diff --git a/spec/std/big/big_rational_spec.cr b/spec/std/big/big_rational_spec.cr
index 0dcc75c10b87..ec8ed3d0621b 100644
--- a/spec/std/big/big_rational_spec.cr
+++ b/spec/std/big/big_rational_spec.cr
@@ -466,12 +466,6 @@ describe BigRational do
     it { br(7, -3).integer?.should be_false }
   end
 
-  it "#hash" do
-    b = br(10, 3)
-    hash = b.hash
-    hash.should eq(b.to_f64.hash)
-  end
-
   it "is a number" do
     br(10, 3).is_a?(Number).should be_true
   end
diff --git a/spec/std/crystal/hasher_spec.cr b/spec/std/crystal/hasher_spec.cr
index 099b49af7755..456f8ce844a1 100644
--- a/spec/std/crystal/hasher_spec.cr
+++ b/spec/std/crystal/hasher_spec.cr
@@ -305,13 +305,14 @@ describe "Crystal::Hasher" do
       Crystal::Hasher.reduce_num(Float64::MAX).should eq(0x1F00_FFFF_FFFF_FFFF_u64)
     end
 
-    pending "reduces BigInt" do
+    it "reduces BigInt" do
       Crystal::Hasher.reduce_num(0.to_big_i).should eq(0_u64)
       Crystal::Hasher.reduce_num(1.to_big_i).should eq(1_u64)
       Crystal::Hasher.reduce_num((-1).to_big_i).should eq(UInt64::MAX)
 
       (1..300).each do |i|
         Crystal::Hasher.reduce_num(2.to_big_i ** i).should eq(1_u64 << (i % 61))
+        Crystal::Hasher.reduce_num(-(2.to_big_i ** i)).should eq(&-(1_u64 << (i % 61)))
       end
     end
 
@@ -324,8 +325,88 @@ describe "Crystal::Hasher" do
 
       (1..300).each do |i|
         Crystal::Hasher.reduce_num(2.to_big_f ** i).should eq(1_u64 << (i % 61))
+        Crystal::Hasher.reduce_num(-(2.to_big_f ** i)).should eq(&-(1_u64 << (i % 61)))
         Crystal::Hasher.reduce_num(0.5.to_big_f ** i).should eq(1_u64 << ((-i) % 61))
+        Crystal::Hasher.reduce_num(-(0.5.to_big_f ** i)).should eq(&-(1_u64 << ((-i) % 61)))
       end
     end
+
+    it "reduces BigDecimal" do
+      Crystal::Hasher.reduce_num(0.to_big_d).should eq(0_u64)
+      Crystal::Hasher.reduce_num(1.to_big_d).should eq(1_u64)
+      Crystal::Hasher.reduce_num((-1).to_big_d).should eq(UInt64::MAX)
+
+      # small inverse powers of 10
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 1)).should eq(0x1CCCCCCCCCCCCCCC_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 2)).should eq(0x0FAE147AE147AE14_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 3)).should eq(0x0E5E353F7CED9168_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 4)).should eq(0x14A305532617C1BD_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 5)).should eq(0x05438088509BF9C6_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 6)).should eq(0x06ED2674080F98FA_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 7)).should eq(0x1A4AEA3ECD9B28E5_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 8)).should eq(0x12A1176CAE291DB0_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 9)).should eq(0x01DCE8BE116A82F8_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 10)).should eq(0x1362E41301BDD9E5_u64)
+
+      # a^(p-1) === 1 (mod p)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFFE_u64)).should eq(1_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFFD_u64)).should eq(10_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFFC_u64)).should eq(100_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFFB_u64)).should eq(1000_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFFA_u64)).should eq(10000_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFF9_u64)).should eq(100000_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFF8_u64)).should eq(1000000_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFF7_u64)).should eq(10000000_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFF6_u64)).should eq(100000000_u64)
+      Crystal::Hasher.reduce_num(BigDecimal.new(1, 0x1FFFFFFFFFFFFFF5_u64)).should eq(1000000000_u64)
+
+      (1..300).each do |i|
+        Crystal::Hasher.reduce_num(2.to_big_d ** i).should eq(1_u64 << (i % 61))
+        Crystal::Hasher.reduce_num(-(2.to_big_d ** i)).should eq(&-(1_u64 << (i % 61)))
+        Crystal::Hasher.reduce_num(0.5.to_big_d ** i).should eq(1_u64 << ((-i) % 61))
+        Crystal::Hasher.reduce_num(-(0.5.to_big_d ** i)).should eq(&-(1_u64 << ((-i) % 61)))
+      end
+    end
+
+    it "reduces BigRational" do
+      Crystal::Hasher.reduce_num(0.to_big_r).should eq(0_u64)
+      Crystal::Hasher.reduce_num(1.to_big_r).should eq(1_u64)
+      Crystal::Hasher.reduce_num((-1).to_big_r).should eq(UInt64::MAX)
+
+      # inverses of small integers
+      Crystal::Hasher.reduce_num(BigRational.new(1, 2)).should eq(0x1000000000000000_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 3)).should eq(0x1555555555555555_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 4)).should eq(0x0800000000000000_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 5)).should eq(0x1999999999999999_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 6)).should eq(0x1AAAAAAAAAAAAAAA_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 7)).should eq(0x1B6DB6DB6DB6DB6D_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 8)).should eq(0x0400000000000000_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 9)).should eq(0x1C71C71C71C71C71_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 10)).should eq(0x1CCCCCCCCCCCCCCC_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 11)).should eq(0x1D1745D1745D1745_u64)
+
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1000000000000000_u64)).should eq(2_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1555555555555555_u64)).should eq(3_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x0800000000000000_u64)).should eq(4_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1999999999999999_u64)).should eq(5_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1AAAAAAAAAAAAAAA_u64)).should eq(6_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1B6DB6DB6DB6DB6D_u64)).should eq(7_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x0400000000000000_u64)).should eq(8_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1C71C71C71C71C71_u64)).should eq(9_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1CCCCCCCCCCCCCCC_u64)).should eq(10_u64)
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1D1745D1745D1745_u64)).should eq(11_u64)
+
+      (1..300).each do |i|
+        Crystal::Hasher.reduce_num(2.to_big_r ** i).should eq(1_u64 << (i % 61))
+        Crystal::Hasher.reduce_num(-(2.to_big_r ** i)).should eq(&-(1_u64 << (i % 61)))
+        Crystal::Hasher.reduce_num(0.5.to_big_r ** i).should eq(1_u64 << ((-i) % 61))
+        Crystal::Hasher.reduce_num(-(0.5.to_big_r ** i)).should eq(&-(1_u64 << ((-i) % 61)))
+      end
+
+      Crystal::Hasher.reduce_num(BigRational.new(1, 0x1FFF_FFFF_FFFF_FFFF_u64)).should eq(Crystal::Hasher::HASH_INF_PLUS)
+      Crystal::Hasher.reduce_num(BigRational.new(-1, 0x1FFF_FFFF_FFFF_FFFF_u64)).should eq(Crystal::Hasher::HASH_INF_MINUS)
+      Crystal::Hasher.reduce_num(BigRational.new(2, 0x1FFF_FFFF_FFFF_FFFF_u64)).should eq(Crystal::Hasher::HASH_INF_PLUS)
+      Crystal::Hasher.reduce_num(BigRational.new(-2, 0x1FFF_FFFF_FFFF_FFFF_u64)).should eq(Crystal::Hasher::HASH_INF_MINUS)
+    end
   end
 end
diff --git a/src/big/big_decimal.cr b/src/big/big_decimal.cr
index ab42c64bf919..a2c95fac2fff 100644
--- a/src/big/big_decimal.cr
+++ b/src/big/big_decimal.cr
@@ -753,10 +753,6 @@ struct BigDecimal < Number
     self
   end
 
-  def hash(hasher)
-    hasher.string(to_s)
-  end
-
   # Returns the *quotient* as absolutely negative if `self` and *other* have
   # different signs, otherwise returns the *quotient*.
   def normalize_quotient(other : BigDecimal, quotient : BigInt) : BigInt
@@ -879,3 +875,22 @@ class String
     BigDecimal.new(self)
   end
 end
+
+# :nodoc:
+struct Crystal::Hasher
+  def self.reduce_num(value : BigDecimal)
+    v = reduce_num(value.value.abs)
+
+    # v = UInt64.mulmod(v, 10_u64.powmod(-scale, HASH_MODULUS), HASH_MODULUS)
+    # TODO: consider #7516 or similar
+    scale = value.scale
+    x = 0x1ccc_cccc_cccc_cccc_u64 # 10^-1 (mod HASH_MODULUS)
+    while scale > 0
+      v = UInt64.mulmod(v, x, HASH_MODULUS) if scale.bits_set?(1)
+      scale = scale.unsafe_shr(1)
+      x = UInt64.mulmod(x, x, HASH_MODULUS)
+    end
+
+    v &* value.sign
+  end
+end
diff --git a/src/big/big_float.cr b/src/big/big_float.cr
index 2cf224d7484a..cadc91282fc1 100644
--- a/src/big/big_float.cr
+++ b/src/big/big_float.cr
@@ -77,9 +77,6 @@ struct BigFloat < Float
     new(mpf)
   end
 
-  # TODO: improve this
-  def_hash to_f64
-
   def self.default_precision
     LibGMP.mpf_get_default_prec
   end
diff --git a/src/big/big_int.cr b/src/big/big_int.cr
index 1e3c3430c542..49738cb8bfbc 100644
--- a/src/big/big_int.cr
+++ b/src/big/big_int.cr
@@ -488,9 +488,6 @@ struct BigInt < Int
     LibGMP.sizeinbase(self, 2).to_i
   end
 
-  # TODO: check hash equality for numbers >= 2**63
-  def_hash to_i64!
-
   def to_s(base : Int = 10, *, precision : Int = 1, upcase : Bool = false) : String
     raise ArgumentError.new("Invalid base #{base}") unless 2 <= base <= 36 || base == 62
     raise ArgumentError.new("upcase must be false for base 62") if upcase && base == 62
@@ -985,18 +982,14 @@ end
 
 # :nodoc:
 struct Crystal::Hasher
-  private HASH_MODULUS_INT_P = BigInt.new((1_u64 << HASH_BITS) - 1)
-  private HASH_MODULUS_INT_N = -BigInt.new((1_u64 << HASH_BITS) - 1)
+  private HASH_MODULUS_INT_P = BigInt.new(HASH_MODULUS)
 
   def self.reduce_num(value : BigInt)
-    # it should calculate `remainder(HASH_MODULUS)`
-    if LibGMP::UI == UInt64
-      v = LibGMP.tdiv_ui(value, HASH_MODULUS).to_i64
-      value < 0 ? -v : v
-    elsif value >= HASH_MODULUS_INT_P || value <= HASH_MODULUS_INT_N
-      value.unsafe_truncated_mod(HASH_MODULUS_INT_P).to_i64
-    else
-      value.to_i64
-    end
+    {% if LibGMP::UI == UInt64 %}
+      v = LibGMP.tdiv_ui(value, HASH_MODULUS)
+      value < 0 ? &-v : v
+    {% else %}
+      value.remainder(HASH_MODULUS_INT_P).to_u64!
+    {% end %}
   end
 end
diff --git a/src/big/big_rational.cr b/src/big/big_rational.cr
index b14c8f4b2d4a..cb4731dc4e95 100644
--- a/src/big/big_rational.cr
+++ b/src/big/big_rational.cr
@@ -285,9 +285,6 @@ struct BigRational < Number
     BigRational.new { |mpq| LibGMP.mpq_abs(mpq, self) }
   end
 
-  # TODO: improve this
-  def_hash to_f64
-
   # Returns the `Float64` representing this rational.
   def to_f : Float64
     to_f64
@@ -459,18 +456,13 @@ end
 
 # :nodoc:
 struct Crystal::Hasher
-  private HASH_MODULUS_RAT_P = BigRational.new((1_u64 << HASH_BITS) - 1)
-  private HASH_MODULUS_RAT_N = -BigRational.new((1_u64 << HASH_BITS) - 1)
-
   def self.reduce_num(value : BigRational)
-    rem = value
-    if value >= HASH_MODULUS_RAT_P || value <= HASH_MODULUS_RAT_N
-      num = value.numerator
-      denom = value.denominator
-      div = num.tdiv(denom)
-      floor = div.tdiv(HASH_MODULUS)
-      rem -= floor * HASH_MODULUS
+    inverse = BigInt.new do |mpz|
+      if LibGMP.invert(mpz, value.denominator, HASH_MODULUS_INT_P) == 0
+        # inverse doesn't exist, i.e. denominator is a multiple of HASH_MODULUS
+        return value >= 0 ? HASH_INF_PLUS : HASH_INF_MINUS
+      end
     end
-    rem.to_big_f.hash
+    UInt64.mulmod(reduce_num(value.numerator.abs), inverse.to_u64!, HASH_MODULUS) &* value.sign
   end
 end
diff --git a/src/big/lib_gmp.cr b/src/big/lib_gmp.cr
index 3cae0de64b77..00834598d9d2 100644
--- a/src/big/lib_gmp.cr
+++ b/src/big/lib_gmp.cr
@@ -141,6 +141,7 @@ lib LibGMP
   fun gcd_ui = __gmpz_gcd_ui(rop : MPZ*, op1 : MPZ*, op2 : UI) : UI
   fun lcm = __gmpz_lcm(rop : MPZ*, op1 : MPZ*, op2 : MPZ*)
   fun lcm_ui = __gmpz_lcm_ui(rop : MPZ*, op1 : MPZ*, op2 : UI)
+  fun invert = __gmpz_invert(rop : MPZ*, op1 : MPZ*, op2 : MPZ*) : Int
   fun remove = __gmpz_remove(rop : MPZ*, op : MPZ*, f : MPZ*) : BitcntT
 
   # # Miscellaneous Functions
diff --git a/src/int.cr b/src/int.cr
index 35ad8c05f642..65a54e4e2fde 100644
--- a/src/int.cr
+++ b/src/int.cr
@@ -193,6 +193,32 @@ struct Int
     {% end %}
   end
 
+  # :nodoc:
+  #
+  # Computes (x * y) % z, but without intermediate overflows.
+  # Precondition: `0 <= x < z && y >= 0`
+  def self.mulmod(x, y, z)
+    result = zero
+    while y > 0
+      if y.bits_set?(1)
+        # result = (result + x) % z
+        if result >= z &- x
+          result &-= z &- x
+        else
+          result &+= x
+        end
+      end
+      # x = (x + x) % z
+      if x >= z &- x
+        x &-= z &- x
+      else
+        x = x.unsafe_shl(1)
+      end
+      y = y.unsafe_shr(1)
+    end
+    result
+  end
+
   # Returns the result of shifting this number's bits *count* positions to the right.
   # Also known as arithmetic right shift.
   #