Step 3: Master polynomial

We are now going to build a "master polynomial" which will encode the gate constraints and the permutation constraints in one, which will be more efficient and equivalent to checking each constrain separately.

Exercise 21

Generate a challenge $\alpha$ and push it to the transcript. Compute the gates constraing on the blinded polynomials:

$$t\mathrm{\_}gates=qM\cdot a\mathrm{\_}blind\cdot b\mathrm{\_}blind+qL\cdot a\mathrm{\_}blind+qR\cdot b\mathrm{\_}blind-c\mathrm{\_}blind$$

Then compute the $L_1$ function as above and the constraints:

$$t\mathrm{\_}perm\mathrm{\_}start=(z\mathrm{\_}poly-1)\cdot L_1$$$$t\mathrm{\_}perm\mathrm{\_}step=z\mathrm{\_}poly\cdot N\mathrm{\_}poly-D\mathrm{\_}poly\cdot z\mathrm{\_}poly(x\cdot \omega )$$

Compute the master polynomial:

$$bigt=t\mathrm{\_}gates+\alpha \cdot t\mathrm{\_}perm\mathrm{\_}start+{\alpha }^2\cdot t\mathrm{\_}perm\mathrm{\_}step$$

and its quotient:

$$quotient\mathrm{\_}poly=\frac{bigt}{ZH}$$

Commit to $quotient\mathrm{\_}poly$ and push to the transcript.

#Solve here!
alpha = 
L1 = 
t_gates     = 
t_perm_start= 
z_shifted   = 
t_perm_step = 
bigt =  
quotient_poly = 
c_t =