Add placeholders for alternative Fp2 implementations

constantine/tower_field_extensions/README.md

@@ -67,6 +67,10 @@ From Ben Edgington, https://hackmd.io/@benjaminion/bls12-381
   Augusto Jun Devegili and Colm Ó hÉigeartaigh and Michael Scott and Ricardo Dahab, 2006\
   https://eprint.iacr.org/2006/471
 
+- Software Implementation of Pairings\
+  D. Hankerson, A. Menezes, and M. Scott, 2009\
+  http://cacr.uwaterloo.ca/~ajmeneze/publications/pairings_software.pdf
+
 - Constructing Tower Extensions for the implementation of Pairing-Based Cryptography\
   Naomi Benger and Michael Scott, 2009\
   https://eprint.iacr.org/2009/556

constantine/tower_field_extensions/fp2_sqrt_minus2.nim

@@ -0,0 +1,72 @@
+# Constantine
+# Copyright (c) 2018-2019    Status Research & Development GmbH
+# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
+# Licensed and distributed under either of
+#   * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
+#   * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
+# at your option. This file may not be copied, modified, or distributed except according to those terms.
+
+# ############################################################
+#
+#        Quadratic Extension field over base field 𝔽p
+#                        𝔽p2 = 𝔽p[√-5]
+#
+# ############################################################
+
+# This implements a quadratic extension field over
+# the base field 𝔽p:
+#   𝔽p2 = 𝔽p[x]
+# with element A of coordinates (a0, a1) represented
+# by a0 + a1 x
+#
+# The irreducible polynomial chosen is
+#   x² - µ with µ = -2
+# i.e. 𝔽p2 = 𝔽p[√-2]
+#
+# Consequently, for this file Fp2 to be valid
+# -2 MUST not be a square in 𝔽p
+#
+# References
+# [1] Software Implementation of Pairings\
+#     D. Hankerson, A. Menezes, and M. Scott, 2009\
+#     http://cacr.uwaterloo.ca/~ajmeneze/publications/pairings_software.pdf
+
+
+import
+  ../arithmetic/finite_fields,
+  ../config/curves,
+  ./abelian_groups
+
+type
+  Fp2*[C: static Curve] = object
+    ## Element of the extension field
+    ## 𝔽p2 = 𝔽p[√-2] of a prime p
+    ##
+    ## with coordinates (c0, c1) such as
+    ## c0 + c1 √-2
+    ##
+    ## This requires -2 to not be a square (mod p)
+    c0*, c1*: Fp[C]
+
+func square*(r: var Fp2, a: Fp2) =
+  ## Return a^2 in 𝔽p2 in ``r``
+  ## ``r`` is initialized/overwritten
+  # (c0, c1)² => (c0 + c1√-2)²
+  #           => c0² + 2 c0 c1√-2 + (c1√-2)²
+  #           => c0² - 2c1² + 2 c0 c1 √-2
+  #           => (c0²-2c1², 2 c0 c1)
+  #
+  # Costs (naive implementation)
+  # - 2 Multiplications 𝔽p
+  # - 1 Squaring 𝔽p
+  # - 1 Doubling 𝔽p
+  # - 1 Substraction 𝔽p
+  # Stack: 6 * ModulusBitSize (4x 𝔽p element + 2 named temporaries + 1 "in-place" mul temporary)
+
+  var c1d, c0s {.noInit.}: typeof(a.c1)
+  c1d.double(a.c1)       # c1d = 2 c1      [1 Dbl]
+  c0s.square(a.c0)       # c0s = c0²       [1 Sqr, 1 Dbl]
+
+  r.c1.prod(c1d, a.c0)   # r.c1 = 2 c1 c0  [1 Mul, 1 Sqr, 1 Dbl]
+  c1d *= a.c1            # c1d = 2 c1²     [2 Mul, 1 Sqr, 1 Dbl] - 1 "in-place" temporary
+  r.c0.diff(c0s, c1d)    # r.c0 = c0²-2c1² [2 Mul, 1 Sqr, 1 Dbl, 1 Sub]

constantine/tower_field_extensions/fp2_sqrt_minus5.nim

@@ -0,0 +1,52 @@
+# Constantine
+# Copyright (c) 2018-2019    Status Research & Development GmbH
+# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
+# Licensed and distributed under either of
+#   * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
+#   * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
+# at your option. This file may not be copied, modified, or distributed except according to those terms.
+
+# ############################################################
+#
+#        Quadratic Extension field over base field 𝔽p
+#                        𝔽p2 = 𝔽p[√-5]
+#
+# ############################################################
+
+# This implements a quadratic extension field over
+# the base field 𝔽p:
+#   𝔽p2 = 𝔽p[x]
+# with element A of coordinates (a0, a1) represented
+# by a0 + a1 x
+#
+# The irreducible polynomial chosen is
+#   x² - µ with µ = -5
+# i.e. 𝔽p2 = 𝔽p[√-5]
+#
+# Consequently, for this file Fp2 to be valid
+# -5 MUST not be a square in 𝔽p
+#
+# References
+# [1] High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves\
+#     Jean-Luc Beuchat and Jorge Enrique González Díaz and Shigeo Mitsunari and Eiji Okamoto and Francisco Rodríguez-Henríquez and Tadanori Teruya, 2010\
+#     https://eprint.iacr.org/2010/354
+
+import
+  ../arithmetic/finite_fields,
+  ../config/curves,
+  ./abelian_groups
+
+type
+  Fp2*[C: static Curve] = object
+    ## Element of the extension field
+    ## 𝔽p2 = 𝔽p[√-5] of a prime p
+    ##
+    ## with coordinates (c0, c1) such as
+    ## c0 + c1 √-5
+    ##
+    ## This requires -5 to not be a square (mod p)
+    c0*, c1*: Fp[C]
+
+# TODO: need fast multiplication by small constant
+#       which probably requires lazy carries
+#       https://github.com/mratsim/constantine/issues/15