Block Qualified Tests

The Test object is at the core of the natively supported Test Credential: it defines the test credential and what is needed to obtain it.

Each credential test contains two distinct components, each forming a Merkle tree formed with the SNARK-friendly Poseidon hash function:

• A multiple choice component, where the answer to each question is part of a given finite set. The resulting Merkle root is named multipleChoiceRoot. The grade for this component is only awarded if the user gets all the answers right: if they know a tree with multipleChoiceRoot at its root.

• An open answer component, where the answer to each question can be any value. The leaves of the tree are the keccak256 hashes of the answers, made compatible with the SNARK scalar modulus. The resulting Merkle root of the correct answers tree is named openAnswersHashesRoot. The grade for this component is awarded incrementally per answer that the user gets right: every matched hash with the correct openAnswerHashes, with the preimage being the user's answer.

The TEST_HEIGHT constant sets the maximum number of questions possible for each component. This value can be either 4, 5 or 6, giving us a maximum of 64 questions per component. It is recommended users choose the smallest test height that can encode all of their questions: this will help reduce proving time. The value of the testRoot, which is the result of hashing together the multipleChoiceRoot and the openAnswersHashesRoot is used to define and identify the test.

If the credential issuer does not define all of the questions for a component, the tree will have to be padded to 64 values. In the bq library, this is done by assigning the default values 0 for multiple choice questions, and keccak256("") for open answer questions.

The final grade of a test is calculated as the weighted sum of these two components, using the following formula:

$$\textrm{grade} = \textrm{result} \cdot \texttt{multipleChoiceWeight} + \\[7pt] + \max((\texttt{nCorrect} + \texttt{nQuestions} - \texttt{maxQuestions}) \cdot \frac{100 - \texttt{multipleChoiceWeight}}{\texttt{nQuestions}}, 0)$$

Where:

• result is either 1 or 0, depending on whether the user solved the multiple choice component or not, respectively.

• multipleChoiceWeight is the percentage of the multiple choice component towards the final grade.

• nCorrect is the number of correct open answers the user got, including to non-defined questions whose answer is keccak256("").

• nQuestions is the number of open answer questions that make up the test.

• maxQuestions is the maximum number of open answer questions the implementation supports, which equals $2 ^ {TEST_HEIGHT}$.

Because of the formula above, nQuestions must always be greater than one.

For tests that only contain a multiple choice component, multipleChoiceWeight must be set to 100, while nQuestions must therefore be set to 1; for tests that only contain an open answer component, multipleChoiceWeight must be set to 0.

When the user's grade is over the defined minimumGrade, they have gained the test credential, and their identity commitment gets added to the credentials group. Otherwise, their identity commitment gets added to the no-credentials group. These parameters that define the criteria to pass a test get encoded into the variable testParameters:

$$\texttt{testParameters} = \textrm{Poseidon}(\texttt{minimumGrade}, \texttt{multipleChoiceWeight}, \texttt{nQuestions})$$