 satisfies 𝛇 -differential privacy, iff for any two neighboring datasets 𝑬 and 𝑬′ and for any output 𝑷 of 𝑩 , Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  The value of 𝜻 decides the degree of privacy protection

Achieving 𝜻 -Differential Privacy Name Gender Age Diabetes Name Gender Age Diabetes Alice F 28 Y Alice F 28 Y Bob M 19 Y Bob M 19 Y Chris M 25 N Chris M 23 Y Doug M 30 N Doug M 30 N  It won’t work if we release the number directly:  𝑬 : the original dataset  𝑬′ : modify an arbitrary patient in 𝑬  Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷 does not hold for any 𝜻 Pr 𝑩 𝑬 = 𝒊 Pr 𝑩 𝑬′ = 𝒊′ 100% 𝒊 ′ = 𝟒 𝟏 𝟐 𝒊 = 𝟑 # of diabetes patients

Achieving 𝜻 -Differential Privacy Name Gender Age Diabetes Name Gender Age Diabetes Alice F 28 Y Alice F 28 Y Bob M 19 Y Bob M 19 Y Chris M 25 N Chris M 23 Y Doug M 30 N Doug M 30 N  Idea:  Perturb the number of diabetes patients to obtain a smooth distribution Pr 𝑩 𝑬 = 𝒊 Pr 𝑩 𝑬′ = 𝒊′ 100% 𝒊 ′ = 𝟒 𝟏 𝟐 𝒊 = 𝟑 # of diabetes patients

Achieving 𝜻 -Differential Privacy Name Gender Age Diabetes Name Gender Age Diabetes Alice F 28 Y Alice F 28 Y Bob M 19 Y Bob M 19 Y Chris M 25 N Chris M 23 Y Doug M 30 N Doug M 30 N  Idea:  Perturb the number of diabetes patients to obtain a smooth distribution Pr 𝑩 𝑬 = 𝒊 Pr 𝑩 𝑬′ = 𝒊′ 100% 𝒊 ′ = 𝟒 𝟏 𝟐 𝒊 = 𝟑 # of diabetes patients

Achieving 𝜻 -Differential Privacy Name Gender Age Diabetes Name Gender Age Diabetes Alice F 28 Y Alice F 28 Y Bob M 19 Y Bob M 19 Y Chris M 25 N Chris M 23 Y Doug M 30 N Doug M 30 N  Idea:  Perturb the number of diabetes patients to obtain a smooth distribution ratio bounded Pr 𝑩 𝑬 = 𝒊 Pr 𝑩 𝑬′ = 𝒊′ 100% 𝒊 ′ = 𝟒 𝟏 𝟐 𝒊 = 𝟑 # of diabetes patients

Laplace Distribution 𝒚  𝑞𝑒𝑔 𝒚 = exp − 2𝝁 ; 𝝁  increase/decrease 𝒚 by 1 1  𝑞𝑒𝑔 𝒚 changes by a factor of exp −  𝝁  𝝁 is referred as the scale 0.5 𝝁 = 1 0.45 𝝁 = 2 0.4 0.35 𝝁 = 4 0.3 0.25 0.2 0.15 0.1 0.05 0 -10 -8 -6 -4 -2 0 2 4 6 8 10

Differential Privacy via Laplace Noise  Dataset: A set of patients Release # of diabetes patients with 𝜻 -differential privacy  Objective: Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  Method: Release the number + Laplace noise 𝑞𝑒𝑔 𝒚 = exp − 𝒚 2𝝁 𝝁  Rationale:  𝑬 : the original dataset; # of diabetes patients = 𝒊  𝑬′ : modify a patient in 𝑬 ; # of diabetes patients = 𝒊′ ratio bounded Pr 𝑩 𝑬 = 𝑷 Pr 𝑩 𝑬′ = 𝑷 𝒊 𝒛 𝒊′ # of diabetes patients

Differential Privacy via Laplace Noise  Dataset: A set of patients Release # of diabetes patients with 𝜻 -differential privacy  Objective: Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  Method: Release the number + Laplace noise 𝑞𝑒𝑔 𝒚 = exp − 𝒚 2𝝁 𝝁  Rationale:  𝑬 : the original dataset; # of diabetes patients = 𝒊  𝑬′ : modify a patient in 𝑬 ; # of diabetes patients = 𝒊′ Pr 𝑩 𝑬 = 𝒛 = 𝑞𝑒𝑔(𝒛 − 𝒊) = exp(−|𝒛 − 𝒊|/𝝁)/2𝝁 Pr 𝑩 𝑬 = 𝑷 𝒊 𝒛 # of diabetes patients

Differential Privacy via Laplace Noise  Dataset: A set of patients Release # of diabetes patients with 𝜻 -differential privacy  Objective: Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  Method: Release the number + Laplace noise 𝑞𝑒𝑔 𝒚 = exp − 𝒚 2𝝁 𝝁  Rationale:  𝑬 : the original dataset; # of diabetes patients = 𝒊  𝑬′ : modify the height of an individual in 𝑬 ; # of diabetes patients = 𝒊′ Pr 𝑩 𝑬′ = 𝒛 = 𝑞𝑒𝑔(𝒛 − 𝒊′) = exp(−|𝒛 − 𝒊′|/𝝁)/2𝝁 Pr 𝑩 𝑬′ = 𝑷 𝒛 𝒊′ # of diabetes patients

Differential Privacy via Laplace Noise  Dataset: A set of patients Release # of diabetes patients with 𝜻 -differential privacy  Objective: Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  Method: Release the number + Laplace noise 𝑞𝑒𝑔 𝒚 = exp − 𝒚 2𝝁 𝝁  Rationale:  𝑬 : the original dataset; # of diabetes patients = 𝒊  𝑬′ : modify the height of an individual in 𝑬 ; # of diabetes patients = 𝒊′ Pr 𝑩 𝑬′ = 𝒛 = 𝑞𝑒𝑔(𝒛 − 𝒊′) = exp(−|𝒛 − 𝒊′|/𝝁)/2𝝁 Pr 𝑩 𝑬 = 𝒛 = 𝑞𝑒𝑔(𝒛 − 𝒊) = exp(−|𝒛 − 𝒊|/𝝁)/2𝝁 Pr 𝑩 𝑬 = 𝑷 Pr 𝑩 𝑬′ = 𝑷 𝒊 𝒛 𝒊′ # of diabetes patients

Differential Privacy via Laplace Noise  Dataset: A set of patients Release # of diabetes patients with 𝜻 -differential privacy  Objective: Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  Method: Release the number + Laplace noise 𝑞𝑒𝑔 𝒚 = exp − 𝒚 2𝝁 𝝁  Rationale:  𝑬 : the original dataset; # of diabetes patients = 𝒊  𝑬′ : modify the height of an individual in 𝑬 ; # of diabetes patients = 𝒊′ Pr 𝑩 𝑬′ = 𝒛 Pr 𝑩 𝑬 = 𝒛 Pr 𝑩 𝑬 = 𝑷 Pr 𝑩 𝑬′ = 𝑷 𝒊 𝒛 𝒊′ # of diabetes patients

Differential Privacy via Laplace Noise  Dataset: A set of patients Release # of diabetes patients with 𝜻 -differential privacy  Objective: Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  Method: Release the number + Laplace noise 𝑞𝑒𝑔 𝒚 = exp − 𝒚 2𝝁 𝝁  Rationale:  𝑬 : the original dataset; # of diabetes patients = 𝒊  𝑬′ : modify the height of an individual in 𝑬 ; # of diabetes patients = 𝒊′ Pr 𝑩 𝑬 = 𝒛 = 𝑞𝑒𝑔(𝒛 − 𝒊 ′ ) Pr 𝑩 𝑬′ = 𝒛 𝑞𝑒𝑔(𝒛 − 𝒊) Pr 𝑩 𝑬 = 𝑷 Pr 𝑩 𝑬′ = 𝑷 𝒊 𝒛 𝒊′ # of diabetes patients

Differential Privacy via Laplace Noise  Dataset: A set of patients Release # of diabetes patients with 𝜻 -differential privacy  Objective: Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  Method: Release the number + Laplace noise 𝑞𝑒𝑔 𝒚 = exp − 𝒚 2𝝁 𝝁  Rationale:  𝑬 : the original dataset; # of diabetes patients = 𝒊  𝑬′ : modify the height of an individual in 𝑬 ; # of diabetes patients = 𝒊′ Pr 𝑩 𝑬 = 𝒛 = 𝑞𝑒𝑔(𝒛 − 𝒊 ′ ) 𝑞𝑒𝑔(𝒛 − 𝒊) = exp(−|𝒛 − 𝒊′|/𝝁)/2𝝁 Pr 𝑩 𝑬′ = 𝒛 exp(−|𝒛 − 𝒊|/𝝁)/2𝝁 Pr 𝑩 𝑬 = 𝑷 Pr 𝑩 𝑬′ = 𝑷 𝒊 𝒛 𝒊′ # of diabetes patients

Differential Privacy via Laplace Noise  Dataset: A set of patients Release # of diabetes patients with 𝜻 -differential privacy  Objective: Pr 𝑩 𝑬 = 𝑷 ≤ exp (𝜻) ∙ Pr 𝑩 𝑬′ = 𝑷  Method: Release the number + Laplace noise 𝑞𝑒𝑔 𝒚 = exp − 𝒚 2𝝁 𝝁  Rationale:  𝑬 : the original dataset; # of diabetes patients = 𝒊  𝑬′ : modify the height of an individual in 𝑬 ; # of diabetes patients = 𝒊′ 𝒊 − 𝒊 ′ Pr 𝑩 𝑬 = 𝒛 = 𝑞𝑒𝑔(𝒛 − 𝒊 ′ ) Pr 𝑩 𝑬′ = 𝒛 𝑞𝑒𝑔(𝒛 − 𝒊) ≤ exp 𝝁 Pr 𝑩 𝑬 = 𝑷 Pr 𝑩 𝑬′ = 𝑷 𝒊 𝒛 𝒊′ # of diabetes patients