A method for establishing cryptographic communications between an attention application terminal and a guardian terminal with a black box accumulator in an attention rewards architecture comprising: receiving an advertising (ad) catalog with N ads at the attention application terminal; receiving an ad policy vector of length N, wherein each index in the ad policy vector corresponds to an ad in the catalog and represents a reward value for interactions with the corresponding ad at the attention application terminal; requesting, from the guardian terminal, a signed BBA tuple τ = C k P k = h 1 id ⋅ h 2 r k P k, where k ← $ Z p is a source of randomness (R = k) supplied by the attention application terminal with all counters initialized at zero, the initialization request including a request for a structure-preserving signature over equivalence classes (SPS-EQ), σ, over the BBA tuple τ; receiving, from the guardian terminal, the signature σ; storing, at the attention application terminal, the signature σ, the BBA tuple τ, and the source of randomness, R, used in the initialization request; matching ads from the ad catalog with a user of the attention application terminal; displaying matched ads to the user at the attention application terminal; detecting user interactions with the matched ads at the attention application terminal; repeatedly and automatically incrementing an ad interaction counter based on the detected user interactions to yield a notice requesting an update to the ad interaction vector; randomizing the BBA tuple τ by computing, at the attention application terminal, τ ′ = τ 1 k ′ τ 2 k ′ where k' is chosen uniformly at random from Z p *; transmitting the randomized BBA tuple τ' to the guardian terminal with the notice requesting the update to the ad interaction vector; receiving, from the guardian terminal, an SPS-EQ over a new BBA tuple (C U, τ 2 ') where C U = τ 1 ′ ⋅ τ 2 ′ sk c, j; upon receipt of the SPS-EQ over the new BBA tuple, storing, at the attention application terminal, the updated randomization R New = R·k'; and verifying the new tuple (C U, τ 2 ') is a correct reflection of the ad interaction state requested in the notice requesting an update to the ad interaction vector. receiving an advertising (ad) catalog with N ads at the attention application terminal; receiving an ad policy vector of length N, wherein each index in the ad policy vector corresponds to an ad in the catalog and represents a reward value for interactions with the corresponding ad at the attention application terminal; requesting, from the guardian terminal, a signed BBA tuple τ = C k P k = h 1 id ⋅ h 2 r k P k, where k ← $ Z p is a source of randomness (R = k) supplied by the attention application terminal with all counters initialized at zero, the initialization request including a request for a structure-preserving signature over equivalence classes (SPS-EQ), σ, over the BBA tuple τ; receiving, from the guardian terminal, the signature σ; storing, at the attention application terminal, the signature σ, the BBA tuple τ, and the source of randomness, R, used in the initialization request; matching ads from the ad catalog with a user of the attention application terminal; displaying matched ads to the user at the attention application terminal; detecting user interactions with the matched ads at the attention application terminal; repeatedly and automatically incrementing an ad interaction counter based on the detected user interactions to yield a notice requesting an update to the ad interaction vector; randomizing the BBA tuple τ by computing, at the attention application terminal, τ ′ = τ 1 k ′ τ 2 k ′ where k' is chosen uniformly at random from Z p *; transmitting the randomized BBA tuple τ' to the guardian terminal with the notice requesting the update to the ad interaction vector; receiving, from the guardian terminal, an SPS-EQ over a new BBA tuple (C U, τ 2 ') where C U = τ 1 ′ ⋅ τ 2 ′ sk c, j; upon receipt of the SPS-EQ over the new BBA tuple, storing, at the attention application terminal, the updated randomization R New = R·k'; and verifying the new tuple (C U, τ 2 ') is a correct reflection of the ad interaction state requested in the notice requesting an update to the ad interaction vector.