From 7ec31cd2f288b56530e72ed4f78b0098f22f757c Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Iv=C3=A1n=20Litteri?= <ilitteri@fi.uba.ar>
Date: Thu, 17 Aug 2023 12:47:43 -0300
Subject: [PATCH 1/5] Finish fp2 arithmetic

---
 .../quadratic_extension_field_arithmetic.py   | 71 +++++++++++++------
 1 file changed, 51 insertions(+), 20 deletions(-)

diff --git a/scripts/quadratic_extension_field_arithmetic.py b/scripts/quadratic_extension_field_arithmetic.py
index 472b234b..c4da2e85 100644
--- a/scripts/quadratic_extension_field_arithmetic.py
+++ b/scripts/quadratic_extension_field_arithmetic.py
@@ -1,22 +1,37 @@
-import montgomery
+import montgomery as monty
 
 # Base field order
 N = 21888242871839275222246405745257275088696311157297823662689037894645226208583
+BETA = monty.ONE
 
-def add(augend0, augend1, addend0, addend1):
-    return montgomery.add(augend0, addend0), montgomery.add(augend1, addend1)
+# Algorithm 5 from https://eprint.iacr.org/2010/354.pdf
+def add(a0, a1, b0, b1):
+    return monty.add(a0, b0), monty.add(a1, b1)
 
-def sub(minuend0, minuend1, subtrahend0, subtrahend1):
-    return montgomery.sub(minuend0, subtrahend0), montgomery.sub(minuend1, subtrahend1)
+# Algorithm 6 from https://eprint.iacr.org/2010/354.pdf
+def sub(a0, a1, b0, b1):
+    return monty.sub(a0, b0), monty.sub(a1, b1)
 
-# [a, ib] * [c, id] = [ac - bd, (ad + bc)i] -> e = ac - bd, f = ad + bc
-def mul(a, b, c, d):
-    e = montgomery.sub(montgomery.mul(a, c), montgomery.mul(b, d))
-    f = montgomery.add(montgomery.mul(a, d), montgomery.mul(b, c))
+# Algorithm 7 from https://eprint.iacr.org/2010/354.pdf
+def scalar_mul(a0, a1, scalar):
+    return monty.mul(a0, scalar), monty.mul(a1, scalar)
+
+def mul(a0, a1, b0, b1):
+    e = monty.sub(monty.mul(a0, b0), monty.mul(a1, b1))
+    f = monty.add(monty.mul(a0, b1), monty.mul(a1, b0))
     return e, f
 
+# Algorithm 8 from https://eprint.iacr.org/2010/354.pdf
+# β = 1
+def inv(a0, a1):
+    t0 = monty.mul(a0, a0)
+    t1 = monty.mul(a1, a1)
+    t0 = monty.sub(t0, monty.mul(BETA, t1))
+    t1 = monty.inv(t0)
+    return monty.mul(a0, t1), monty.sub(0, monty.mul(a1, t1))
+
 def exp(base0, base1, exponent):
-    pow0 = montgomery.ONE
+    pow0 = monty.ONE
     pow1 = 0
     while exponent > 0:
         if exponent % 2 == 1:
@@ -27,27 +42,43 @@ def exp(base0, base1, exponent):
 
 def main():
     # (1 + 2i) * (2 + 2i) = [ac - bd, (ad + bc)i] = -2 + 6i
-    fp2_a = montgomery.ONE, montgomery.TWO
-    fp2_b = montgomery.TWO, montgomery.TWO
+    fp2_a = monty.ONE, monty.TWO
+    fp2_b = monty.TWO, monty.TWO
     fp2_ab = mul(*fp2_a, *fp2_b)
 
-    assert(montgomery.out_of(fp2_ab[0]) == N - 2)
-    assert(montgomery.out_of(fp2_ab[1]) == 6)
+    assert(monty.out_of(fp2_ab[0]) == N - 2)
+    assert(monty.out_of(fp2_ab[1]) == 6)
 
     # (1 + 2i) ^ 0 = 1
     fp2_one = exp(*fp2_a, 0)
-    assert(montgomery.out_of(fp2_one[0]) == 1)
-    assert(montgomery.out_of(fp2_one[1]) == 0)
+    assert(monty.out_of(fp2_one[0]) == 1)
+    assert(monty.out_of(fp2_one[1]) == 0)
 
     # (1 + 2i) ^ 2 = -3 + 4i
     fp2_a_squared = exp(*fp2_a, 2)
-    assert(montgomery.out_of(fp2_a_squared[0]) == N - 3)
-    assert(montgomery.out_of(fp2_a_squared[1]) == 4)
+    assert(monty.out_of(fp2_a_squared[0]) == N - 3)
+    assert(monty.out_of(fp2_a_squared[1]) == 4)
 
     # (1 + 2i) ^ 3 = (1 + 2i) * (-3 + 4i) = [ac - bd, (ad + bc)i] = -11 - 2i
     fp2_a_cubed = exp(*fp2_a, 3)
-    assert(montgomery.out_of(fp2_a_cubed[0]) == N - 11)
-    assert(montgomery.out_of(fp2_a_cubed[1]) == N - 2)
+    assert(monty.out_of(fp2_a_cubed[0]) == N - 11)
+    assert(monty.out_of(fp2_a_cubed[1]) == N - 2)
+
+    # (1 + 2i) * 0 = 0
+    fp2_zero = scalar_mul(*fp2_a, 0)
+    assert(fp2_zero == (0, 0))
+
+    # (1 + 2i) * 1 = 1 + 2i
+    fp2_one = scalar_mul(*fp2_a, monty.ONE)
+    assert(fp2_one == fp2_a)
+
+    # (1 + 2i) * 2 = 2 + 4i
+    fp2_two = scalar_mul(*fp2_a, monty.TWO)
+    assert(fp2_two == (monty.TWO, monty.FOUR))
 
+    # (1 + 2i) * 3 = 3 + 6i
+    fp2_three = scalar_mul(*fp2_a, monty.THREE)
+    assert(fp2_three == (monty.THREE, monty.SIX))
+ 
 if __name__ == '__main__':
     main()
\ No newline at end of file

From 0b96358b1e0d6b659109d09babdad8e4ae6b5503 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Iv=C3=A1n=20Litteri?= <ilitteri@fi.uba.ar>
Date: Thu, 17 Aug 2023 12:47:57 -0300
Subject: [PATCH 2/5] Add more precomputed values

---
 scripts/montgomery.py | 7 +++++++
 1 file changed, 7 insertions(+)

diff --git a/scripts/montgomery.py b/scripts/montgomery.py
index 4d6487a8..ad4654ed 100644
--- a/scripts/montgomery.py
+++ b/scripts/montgomery.py
@@ -18,6 +18,13 @@
 
 ONE = 6350874878119819312338956282401532409788428879151445726012394534686998597021
 TWO = 12701749756239638624677912564803064819576857758302891452024789069373997194042
+THREE = 19052624634359457937016868847204597229365286637454337178037183604060995791063
+FOUR = 3515256640640002027109419384348854550457404359307959241360540244102768179501
+FIVE = 9866131518759821339448375666750386960245833238459404967372934778789766776522
+SIX = 16217006396879640651787331949151919370034262117610850693385329313476765373543
+SEVEN = 679638403160184741879882486296176691126379839464472756708685953518537761981
+EIGHT = 7030513281280004054218838768697709100914808718615918482721080488205536359002
+NINE = 13381388159399823366557795051099241510703237597767364208733475022892534956023
 
 # Extended euclidean algorithm to find modular inverses for integers.
 def prime_field_inv(a, modulus):

From 2d9295bd2862a19b0ba4acb6a19749a55b39fdd2 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Iv=C3=A1n=20Litteri?= <ilitteri@fi.uba.ar>
Date: Thu, 17 Aug 2023 17:32:31 -0300
Subject: [PATCH 3/5] Fix fp2 inv

---
 scripts/quadratic_extension_field_arithmetic.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/scripts/quadratic_extension_field_arithmetic.py b/scripts/quadratic_extension_field_arithmetic.py
index c4da2e85..d89dec4b 100644
--- a/scripts/quadratic_extension_field_arithmetic.py
+++ b/scripts/quadratic_extension_field_arithmetic.py
@@ -2,7 +2,6 @@
 
 # Base field order
 N = 21888242871839275222246405745257275088696311157297823662689037894645226208583
-BETA = monty.ONE
 
 # Algorithm 5 from https://eprint.iacr.org/2010/354.pdf
 def add(a0, a1, b0, b1):
@@ -22,11 +21,12 @@ def mul(a0, a1, b0, b1):
     return e, f
 
 # Algorithm 8 from https://eprint.iacr.org/2010/354.pdf
-# β = 1
+# β = -1
 def inv(a0, a1):
     t0 = monty.mul(a0, a0)
     t1 = monty.mul(a1, a1)
-    t0 = monty.sub(t0, monty.mul(BETA, t1))
+    # This step is actually to - β * t1 but β = -1 so we can just add t1 to t0.
+    t0 = monty.add(t0, t1)
     t1 = monty.inv(t0)
     return monty.mul(a0, t1), monty.sub(0, monty.mul(a1, t1))
 

From bc45ea8bd1812c47c39d2caf62fbc13234d0130e Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Iv=C3=A1n=20Litteri?= <ilitteri@fi.uba.ar>
Date: Thu, 17 Aug 2023 17:39:55 -0300
Subject: [PATCH 4/5] Add EOF

---
 scripts/quadratic_extension_field_arithmetic.py | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/scripts/quadratic_extension_field_arithmetic.py b/scripts/quadratic_extension_field_arithmetic.py
index d89dec4b..9c212594 100644
--- a/scripts/quadratic_extension_field_arithmetic.py
+++ b/scripts/quadratic_extension_field_arithmetic.py
@@ -81,4 +81,5 @@ def main():
     assert(fp2_three == (monty.THREE, monty.SIX))
  
 if __name__ == '__main__':
-    main()
\ No newline at end of file
+    main()
+    
\ No newline at end of file

From 5b17ebca7e0259e2fb26e2f368814550874d611a Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Iv=C3=A1n=20Litteri?= <ilitteri@fi.uba.ar>
Date: Thu, 17 Aug 2023 17:50:31 -0300
Subject: [PATCH 5/5] Rename module

---
 scripts/{quadratic_extension_field_arithmetic.py => fp2.py} | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 rename scripts/{quadratic_extension_field_arithmetic.py => fp2.py} (100%)

diff --git a/scripts/quadratic_extension_field_arithmetic.py b/scripts/fp2.py
similarity index 100%
rename from scripts/quadratic_extension_field_arithmetic.py
rename to scripts/fp2.py