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+# We'll use y^2 = x^3 + 3 for our curve, over F_101
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+p = 101
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+F = FiniteField(p)
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+R.<x> = F[]
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+E = EllipticCurve(F, [0, 3])
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+print(E)
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+
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+e_points = E.points()
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+# Our generator G_1 = E(1, 2)
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+G_1 = e_points[1]
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+
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+# Let's find a subgroup with this generator.
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+print(f"Finding a subgroup with generator {G_1} ...")
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+r = 1
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+elem = G_1
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+while True:
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+ elem += G_1
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+ r += 1
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+ if elem == e_points[0]:
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+ break
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+
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+print(f"Found subgroup of order r={r} using generator {G_1}")
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+
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+# Now let's find the embedding degree.
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+# The embedding degree is the smallest k such that r|p^k - 1
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+# In other words: p^k == 1 mod r
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+k = 1
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+print(f"Finding embedding degree for {p}^k mod {r} ...")
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+while True:
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+ if p ^ k % r == 1:
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+ break
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+ k += 1
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+
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+print(f"Found embedding degree: k={k}")
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+
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+# Our extension field. The polynomial x^2+2 is irreducible in F_101.
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+F2.<u> = F.extension(x^2+2, 'u')
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+assert u^2 == -2
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+print(F2)
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+E2 = EllipticCurve(F2, [0, 3])
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+print(E2)
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+# One of the generators for this curve we can use is (36, 31u)
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+G_2 = E2(36, 31*u)
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+
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+# Now we build the trusted setup. The SRS is a list of EC points
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+# parameterized by a randomly generated secret number s.
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+# According to the PLONK protocol paper, a circuit with n gates requires
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+# an SRS with at least n+5 elements.
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+
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+# We choose 2 as our random number for demo purposes.
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+s = 2
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+# Our circuit will have 4 gates.
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+n = 4
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+
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+SRS = []
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+for i in range(0, n+3):
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+ SRS.append(s^i * G_1)
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+for i in range(0, 2):
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+ SRS.append(s^i * G_2)
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+
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+# Composing our circuit. We'll test a^2 + b^2 = c^2:
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+# x_1 * x_1 = x_2
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+# x_3 * x_3 = x_4
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+# x_5 * x_5 = x_6
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+# x_2 + x_4 = x_6
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+#
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+# In order to satisfy these constraints, we need to supply six numbers
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+# as wire values that make all of the equations correct.
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+# e.g. x=(3, 9, 4, 16, 5, 25) would work.
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+
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+# A full PLONK gate looks like this:
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+# (q_L)*a + (q_R)*b + (q_O)*c + (q_M)*a*b + q_C = 0
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+#
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+# Where a, b, c are the left, right, output wires of the gate.
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+#
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+# a + b = c ---> q_L=1 , q_R=1, q_O=-1, q_M=0, q_C = 0
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+# a * b = c ---> q_O=-1, q_M=1, and the rest = 0
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+#
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+# To bind a variable to a public value:
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+# q_R = q_O = q_M = 0
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+# q_L = 1
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+# q_C = public_value
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+#
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+# Considering all inputs as private, we get these four PLONK gates
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+# representing our circuit:
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+# 0*a_1 + 0*b_1 + (-1)*c_1 + 1*a_1*b_1 + 0 = 0 (a_1 * b_1 = c_1)
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+# 0*a_2 + 0*b_2 + (-1)*c_2 + 1*a_2*b_2 + 0 = 0 (a_2 * b_2 = c_2)
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+# 0*a_3 + 0*b_3 + (-1)*c_3 + 1*a_3*b_3 + 0 = 0 (a_3 * b_3 = c_3)
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+# 1*a_4 + 1*b_4 + (-1)*c_4 + 0*a_4*b_4 + 0 = 0 (a_4 + b_4 = c_4)
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+#
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+# So let's test with (3, 4, 5)
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+# a_i (left) values will be (3, 4, 5, 9)
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+# b_i (right) values will be (3, 4, 5, 16)
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+# c_i (output) values will be (9, 16, 25, 25)
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+
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+# Selectors
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+q_L = vector([0, 0, 0, 1])
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+q_R = vector([0, 0, 0, 1])
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+q_O = vector([-1, -1, -1, -1])
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+q_M = vector([1, 1, 1, 0])
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+q_C = vector([0, 0, 0, 0])
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+# Assignments
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+a = vector([3, 4, 5, 9])
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+b = vector([3, 4, 5, 16])
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+c = vector([9, 16, 25, 25])
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parazyd