Wiki: Uniform Spartan + R1CS Info #432
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moodlezoup
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Jul 26, 2024
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| Notice that the matrix follows a sparse-diagonal pattern. We can use this to define $A_{small}$. | ||
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imo it's more intuitive to introduce A_small first, as it clearly captures the uniform constraints Q = R and R = S
| ## Uniform Spartan | ||
| Jolt's Spartan instance is highly patterned. We run the same R1CS for each cycle of the RISC-V program. We can leverage this property to increase prover and verifier efficiency. | ||
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| Think of a simple constraint system with the following inputs: $Q, R, S$. Each of these inputs is repeated $nc = num\_cycles$ times. |
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num_cycles formatting is messed up
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| Note that Z is also padded to a power of two, so the full Z vector is $[Q_0, Q_1, Q_2, 0, R_0, R_1, R_2, 0, S_0, S_1, S_2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]$ | ||
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| $v_0$ indexes constant vs non-constant columns. |
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Move this into the bullet point - $v_0, v_1, v_2$ index the variable
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| > [!NOTE] | ||
| > Syntax: For bitvector $x=x_0, x_1, ... x_n$, $x_0$ represents the most-significant bit, and $x_n$ represents the least significant bit. By default sumchecks will bind from the "top", where they bind the MSB first. For some sumchecks, notably sparse ones we'll bind from the "bottom" for algorithmic efficiency reasons. |
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I'm hoping to switch all sumchecks to bind from the bottom so this note may soon become out-of-date
| $v_0$ indexes constant vs non-constant columns. | ||
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| The $A, B, C$ matrices follow a similar pattern where the constraints are repeated $nc$ times and the variables are repeated $nc$ times. For example, given a 2 constraint system over the variables $Q, R, S$. | ||
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maybe add a legend for what the different colors mean. assuming yellow = padding, then the four columns before C should also be yellow, right?
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Fills in missing details on our instantiation of Spartan.
Left the Cross-Eq constraints and Verifier MLE evals empty for now as we work on #347.