From 4c039b7bbe74fbce48c465d8988f73d2d0a6cafb Mon Sep 17 00:00:00 2001
From: "Wonyong Kim(Ryan Kim)" <ryan@lightscale.io>
Date: Tue, 26 Sep 2023 14:58:22 +0900
Subject: [PATCH 1/2] refac(math): remove `Evaluate()` and `Degree()` from
 `Polynomial<F>`

A function definition in `Polynomial<F>` doesn't help in any respects.
It's just a code duplication. In addition, a function signature of `Evaluate()`
is not matched to one in `MultivariatePolynomial<F>`.
---
 .../multivariate/multivariate_polynomial.h    | 24 +++++--------------
 tachyon/math/polynomials/polynomial.h         | 22 +----------------
 .../univariate/univariate_polynomial.h        | 20 +++++-----------
 3 files changed, 13 insertions(+), 53 deletions(-)

diff --git a/tachyon/math/polynomials/multivariate/multivariate_polynomial.h b/tachyon/math/polynomials/multivariate/multivariate_polynomial.h
index f87a94efd..b3563f65e 100644
--- a/tachyon/math/polynomials/multivariate/multivariate_polynomial.h
+++ b/tachyon/math/polynomials/multivariate/multivariate_polynomial.h
@@ -25,10 +25,6 @@ class MultivariatePolynomial
   constexpr MultivariatePolynomial(Coefficients&& coefficients)
       : coefficients_(std::move(coefficients)) {}
 
-  constexpr Field Evaluate(const std::vector<Field>& points) const {
-    return DoEvaluate(points);
-  }
-
   constexpr static MultivariatePolynomial Zero() {
     return MultivariatePolynomial(Coefficients::Zero());
   }
@@ -67,6 +63,12 @@ class MultivariatePolynomial
     return coefficients_.GetLeadingCoefficient();
   }
 
+  constexpr size_t Degree() const { return coefficients_.Degree(); }
+
+  constexpr Field Evaluate(const std::vector<Field>& points) const {
+    return coefficients_.Evaluate(points);
+  }
+
   std::string ToString() const { return coefficients_.ToString(); }
 
 #define OPERATION_METHOD(Name)                                              \
@@ -102,16 +104,8 @@ class MultivariatePolynomial
 #undef OPERATION_METHOD
 
  private:
-  friend class Polynomial<MultivariatePolynomial<Coefficients>>;
   friend class internal::MultivariatePolynomialOp<Coefficients>;
 
-  // Polynomial methods
-  constexpr size_t DoDegree() const { return coefficients_.Degree(); }
-
-  constexpr Field DoEvaluate(const std::vector<Field>& points) const {
-    return coefficients_.Evaluate(points);
-  }
-
   Coefficients coefficients_;
 };
 
@@ -121,12 +115,6 @@ std::ostream& operator<<(std::ostream& os,
   return os << p.ToString();
 }
 
-template <typename Coefficients>
-class CoefficientsTraits<MultivariatePolynomial<Coefficients>> {
- public:
-  using Field = typename Coefficients::Field;
-};
-
 template <typename F, size_t MaxDegree>
 using SparseMultivariatePolynomial =
     MultivariatePolynomial<SparseCoefficients<F, MaxDegree>>;
diff --git a/tachyon/math/polynomials/polynomial.h b/tachyon/math/polynomials/polynomial.h
index c61bfc4c1..92c711af4 100644
--- a/tachyon/math/polynomials/polynomial.h
+++ b/tachyon/math/polynomials/polynomial.h
@@ -1,34 +1,14 @@
 #ifndef TACHYON_MATH_POLYNOMIALS_POLYNOMIAL_H_
 #define TACHYON_MATH_POLYNOMIALS_POLYNOMIAL_H_
 
-#include <stddef.h>
-
 #include "tachyon/base/no_destructor.h"
 #include "tachyon/math/base/identities.h"
 #include "tachyon/math/base/rings.h"
 
 namespace tachyon::math {
 
-template <typename T, typename SFINAE = void>
-class CoefficientsTraits;
-
 template <typename Derived>
-class Polynomial : public Ring<Derived> {
- public:
-  constexpr static const size_t kMaxDegree = Derived::kMaxDegree;
-
-  using Field = typename CoefficientsTraits<Derived>::Field;
-
-  constexpr size_t Degree() const {
-    const Derived* derived = static_cast<const Derived*>(this);
-    return derived->DoDegree();
-  }
-
-  constexpr Field Evaluate(const Field& point) const {
-    const Derived* derived = static_cast<const Derived*>(this);
-    return derived->DoEvaluate(point);
-  }
-};
+class Polynomial : public Ring<Derived> {};
 
 template <typename Derived>
 class MultiplicativeIdentity<Polynomial<Derived>> {
diff --git a/tachyon/math/polynomials/univariate/univariate_polynomial.h b/tachyon/math/polynomials/univariate/univariate_polynomial.h
index ef3991291..6dabf3ad4 100644
--- a/tachyon/math/polynomials/univariate/univariate_polynomial.h
+++ b/tachyon/math/polynomials/univariate/univariate_polynomial.h
@@ -62,6 +62,12 @@ class UnivariatePolynomial
     return coefficients_.GetLeadingCoefficient();
   }
 
+  constexpr size_t Degree() const { return coefficients_.Degree(); }
+
+  constexpr Field Evaluate(const Field& point) const {
+    return coefficients_.Evaluate(point);
+  }
+
   auto ToSparse() const {
     return internal::UnivariatePolynomialOp<Coefficients>::ToSparsePolynomial(
         *this);
@@ -149,16 +155,8 @@ class UnivariatePolynomial
   }
 
  private:
-  friend class Polynomial<UnivariatePolynomial<Coefficients>>;
   friend class internal::UnivariatePolynomialOp<Coefficients>;
 
-  // Polynomial methods
-  constexpr size_t DoDegree() const { return coefficients_.Degree(); }
-
-  constexpr Field DoEvaluate(const Field& point) const {
-    return coefficients_.Evaluate(point);
-  }
-
   Coefficients coefficients_;
 };
 
@@ -168,12 +166,6 @@ std::ostream& operator<<(std::ostream& os,
   return os << p.ToString();
 }
 
-template <typename Coefficients>
-class CoefficientsTraits<UnivariatePolynomial<Coefficients>> {
- public:
-  using Field = typename Coefficients::Field;
-};
-
 template <typename F, size_t MaxDegree>
 using DenseUnivariatePolynomial =
     UnivariatePolynomial<DenseCoefficients<F, MaxDegree>>;

From 829209eb25c01c17a6db32fd8ef8355a4f8748ea Mon Sep 17 00:00:00 2001
From: "Wonyong Kim(Ryan Kim)" <ryan@lightscale.io>
Date: Tue, 26 Sep 2023 15:04:33 +0900
Subject: [PATCH 2/2] refac(math): remove unused
 `Additive|MultiplicativeIdentity`

They are added to support some features such as `Matrix::Identity()`
in the past. But thanks to #23, we don't need this anymore.
Without this, you can use `Eigen::Matrix<BaseField, 2, 2>::Identity()`
for identity matrix.

Related: #23
---
 tachyon/math/base/BUILD.bazel                 |  5 ---
 tachyon/math/base/gmp/BUILD.bazel             | 12 -------
 tachyon/math/base/gmp/gmp_identities.h        | 35 -------------------
 .../math/base/gmp/gmp_identities_unittest.cc  | 19 ----------
 tachyon/math/base/identities.h                | 34 ------------------
 tachyon/math/elliptic_curves/BUILD.bazel      |  5 +--
 tachyon/math/elliptic_curves/affine_point.h   | 26 --------------
 tachyon/math/elliptic_curves/jacobian_point.h | 27 --------------
 tachyon/math/elliptic_curves/msm/BUILD.bazel  |  1 -
 tachyon/math/elliptic_curves/msm/glv.h        |  1 -
 tachyon/math/elliptic_curves/point_xyzz.h     | 27 --------------
 .../math/elliptic_curves/projective_point.h   | 27 --------------
 tachyon/math/finite_fields/BUILD.bazel        |  2 --
 tachyon/math/finite_fields/prime_field.h      | 27 --------------
 tachyon/math/finite_fields/prime_field_gmp.h  | 26 --------------
 tachyon/math/finite_fields/prime_field_gpu.h  | 27 --------------
 .../finite_fields/prime_field_gpu_debug.h     | 27 --------------
 tachyon/math/polynomials/BUILD.bazel          |  6 ++--
 .../math/polynomials/multivariate/BUILD.bazel |  1 -
 tachyon/math/polynomials/polynomial.h         | 28 ---------------
 .../math/polynomials/univariate/BUILD.bazel   |  1 -
 21 files changed, 3 insertions(+), 361 deletions(-)
 delete mode 100644 tachyon/math/base/gmp/gmp_identities.h
 delete mode 100644 tachyon/math/base/gmp/gmp_identities_unittest.cc
 delete mode 100644 tachyon/math/base/identities.h

diff --git a/tachyon/math/base/BUILD.bazel b/tachyon/math/base/BUILD.bazel
index 0e3725e58..33e170881 100644
--- a/tachyon/math/base/BUILD.bazel
+++ b/tachyon/math/base/BUILD.bazel
@@ -64,11 +64,6 @@ tachyon_cc_library(
     deps = [":semigroups"],
 )
 
-tachyon_cc_library(
-    name = "identities",
-    hdrs = ["identities.h"],
-)
-
 tachyon_cc_library(
     name = "rings",
     hdrs = ["rings.h"],
diff --git a/tachyon/math/base/gmp/BUILD.bazel b/tachyon/math/base/gmp/BUILD.bazel
index 8f3a631d2..260fd3c74 100644
--- a/tachyon/math/base/gmp/BUILD.bazel
+++ b/tachyon/math/base/gmp/BUILD.bazel
@@ -20,16 +20,6 @@ tachyon_cc_library(
     ],
 )
 
-tachyon_cc_library(
-    name = "gmp_identities",
-    hdrs = ["gmp_identities.h"],
-    deps = [
-        "//tachyon/base:no_destructor",
-        "//tachyon/math/base:identities",
-        "@local_config_gmp//:gmp",
-    ],
-)
-
 tachyon_cc_library(
     name = "gmp_util",
     srcs = ["gmp_util.cc"],
@@ -56,11 +46,9 @@ tachyon_cc_test(
     name = "gmp_unittests",
     size = "small",
     srcs = [
-        "gmp_identities_unittest.cc",
         "gmp_util_unittest.cc",
     ],
     deps = [
-        ":gmp_identities",
         ":gmp_util",
         "//tachyon/build:build_config",
     ],
diff --git a/tachyon/math/base/gmp/gmp_identities.h b/tachyon/math/base/gmp/gmp_identities.h
deleted file mode 100644
index 22ca07ad0..000000000
--- a/tachyon/math/base/gmp/gmp_identities.h
+++ /dev/null
@@ -1,35 +0,0 @@
-#ifndef TACHYON_MATH_BASE_GMP_GMP_IDENTITIES_H_
-#define TACHYON_MATH_BASE_GMP_GMP_IDENTITIES_H_
-
-#include "third_party/gmp/include/gmpxx.h"
-
-#include "tachyon/base/no_destructor.h"
-#include "tachyon/math/base/identities.h"
-
-namespace tachyon::math {
-
-template <>
-class MultiplicativeIdentity<mpz_class> {
- public:
-  static const mpz_class& One() {
-    static base::NoDestructor<mpz_class> one(1);
-    return *one;
-  }
-
-  static bool IsOne(const mpz_class& value) { return value == One(); }
-};
-
-template <>
-class AdditiveIdentity<mpz_class> {
- public:
-  static const mpz_class& Zero() {
-    static base::NoDestructor<mpz_class> zero(0);
-    return *zero;
-  }
-
-  static bool IsZero(const mpz_class& value) { return value == Zero(); }
-};
-
-}  // namespace tachyon::math
-
-#endif  // TACHYON_MATH_BASE_GMP_GMP_IDENTITIES_H_
diff --git a/tachyon/math/base/gmp/gmp_identities_unittest.cc b/tachyon/math/base/gmp/gmp_identities_unittest.cc
deleted file mode 100644
index 2695921cd..000000000
--- a/tachyon/math/base/gmp/gmp_identities_unittest.cc
+++ /dev/null
@@ -1,19 +0,0 @@
-#include "tachyon/math/base/gmp/gmp_identities.h"
-
-#include "gtest/gtest.h"
-
-namespace tachyon::math {
-
-TEST(GmpIdentities, MultiplicativeIdentity) {
-  EXPECT_EQ(One<mpz_class>(), mpz_class(1));
-  EXPECT_TRUE(IsOne<mpz_class>(mpz_class(1)));
-  EXPECT_FALSE(IsOne<mpz_class>(mpz_class(0)));
-}
-
-TEST(GmpIdentities, AdditiveIdentity) {
-  EXPECT_EQ(Zero<mpz_class>(), mpz_class(0));
-  EXPECT_TRUE(IsZero<mpz_class>(mpz_class(0)));
-  EXPECT_FALSE(IsZero<mpz_class>(mpz_class(1)));
-}
-
-}  // namespace tachyon::math
diff --git a/tachyon/math/base/identities.h b/tachyon/math/base/identities.h
deleted file mode 100644
index b5fc345fb..000000000
--- a/tachyon/math/base/identities.h
+++ /dev/null
@@ -1,34 +0,0 @@
-#ifndef TACHYON_MATH_BASE_IDENTITIES_H_
-#define TACHYON_MATH_BASE_IDENTITIES_H_
-
-namespace tachyon::math {
-
-template <typename T, typename SFINAE = void>
-class MultiplicativeIdentity;
-
-template <typename T>
-const T& One() {
-  return MultiplicativeIdentity<T>::One();
-}
-
-template <typename T>
-constexpr bool IsOne(const T& value) {
-  return MultiplicativeIdentity<T>::IsOne(value);
-}
-
-template <typename T, typename SFINAE = void>
-class AdditiveIdentity;
-
-template <typename T>
-const T& Zero() {
-  return AdditiveIdentity<T>::Zero();
-}
-
-template <typename T>
-constexpr bool IsZero(const T& value) {
-  return AdditiveIdentity<T>::IsZero(value);
-}
-
-}  // namespace tachyon::math
-
-#endif  // TACHYON_MATH_BASE_IDENTITIES_H_
diff --git a/tachyon/math/elliptic_curves/BUILD.bazel b/tachyon/math/elliptic_curves/BUILD.bazel
index 2400cc8d7..0eedb6096 100644
--- a/tachyon/math/elliptic_curves/BUILD.bazel
+++ b/tachyon/math/elliptic_curves/BUILD.bazel
@@ -14,8 +14,5 @@ tachyon_cc_library(
         "projective_point.h",
         "semigroups.h",
     ],
-    deps = [
-        "//tachyon/base:no_destructor",
-        "//tachyon/math/base:identities",
-    ],
+    deps = ["//tachyon/base:no_destructor"],
 )
diff --git a/tachyon/math/elliptic_curves/affine_point.h b/tachyon/math/elliptic_curves/affine_point.h
index 59e703506..ec7080468 100644
--- a/tachyon/math/elliptic_curves/affine_point.h
+++ b/tachyon/math/elliptic_curves/affine_point.h
@@ -20,32 +20,6 @@ JacobianPoint<Curve> operator*(const ScalarField& v,
   return point * v;
 }
 
-template <typename Curve>
-class MultiplicativeIdentity<AffinePoint<Curve>> {
- public:
-  using P = AffinePoint<Curve>;
-
-  static const P& One() {
-    static base::NoDestructor<P> one(P::One());
-    return *one;
-  }
-
-  constexpr static bool IsOne(const P& value) { return value.IsOne(); }
-};
-
-template <typename Curve>
-class AdditiveIdentity<AffinePoint<Curve>> {
- public:
-  using P = AffinePoint<Curve>;
-
-  static const P& Zero() {
-    static base::NoDestructor<P> zero(P::Zero());
-    return *zero;
-  }
-
-  constexpr static bool IsZero(const P& value) { return value.IsZero(); }
-};
-
 template <typename Curve>
 struct PointConversions<AffinePoint<Curve>, AffinePoint<Curve>> {
   constexpr static const AffinePoint<Curve>& Convert(
diff --git a/tachyon/math/elliptic_curves/jacobian_point.h b/tachyon/math/elliptic_curves/jacobian_point.h
index 6477f3489..2542df0c9 100644
--- a/tachyon/math/elliptic_curves/jacobian_point.h
+++ b/tachyon/math/elliptic_curves/jacobian_point.h
@@ -4,7 +4,6 @@
 #include <ostream>
 
 #include "tachyon/base/no_destructor.h"
-#include "tachyon/math/base/identities.h"
 #include "tachyon/math/elliptic_curves/point_conversions_forward.h"
 
 namespace tachyon::math {
@@ -23,32 +22,6 @@ JacobianPoint<Curve> operator*(const ScalarField& v,
   return point * v;
 }
 
-template <typename Curve>
-class MultiplicativeIdentity<JacobianPoint<Curve>> {
- public:
-  using P = JacobianPoint<Curve>;
-
-  static const P& One() {
-    static base::NoDestructor<P> one(P::One());
-    return *one;
-  }
-
-  constexpr static bool IsOne(const P& value) { return value.IsOne(); }
-};
-
-template <typename Curve>
-class AdditiveIdentity<JacobianPoint<Curve>> {
- public:
-  using P = JacobianPoint<Curve>;
-
-  static const P& Zero() {
-    static base::NoDestructor<P> zero(P::Zero());
-    return *zero;
-  }
-
-  constexpr static bool IsZero(const P& value) { return value.IsZero(); }
-};
-
 template <typename Curve>
 struct PointConversions<JacobianPoint<Curve>, JacobianPoint<Curve>> {
   constexpr static const JacobianPoint<Curve>& Convert(
diff --git a/tachyon/math/elliptic_curves/msm/BUILD.bazel b/tachyon/math/elliptic_curves/msm/BUILD.bazel
index 1c3d8eaec..532df805a 100644
--- a/tachyon/math/elliptic_curves/msm/BUILD.bazel
+++ b/tachyon/math/elliptic_curves/msm/BUILD.bazel
@@ -15,7 +15,6 @@ tachyon_cc_library(
         "//tachyon/base:static_storage",
         "//tachyon/math/base:bit_iterator",
         "//tachyon/math/base/gmp:bit_traits",
-        "//tachyon/math/base/gmp:gmp_identities",
         "//tachyon/math/base/gmp:signed_value",
         "//tachyon/math/elliptic_curves:points",
         "//tachyon/math/matrix:gmp_num_traits",
diff --git a/tachyon/math/elliptic_curves/msm/glv.h b/tachyon/math/elliptic_curves/msm/glv.h
index 9ffe94e44..cc2802ba3 100644
--- a/tachyon/math/elliptic_curves/msm/glv.h
+++ b/tachyon/math/elliptic_curves/msm/glv.h
@@ -4,7 +4,6 @@
 #include "tachyon/base/static_storage.h"
 #include "tachyon/math/base/bit_iterator.h"
 #include "tachyon/math/base/gmp/bit_traits.h"
-#include "tachyon/math/base/gmp/gmp_identities.h"
 #include "tachyon/math/base/gmp/signed_value.h"
 #include "tachyon/math/elliptic_curves/affine_point.h"
 #include "tachyon/math/elliptic_curves/jacobian_point.h"
diff --git a/tachyon/math/elliptic_curves/point_xyzz.h b/tachyon/math/elliptic_curves/point_xyzz.h
index 2c8cde643..3ad34b384 100644
--- a/tachyon/math/elliptic_curves/point_xyzz.h
+++ b/tachyon/math/elliptic_curves/point_xyzz.h
@@ -4,7 +4,6 @@
 #include <ostream>
 
 #include "tachyon/base/no_destructor.h"
-#include "tachyon/math/base/identities.h"
 #include "tachyon/math/elliptic_curves/point_conversions_forward.h"
 
 namespace tachyon::math {
@@ -23,32 +22,6 @@ PointXYZZ<Curve> operator*(const ScalarField& v,
   return point * v;
 }
 
-template <typename Curve>
-class MultiplicativeIdentity<PointXYZZ<Curve>> {
- public:
-  using P = PointXYZZ<Curve>;
-
-  static const P& One() {
-    static base::NoDestructor<P> one(P::One());
-    return *one;
-  }
-
-  constexpr static bool IsOne(const P& value) { return value.IsOne(); }
-};
-
-template <typename Curve>
-class AdditiveIdentity<PointXYZZ<Curve>> {
- public:
-  using P = PointXYZZ<Curve>;
-
-  static const P& Zero() {
-    static base::NoDestructor<P> zero(P::Zero());
-    return *zero;
-  }
-
-  constexpr static bool IsZero(const P& value) { return value.IsZero(); }
-};
-
 template <typename Curve>
 struct PointConversions<PointXYZZ<Curve>, PointXYZZ<Curve>> {
   constexpr static const PointXYZZ<Curve>& Convert(
diff --git a/tachyon/math/elliptic_curves/projective_point.h b/tachyon/math/elliptic_curves/projective_point.h
index e9696d926..ef7f5d339 100644
--- a/tachyon/math/elliptic_curves/projective_point.h
+++ b/tachyon/math/elliptic_curves/projective_point.h
@@ -4,7 +4,6 @@
 #include <ostream>
 
 #include "tachyon/base/no_destructor.h"
-#include "tachyon/math/base/identities.h"
 #include "tachyon/math/elliptic_curves/point_conversions_forward.h"
 
 namespace tachyon::math {
@@ -24,32 +23,6 @@ ProjectivePoint<Curve> operator*(const ScalarField& v,
   return point * v;
 }
 
-template <typename Curve>
-class MultiplicativeIdentity<ProjectivePoint<Curve>> {
- public:
-  using P = ProjectivePoint<Curve>;
-
-  static const P& One() {
-    static base::NoDestructor<P> one(P::One());
-    return *one;
-  }
-
-  constexpr static bool IsOne(const P& value) { return value.IsOne(); }
-};
-
-template <typename Curve>
-class AdditiveIdentity<ProjectivePoint<Curve>> {
- public:
-  using P = ProjectivePoint<Curve>;
-
-  static const P& Zero() {
-    static base::NoDestructor<P> zero(P::Zero());
-    return *zero;
-  }
-
-  constexpr static bool IsZero(const P& value) { return value.IsZero(); }
-};
-
 template <typename Curve>
 struct PointConversions<ProjectivePoint<Curve>, ProjectivePoint<Curve>> {
   constexpr static const ProjectivePoint<Curve>& Convert(
diff --git a/tachyon/math/finite_fields/BUILD.bazel b/tachyon/math/finite_fields/BUILD.bazel
index ac56d0cb9..b60671788 100644
--- a/tachyon/math/finite_fields/BUILD.bazel
+++ b/tachyon/math/finite_fields/BUILD.bazel
@@ -56,7 +56,6 @@ tachyon_cc_library(
         "//tachyon/base:no_destructor",
         "//tachyon/base/strings:string_util",
         "//tachyon/math/base:field",
-        "//tachyon/math/base:identities",
         "//tachyon/math/base/gmp:gmp_util",
     ],
 )
@@ -72,7 +71,6 @@ tachyon_cc_library(
         "//tachyon/base/containers:adapters",
         "//tachyon/base/strings:string_util",
         "//tachyon/math/base:arithmetics",
-        "//tachyon/math/base:identities",
         "//tachyon/math/base/gmp:gmp_util",
         "@com_google_googletest//:gtest_prod",
     ],
diff --git a/tachyon/math/finite_fields/prime_field.h b/tachyon/math/finite_fields/prime_field.h
index 0a0d5c709..3750f4085 100644
--- a/tachyon/math/finite_fields/prime_field.h
+++ b/tachyon/math/finite_fields/prime_field.h
@@ -12,7 +12,6 @@
 #include "tachyon/math/base/arithmetics.h"
 #include "tachyon/math/base/big_int.h"
 #include "tachyon/math/base/gmp/gmp_util.h"
-#include "tachyon/math/base/identities.h"
 #include "tachyon/math/finite_fields/modulus.h"
 #include "tachyon/math/finite_fields/prime_field_base.h"
 #include "tachyon/math/finite_fields/prime_field_forward.h"
@@ -297,32 +296,6 @@ std::ostream& operator<<(std::ostream& os, const PrimeField<Config>& f) {
   return os << f.ToString();
 }
 
-template <typename Config>
-class MultiplicativeIdentity<PrimeField<Config>> {
- public:
-  using F = PrimeField<Config>;
-
-  static const F& One() {
-    static F one(F::One());
-    return one;
-  }
-
-  constexpr static bool IsOne(const F& value) { return value.IsOne(); }
-};
-
-template <typename Config>
-class AdditiveIdentity<PrimeField<Config>> {
- public:
-  using F = PrimeField<Config>;
-
-  static const F& Zero() {
-    static F zero(F::Zero());
-    return zero;
-  }
-
-  constexpr static bool IsZero(const F& value) { return value.IsZero(); }
-};
-
 }  // namespace tachyon::math
 
 #endif  // TACHYON_MATH_FINITE_FIELDS_PRIME_FIELD_H_
diff --git a/tachyon/math/finite_fields/prime_field_gmp.h b/tachyon/math/finite_fields/prime_field_gmp.h
index acc3c7123..e53f30b8e 100644
--- a/tachyon/math/finite_fields/prime_field_gmp.h
+++ b/tachyon/math/finite_fields/prime_field_gmp.h
@@ -241,32 +241,6 @@ std::ostream& operator<<(std::ostream& os, const PrimeFieldGmp<Config>& f) {
   return os << f.ToString();
 }
 
-template <typename Config>
-class MultiplicativeIdentity<PrimeFieldGmp<Config>> {
- public:
-  using F = PrimeFieldGmp<Config>;
-
-  static const F& One() {
-    static base::NoDestructor<F> one(F::One());
-    return *one;
-  }
-
-  constexpr static bool IsOne(const F& value) { return value.IsOne(); }
-};
-
-template <typename Config>
-class AdditiveIdentity<PrimeFieldGmp<Config>> {
- public:
-  using F = PrimeFieldGmp<Config>;
-
-  static const F& Zero() {
-    static base::NoDestructor<F> zero(F::Zero());
-    return *zero;
-  }
-
-  constexpr static bool IsZero(const F& value) { return value.IsZero(); }
-};
-
 }  // namespace tachyon::math
 
 #endif  // TACHYON_MATH_FINITE_FIELDS_PRIME_FIELD_GMP_H_
diff --git a/tachyon/math/finite_fields/prime_field_gpu.h b/tachyon/math/finite_fields/prime_field_gpu.h
index e9bd71dd3..6082b7d63 100644
--- a/tachyon/math/finite_fields/prime_field_gpu.h
+++ b/tachyon/math/finite_fields/prime_field_gpu.h
@@ -14,7 +14,6 @@
 #include "tachyon/math/base/arithmetics.h"
 #include "tachyon/math/base/big_int.h"
 #include "tachyon/math/base/gmp/gmp_util.h"
-#include "tachyon/math/base/identities.h"
 #include "tachyon/math/finite_fields/kernels/carry_chain.h"
 #include "tachyon/math/finite_fields/modulus.h"
 #include "tachyon/math/finite_fields/prime_field.h"
@@ -416,32 +415,6 @@ std::ostream& operator<<(std::ostream& os, const PrimeFieldGpu<Config>& f) {
   return os << f.ToString();
 }
 
-template <typename Config>
-class MultiplicativeIdentity<PrimeFieldGpu<Config>> {
- public:
-  using F = PrimeFieldGpu<Config>;
-
-  static const F& One() {
-    static F one(F::One());
-    return one;
-  }
-
-  constexpr static bool IsOne(const F& value) { return value.IsOne(); }
-};
-
-template <typename Config>
-class AdditiveIdentity<PrimeFieldGpu<Config>> {
- public:
-  using F = PrimeFieldGpu<Config>;
-
-  static const F& Zero() {
-    static F zero(F::Zero());
-    return zero;
-  }
-
-  constexpr static bool IsZero(const F& value) { return value.IsZero(); }
-};
-
 }  // namespace tachyon::math
 
 #endif  // TACHYON_MATH_FINITE_FIELDS_PRIME_FIELD_CUDA_H_
diff --git a/tachyon/math/finite_fields/prime_field_gpu_debug.h b/tachyon/math/finite_fields/prime_field_gpu_debug.h
index bafc0dd28..bd5925ad2 100644
--- a/tachyon/math/finite_fields/prime_field_gpu_debug.h
+++ b/tachyon/math/finite_fields/prime_field_gpu_debug.h
@@ -11,7 +11,6 @@
 #include "tachyon/math/base/arithmetics.h"
 #include "tachyon/math/base/big_int.h"
 #include "tachyon/math/base/gmp/gmp_util.h"
-#include "tachyon/math/base/identities.h"
 #include "tachyon/math/finite_fields/carry_chain.h"
 #include "tachyon/math/finite_fields/modulus.h"
 #include "tachyon/math/finite_fields/prime_field.h"
@@ -377,32 +376,6 @@ std::ostream& operator<<(std::ostream& os,
   return os << f.ToString();
 }
 
-template <typename Config>
-class MultiplicativeIdentity<PrimeFieldGpuDebug<Config>> {
- public:
-  using F = PrimeFieldGpuDebug<Config>;
-
-  static const F& One() {
-    static F one(F::One());
-    return one;
-  }
-
-  constexpr static bool IsOne(const F& value) { return value.IsOne(); }
-};
-
-template <typename Config>
-class AdditiveIdentity<PrimeFieldGpuDebug<Config>> {
- public:
-  using F = PrimeFieldGpuDebug<Config>;
-
-  static const F& Zero() {
-    static F zero(F::Zero());
-    return zero;
-  }
-
-  constexpr static bool IsZero(const F& value) { return value.IsZero(); }
-};
-
 }  // namespace tachyon::math
 
 #endif  // TACHYON_MATH_FINITE_FIELDS_PRIME_FIELD_GPU_DEBUG_H_
diff --git a/tachyon/math/polynomials/BUILD.bazel b/tachyon/math/polynomials/BUILD.bazel
index 5816074c1..22868c9f6 100644
--- a/tachyon/math/polynomials/BUILD.bazel
+++ b/tachyon/math/polynomials/BUILD.bazel
@@ -1,11 +1,9 @@
-load("//bazel:tachyon_cc.bzl", "tachyon_cc_library", "tachyon_cc_test")
+load("//bazel:tachyon_cc.bzl", "tachyon_cc_library")
 
 package(default_visibility = ["//visibility:public"])
 
 tachyon_cc_library(
     name = "polynomial",
     hdrs = ["polynomial.h"],
-    deps = [
-        "//tachyon/math/base:rings",
-    ],
+    deps = ["//tachyon/math/base:rings"],
 )
diff --git a/tachyon/math/polynomials/multivariate/BUILD.bazel b/tachyon/math/polynomials/multivariate/BUILD.bazel
index 48d36e5ee..1bf04bcd1 100644
--- a/tachyon/math/polynomials/multivariate/BUILD.bazel
+++ b/tachyon/math/polynomials/multivariate/BUILD.bazel
@@ -16,7 +16,6 @@ tachyon_cc_library(
         "//tachyon/base/containers:cxx20_erase",
         "//tachyon/base/ranges:algorithm",
         "//tachyon/base/strings:string_util",
-        "//tachyon/math/base:identities",
         "//tachyon/math/polynomials:polynomial",
         "@com_google_absl//absl/numeric:bits",
         "@com_google_absl//absl/types:span",
diff --git a/tachyon/math/polynomials/polynomial.h b/tachyon/math/polynomials/polynomial.h
index 92c711af4..a443f30aa 100644
--- a/tachyon/math/polynomials/polynomial.h
+++ b/tachyon/math/polynomials/polynomial.h
@@ -1,8 +1,6 @@
 #ifndef TACHYON_MATH_POLYNOMIALS_POLYNOMIAL_H_
 #define TACHYON_MATH_POLYNOMIALS_POLYNOMIAL_H_
 
-#include "tachyon/base/no_destructor.h"
-#include "tachyon/math/base/identities.h"
 #include "tachyon/math/base/rings.h"
 
 namespace tachyon::math {
@@ -10,32 +8,6 @@ namespace tachyon::math {
 template <typename Derived>
 class Polynomial : public Ring<Derived> {};
 
-template <typename Derived>
-class MultiplicativeIdentity<Polynomial<Derived>> {
- public:
-  using P = Polynomial<Derived>;
-
-  static const P& One() {
-    static base::NoDestructor<P> one(P::One());
-    return *one;
-  }
-
-  constexpr static bool IsOne(const P& value) { return value.IsOne(); }
-};
-
-template <typename Derived>
-class AdditiveIdentity<Polynomial<Derived>> {
- public:
-  using P = Polynomial<Derived>;
-
-  static const P& Zero() {
-    static base::NoDestructor<P> zero(P::Zero());
-    return *zero;
-  }
-
-  constexpr static bool IsZero(const P& value) { return value.IsZero(); }
-};
-
 }  // namespace tachyon::math
 
 #endif  // TACHYON_MATH_POLYNOMIALS_POLYNOMIAL_H_
diff --git a/tachyon/math/polynomials/univariate/BUILD.bazel b/tachyon/math/polynomials/univariate/BUILD.bazel
index 194a912a0..57b6af5db 100644
--- a/tachyon/math/polynomials/univariate/BUILD.bazel
+++ b/tachyon/math/polynomials/univariate/BUILD.bazel
@@ -17,7 +17,6 @@ tachyon_cc_library(
         "//tachyon/base/ranges:algorithm",
         "//tachyon/base/strings:string_util",
         "//tachyon/math/base:arithmetics_results",
-        "//tachyon/math/base:identities",
         "//tachyon/math/polynomials:polynomial",
         "@com_google_absl//absl/numeric:bits",
         "@com_google_absl//absl/types:span",