diff --git a/src/Groups.jl b/src/Groups.jl
index 9d60ad1..fcfdd3e 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -23,6 +23,7 @@ include("findreplace.jl")
 
 module New
 include("new_types.jl")
+include("new_hashing.jl")
 include("normalform.jl")
 
 end # module New
diff --git a/src/new_hashing.jl b/src/new_hashing.jl
new file mode 100644
index 0000000..20dceab
--- /dev/null
+++ b/src/new_hashing.jl
@@ -0,0 +1,49 @@
+## Hashing
+
+_hashing_data(g::FPGroupElement) = word(g)
+
+bitget(h::UInt, n::Int) = Bool((h & (1 << n)) >> n)
+bitclear(h::UInt, n::Int) = h & ~(1 << n)
+bitset(h::UInt, n::Int) = h | (1 << n)
+bitset(h::UInt, v::Bool, n::Int) = v ? bitset(h, n) : bitclear(h, n)
+
+# We store hash of a word in field `savedhash` to use it as cheap proxy to
+# determine non-equal words. Additionally bits of `savehash` store boolean
+# information as follows
+# * `savedhash & 1` (the first bit): is word in normal form?
+# * `savedhash & 2` (the second bit): is the hash valid?
+const __BITFLAGS_MASK = ~(~(UInt(0)) << 2)
+
+isnormalform(g::FPGroupElement) = bitget(g.savedhash, 0)
+_isvalidhash(g::FPGroupElement) = bitget(g.savedhash, 1)
+
+_setnormalform(h::UInt, v::Bool) = bitset(h, v, 0)
+_setvalidhash(h::UInt, v::Bool) = bitset(h, v, 1)
+
+_setnormalform!(g::FPGroupElement, v::Bool) =
+    g.savedhash = _setnormalform(g.savedhash, v)
+_setvalidhash!(g::FPGroupElement, v::Bool) =
+    g.savedhash = _setvalidhash(g.savedhash, v)
+
+# To update hash use this internal method, possibly only after computing the
+# normal form of `g`:
+function _update_savedhash!(g::FPGroupElement, data=_hashing_data(g))
+    h = hash(data, hash(parent(g)))
+    h = (h << count_ones(__BITFLAGS_MASK)) | (__BITFLAGS_MASK & g.savedhash)
+    g.savedhash = _setvalidhash(h, true)
+    return g
+end
+
+function Base.hash(g::FPGroupElement, h::UInt)
+    normalform!(g)
+    _isvalidhash(g) || _update_savedhash!(g)
+    return hash(g.savedhash >> count_ones(__BITFLAGS_MASK) , h)
+end
+
+function Base.copyto!(res::FPGroupElement, g::FPGroupElement)
+    @assert parent(res) === parent(g)
+    resize!(word(res), length(word(g)))
+    copyto!(word(res), word(g))
+    res.savedhash = g.savedhash
+    return res
+end
diff --git a/src/new_types.jl b/src/new_types.jl
index 6bd36e3..d05b030 100644
--- a/src/new_types.jl
+++ b/src/new_types.jl
@@ -132,38 +132,6 @@ function Base.show(io::IO, f::FPGroupElement)
     print(io, KnuthBendix.string_repr(word(f), alphabet(f)))
 end
 
-## Hashing
-
-# We store hash of a word in field `savedhash` to use it as cheap replacement
-# for equality checking. In the last bit of `savehash` we store the information
-# whether the word is already reduced (in normal form) or not. Hence
-isnormalform(g::FPGroupElement) = Bool(g.savedhash & 1)
-
-# To update hash use this internal method, possibly only after computing the
-# normal form of `g`:
-_update_savedhash!(g::FPGroupElement, h = hash(word(g))) =
-    (g.savedhash = h | 1; return g)
-
-# If `g` has been mutated (e.g. `word(g)` has been modified, `_invalidate_hash`
-# must be called.
-_invalidate_hash!(g::FPGroupElement) = g.savedhash = g.savedhash & 0
-
-# Accessor for the saved hash
-@inline _savedhash(f::FPGroupElement) = f.savedhash
-
-function Base.hash(g::FPGroupElement, h::UInt)
-    normalform!(g)
-    # compute the best approximation of the normal form
-    return hash(parent(g), _savedhash(g) ⊻ h)
-end
-
-function Base.copyto!(res::FPGroupElement, g::FPGroupElement)
-    resize!(word(res), length(word(g)))
-    copyto!(word(res), word(g))
-    _invalidate_hash!(res)
-    return res
-end
-
 ## GroupElement Interface for FPGroupElement
 
 Base.parent(f::FPGroupElement) = f.parent
@@ -171,13 +139,14 @@ GroupsCore.parent_type(::Type{<:FPGroupElement{G}}) where G = G
 
 function Base.:(==)(g::FPGroupElement, h::FPGroupElement)
     @boundscheck @assert parent(g) === parent(h)
+    normalform!(g)
+    normalform!(h)
     hash(g) != hash(h) && return false
-    # hash reduces to normal form behind the scenes
     return word(g) == word(h)
 end
 
 function Base.deepcopy_internal(g::FPGroupElement, stackdict::IdDict)
-    return FPGroupElement(copy(word(g)), _savedhash(g), parent(g))
+    return FPGroupElement(copy(word(g)), g.savedhash, parent(g))
 end
 
 Base.inv(g::FPGroupElement) =
diff --git a/src/normalform.jl b/src/normalform.jl
index 76ef5dd..31cde79 100644
--- a/src/normalform.jl
+++ b/src/normalform.jl
@@ -11,7 +11,8 @@ Compute the normal form of `g`, possibly modifying `g` in-place.
         copyto!(word(g), w)
     end
 
-    g = _update_savedhash!(g)
+    _setnormalform!(g, true)
+    _setvalidhash!(g, false)
     @assert isnormalform(g)
     return g
 end
@@ -24,11 +25,11 @@ function normalform!(res::GEl, g::GEl) where GEl<:FPGroupElement
     @boundscheck @assert parent(res) === parent(g)
     if isnormalform(g)
         copyto!(res, g)
-        _update_savedhash!(res, g.savedhash)
     else
         resize!(word(res), 0)
         normalform!(word(res), g)
-        res = _update_savedhash!(res)
+        _setnormalform!(res, true)
+        _setvalidhash!(res, false)
     end
     return res
 end