small fixes

arithmetics/README.md

@@ -80,7 +80,7 @@ TODO
 
 There are many many implementations out there, but here is one for Circom:
 
-```cs
+```cpp
 template Num2Bits(n) {
   assert(n < 254); // due to BN128
   signal input in;

elliptic-curves/README.md

@@ -392,6 +392,12 @@ TODO
 
 TODO
 
+## Exercise 82
+
+> Consider the small prime factor 2 of the TinyJubJub curve. Compute the full 2-torsion group of $TJJ_{13}$ and then compute the groups $\mathbb{G}_1[2]$ and $\mathbb{G}_2[2]$
+
+TODO
+
 ## Exercise 83
 
 > Consider `alt_bn128` curve and and it's curve extension. Write a Sage program that computes a generator for each of the torsion group $\mathbb{G}_1[p]$ and $\mathbb{G}_2[p]$.

elliptic-curves/full-torsion-groups.sage

@@ -34,7 +34,7 @@ def example_91():
 
   # generate extension
   FPt.<t> = FP[] # type: ignore
-  P_MOD_M = FPt(t^k + 2)
+  P_MOD_M = FPt(t^k + 2) # type: ignore
   assert(P_MOD_M.is_irreducible())
 
   FP_M = GF(p^k, name='t', modulus=P_MOD_M)

statements/qap.sage

@@ -23,7 +23,7 @@ def qap(r1cs, p: int):
 
     # pick k random elements, unique & invertible
     elems = []
-    for _ in range(k):
+    while len(elems) < k:
         rand_elem = Fp.random_element()
         if rand_elem not in elems:
             elems.append(rand_elem)