Bob does not like Key Generator

April 1, 2021
Powerful Alice

Let

be a vector containing confidential information that Alice possesses. Let's assume Bob has another friend who has a confidential vector
. Yeah, it's about the number of shoes again.

Having encryption of Alice's vector

and the functional encryption key
for a publicly known vector
, Bob can compute
.

And having an encryption of an imaginary friend's vector

and the functional encryption key
for a publicly known vector
, Bob can compute
.

sequenceDiagram; participant A as Alice; participant I as Imaginary friend; participant B as Bob; participant K as Key Generator; K->>B: functional encryption key k_y fooooooooooooooooooo; A->>A: encrypts xxxxxxxxxx; A->>B: send ciphertext; B->>B: computes inner-product by using functional encryption key; I->>I: encrypts xxxxxxx; I->>B: send ciphertext; B->>B: computes inner-product by using functional encryption key;

At this point, Bob has read enough cryptographic papers to know it is generally good to avoid a trusted third party since it represents a single point of failure. Note that the Key Generator can fully decrypt all messages.

It might not solve the single point of failure issue, but Bob suddenly wants to delegate the responsibility for key generation to Alice. He trusts Alice. And this would make the whole system simpler, since fewer subjects would be involved.

sequenceDiagram; participant A as Alice; participant I as Imaginary friend; participant B as Bob; A->>B: functional encryption key k_y fooooooooooooooooooo; A->>A: encrypts xxxxxxxxxx; A->>B: send ciphertext; B->>B: computes inner-product by using functional encryption key; I->>I: encrypts xxxxxxx; I->>B: send ciphertext; B->>B: computes inner-product by using functional encryption key;

However, in this case, Alice could decrypt any message she would intercept. For example, if Alice intercepts the encryption of an imaginary friend's vector

, she could decrypt not only
but also
.

Bob's friends might not want to tolerate such a powerful Alice. Bob knows this and decides to stick to a Key Generator that is not controlled by Alice.

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