diff --git a/news/8792.bugfix b/news/8792.bugfix
new file mode 100644
index 000000000..e83bdb09c
--- /dev/null
+++ b/news/8792.bugfix
@@ -0,0 +1,2 @@
+New resolver: Pick up hash declarations in constraints files and use them to
+filter available distributions.
diff --git a/news/8839.bugfix b/news/8839.bugfix
new file mode 100644
index 000000000..987b801e9
--- /dev/null
+++ b/news/8839.bugfix
@@ -0,0 +1,3 @@
+New resolver: If a package appears multiple times in user specification with
+different ``--hash`` options, only hashes that present in all specifications
+should be allowed.
diff --git a/src/pip/_internal/resolution/resolvelib/base.py b/src/pip/_internal/resolution/resolvelib/base.py
index 9245747bf..7c09cd70b 100644
--- a/src/pip/_internal/resolution/resolvelib/base.py
+++ b/src/pip/_internal/resolution/resolvelib/base.py
@@ -1,5 +1,8 @@
+from pip._vendor.packaging.specifiers import SpecifierSet
 from pip._vendor.packaging.utils import canonicalize_name
 
+from pip._internal.req.req_install import InstallRequirement
+from pip._internal.utils.hashes import Hashes
 from pip._internal.utils.typing import MYPY_CHECK_RUNNING
 
 if MYPY_CHECK_RUNNING:
@@ -8,7 +11,6 @@ if MYPY_CHECK_RUNNING:
     from pip._vendor.packaging.version import _BaseVersion
 
     from pip._internal.models.link import Link
-    from pip._internal.req.req_install import InstallRequirement
 
     CandidateLookup = Tuple[
         Optional["Candidate"],
@@ -24,6 +26,39 @@ def format_name(project, extras):
     return "{}[{}]".format(project, ",".join(canonical_extras))
 
 
+class Constraint(object):
+    def __init__(self, specifier, hashes):
+        # type: (SpecifierSet, Hashes) -> None
+        self.specifier = specifier
+        self.hashes = hashes
+
+    @classmethod
+    def empty(cls):
+        # type: () -> Constraint
+        return Constraint(SpecifierSet(), Hashes())
+
+    @classmethod
+    def from_ireq(cls, ireq):
+        # type: (InstallRequirement) -> Constraint
+        return Constraint(ireq.specifier, ireq.hashes(trust_internet=False))
+
+    def __nonzero__(self):
+        # type: () -> bool
+        return bool(self.specifier) or bool(self.hashes)
+
+    def __bool__(self):
+        # type: () -> bool
+        return self.__nonzero__()
+
+    def __and__(self, other):
+        # type: (InstallRequirement) -> Constraint
+        if not isinstance(other, InstallRequirement):
+            return NotImplemented
+        specifier = self.specifier & other.specifier
+        hashes = self.hashes & other.hashes(trust_internet=False)
+        return Constraint(specifier, hashes)
+
+
 class Requirement(object):
     @property
     def name(self):
diff --git a/src/pip/_internal/resolution/resolvelib/factory.py b/src/pip/_internal/resolution/resolvelib/factory.py
index dab23aa09..172f054fa 100644
--- a/src/pip/_internal/resolution/resolvelib/factory.py
+++ b/src/pip/_internal/resolution/resolvelib/factory.py
@@ -22,6 +22,7 @@ from pip._internal.utils.misc import (
 from pip._internal.utils.typing import MYPY_CHECK_RUNNING
 from pip._internal.utils.virtualenv import running_under_virtualenv
 
+from .base import Constraint
 from .candidates import (
     AlreadyInstalledCandidate,
     EditableCandidate,
@@ -154,6 +155,7 @@ class Factory(object):
         self,
         ireqs,  # type: Sequence[InstallRequirement]
         specifier,  # type: SpecifierSet
+        hashes,  # type: Hashes
     ):
         # type: (...) -> Iterable[Candidate]
         if not ireqs:
@@ -166,11 +168,10 @@ class Factory(object):
         template = ireqs[0]
         name = canonicalize_name(template.req.name)
 
-        hashes = Hashes()
         extras = frozenset()  # type: FrozenSet[str]
         for ireq in ireqs:
             specifier &= ireq.req.specifier
-            hashes |= ireq.hashes(trust_internet=False)
+            hashes &= ireq.hashes(trust_internet=False)
             extras |= frozenset(ireq.extras)
 
         # We use this to ensure that we only yield a single candidate for
@@ -220,7 +221,7 @@ class Factory(object):
         return six.itervalues(candidates)
 
     def find_candidates(self, requirements, constraint):
-        # type: (Sequence[Requirement], SpecifierSet) -> Iterable[Candidate]
+        # type: (Sequence[Requirement], Constraint) -> Iterable[Candidate]
         explicit_candidates = set()  # type: Set[Candidate]
         ireqs = []  # type: List[InstallRequirement]
         for req in requirements:
@@ -233,7 +234,11 @@ class Factory(object):
         # If none of the requirements want an explicit candidate, we can ask
         # the finder for candidates.
         if not explicit_candidates:
-            return self._iter_found_candidates(ireqs, constraint)
+            return self._iter_found_candidates(
+                ireqs,
+                constraint.specifier,
+                constraint.hashes,
+            )
 
         if constraint:
             name = explicit_candidates.pop().name
diff --git a/src/pip/_internal/resolution/resolvelib/provider.py b/src/pip/_internal/resolution/resolvelib/provider.py
index b2eb9d06e..80577a61c 100644
--- a/src/pip/_internal/resolution/resolvelib/provider.py
+++ b/src/pip/_internal/resolution/resolvelib/provider.py
@@ -1,8 +1,9 @@
-from pip._vendor.packaging.specifiers import SpecifierSet
 from pip._vendor.resolvelib.providers import AbstractProvider
 
 from pip._internal.utils.typing import MYPY_CHECK_RUNNING
 
+from .base import Constraint
+
 if MYPY_CHECK_RUNNING:
     from typing import (
         Any,
@@ -41,7 +42,7 @@ class PipProvider(AbstractProvider):
     def __init__(
         self,
         factory,  # type: Factory
-        constraints,  # type: Dict[str, SpecifierSet]
+        constraints,  # type: Dict[str, Constraint]
         ignore_dependencies,  # type: bool
         upgrade_strategy,  # type: str
         user_requested,  # type: Set[str]
@@ -134,7 +135,7 @@ class PipProvider(AbstractProvider):
         if not requirements:
             return []
         constraint = self._constraints.get(
-            requirements[0].name, SpecifierSet(),
+            requirements[0].name, Constraint.empty(),
         )
         candidates = self._factory.find_candidates(requirements, constraint)
         return reversed(self._sort_matches(candidates))
diff --git a/src/pip/_internal/resolution/resolvelib/resolver.py b/src/pip/_internal/resolution/resolvelib/resolver.py
index aecddb113..449cfea28 100644
--- a/src/pip/_internal/resolution/resolvelib/resolver.py
+++ b/src/pip/_internal/resolution/resolvelib/resolver.py
@@ -14,12 +14,12 @@ from pip._internal.resolution.resolvelib.provider import PipProvider
 from pip._internal.utils.misc import dist_is_editable
 from pip._internal.utils.typing import MYPY_CHECK_RUNNING
 
+from .base import Constraint
 from .factory import Factory
 
 if MYPY_CHECK_RUNNING:
     from typing import Dict, List, Optional, Set, Tuple
 
-    from pip._vendor.packaging.specifiers import SpecifierSet
     from pip._vendor.resolvelib.resolvers import Result
     from pip._vendor.resolvelib.structs import Graph
 
@@ -81,7 +81,7 @@ class Resolver(BaseResolver):
     def resolve(self, root_reqs, check_supported_wheels):
         # type: (List[InstallRequirement], bool) -> RequirementSet
 
-        constraints = {}  # type: Dict[str, SpecifierSet]
+        constraints = {}  # type: Dict[str, Constraint]
         user_requested = set()  # type: Set[str]
         requirements = []
         for req in root_reqs:
@@ -94,9 +94,9 @@ class Resolver(BaseResolver):
                     continue
                 name = canonicalize_name(req.name)
                 if name in constraints:
-                    constraints[name] = constraints[name] & req.specifier
+                    constraints[name] &= req
                 else:
-                    constraints[name] = req.specifier
+                    constraints[name] = Constraint.from_ireq(req)
             else:
                 if req.user_supplied and req.name:
                     user_requested.add(canonicalize_name(req.name))
diff --git a/src/pip/_internal/utils/hashes.py b/src/pip/_internal/utils/hashes.py
index d1b062fed..d9f74a640 100644
--- a/src/pip/_internal/utils/hashes.py
+++ b/src/pip/_internal/utils/hashes.py
@@ -46,16 +46,24 @@ class Hashes(object):
         """
         self._allowed = {} if hashes is None else hashes
 
-    def __or__(self, other):
+    def __and__(self, other):
         # type: (Hashes) -> Hashes
         if not isinstance(other, Hashes):
             return NotImplemented
-        new = self._allowed.copy()
+
+        # If either of the Hashes object is entirely empty (i.e. no hash
+        # specified at all), all hashes from the other object are allowed.
+        if not other:
+            return self
+        if not self:
+            return other
+
+        # Otherwise only hashes that present in both objects are allowed.
+        new = {}
         for alg, values in iteritems(other._allowed):
-            try:
-                new[alg] += values
-            except KeyError:
-                new[alg] = values
+            if alg not in self._allowed:
+                continue
+            new[alg] = [v for v in values if v in self._allowed[alg]]
         return Hashes(new)
 
     @property
diff --git a/tests/functional/test_new_resolver_hashes.py b/tests/functional/test_new_resolver_hashes.py
new file mode 100644
index 000000000..4b13ebc30
--- /dev/null
+++ b/tests/functional/test_new_resolver_hashes.py
@@ -0,0 +1,213 @@
+import collections
+import hashlib
+
+import pytest
+
+from pip._internal.utils.urls import path_to_url
+from tests.lib import (
+    create_basic_sdist_for_package,
+    create_basic_wheel_for_package,
+)
+
+_FindLinks = collections.namedtuple(
+    "_FindLinks", "index_html sdist_hash wheel_hash",
+)
+
+
+def _create_find_links(script):
+    sdist_path = create_basic_sdist_for_package(script, "base", "0.1.0")
+    wheel_path = create_basic_wheel_for_package(script, "base", "0.1.0")
+
+    sdist_hash = hashlib.sha256(sdist_path.read_bytes()).hexdigest()
+    wheel_hash = hashlib.sha256(wheel_path.read_bytes()).hexdigest()
+
+    index_html = script.scratch_path / "index.html"
+    index_html.write_text(
+        """
+        <a href="{sdist_url}#sha256={sdist_hash}">{sdist_path.stem}</a>
+        <a href="{wheel_url}#sha256={wheel_hash}">{wheel_path.stem}</a>
+        """.format(
+            sdist_url=path_to_url(sdist_path),
+            sdist_hash=sdist_hash,
+            sdist_path=sdist_path,
+            wheel_url=path_to_url(wheel_path),
+            wheel_hash=wheel_hash,
+            wheel_path=wheel_path,
+        )
+    )
+
+    return _FindLinks(index_html, sdist_hash, wheel_hash)
+
+
+@pytest.mark.parametrize(
+    "requirements_template, message",
+    [
+        (
+            """
+            base==0.1.0 --hash=sha256:{sdist_hash} --hash=sha256:{wheel_hash}
+            base==0.1.0 --hash=sha256:{sdist_hash} --hash=sha256:{wheel_hash}
+            """,
+            "Checked 2 links for project {name!r} against 2 hashes "
+            "(2 matches, 0 no digest): discarding no candidates",
+        ),
+        (
+            # Different hash lists are intersected.
+            """
+            base==0.1.0 --hash=sha256:{sdist_hash} --hash=sha256:{wheel_hash}
+            base==0.1.0 --hash=sha256:{sdist_hash}
+            """,
+            "Checked 2 links for project {name!r} against 1 hashes "
+            "(1 matches, 0 no digest): discarding 1 non-matches",
+        ),
+    ],
+    ids=["identical", "intersect"],
+)
+def test_new_resolver_hash_intersect(script, requirements_template, message):
+    find_links = _create_find_links(script)
+
+    requirements_txt = script.scratch_path / "requirements.txt"
+    requirements_txt.write_text(
+        requirements_template.format(
+            sdist_hash=find_links.sdist_hash,
+            wheel_hash=find_links.wheel_hash,
+        ),
+    )
+
+    result = script.pip(
+        "install",
+        "--use-feature=2020-resolver",
+        "--no-cache-dir",
+        "--no-deps",
+        "--no-index",
+        "--find-links", find_links.index_html,
+        "--verbose",
+        "--requirement", requirements_txt,
+    )
+
+    assert message.format(name=u"base") in result.stdout, str(result)
+
+
+def test_new_resolver_hash_intersect_from_constraint(script):
+    find_links = _create_find_links(script)
+
+    constraints_txt = script.scratch_path / "constraints.txt"
+    constraints_txt.write_text(
+        "base==0.1.0 --hash=sha256:{sdist_hash}".format(
+            sdist_hash=find_links.sdist_hash,
+        ),
+    )
+    requirements_txt = script.scratch_path / "requirements.txt"
+    requirements_txt.write_text(
+        """
+        base==0.1.0 --hash=sha256:{sdist_hash} --hash=sha256:{wheel_hash}
+        """.format(
+            sdist_hash=find_links.sdist_hash,
+            wheel_hash=find_links.wheel_hash,
+        ),
+    )
+
+    result = script.pip(
+        "install",
+        "--use-feature=2020-resolver",
+        "--no-cache-dir",
+        "--no-deps",
+        "--no-index",
+        "--find-links", find_links.index_html,
+        "--verbose",
+        "--constraint", constraints_txt,
+        "--requirement", requirements_txt,
+    )
+
+    message = (
+        "Checked 2 links for project {name!r} against 1 hashes "
+        "(1 matches, 0 no digest): discarding 1 non-matches"
+    ).format(name=u"base")
+    assert message in result.stdout, str(result)
+
+
+@pytest.mark.parametrize(
+    "requirements_template, constraints_template",
+    [
+        (
+            """
+            base==0.1.0 --hash=sha256:{sdist_hash}
+            base==0.1.0 --hash=sha256:{wheel_hash}
+            """,
+            "",
+        ),
+        (
+            "base==0.1.0 --hash=sha256:{sdist_hash}",
+            "base==0.1.0 --hash=sha256:{wheel_hash}",
+        ),
+    ],
+    ids=["both-requirements", "one-each"],
+)
+def test_new_resolver_hash_intersect_empty(
+    script, requirements_template, constraints_template,
+):
+    find_links = _create_find_links(script)
+
+    constraints_txt = script.scratch_path / "constraints.txt"
+    constraints_txt.write_text(
+        constraints_template.format(
+            sdist_hash=find_links.sdist_hash,
+            wheel_hash=find_links.wheel_hash,
+        ),
+    )
+
+    requirements_txt = script.scratch_path / "requirements.txt"
+    requirements_txt.write_text(
+        requirements_template.format(
+            sdist_hash=find_links.sdist_hash,
+            wheel_hash=find_links.wheel_hash,
+        ),
+    )
+
+    result = script.pip(
+        "install",
+        "--use-feature=2020-resolver",
+        "--no-cache-dir",
+        "--no-deps",
+        "--no-index",
+        "--find-links", find_links.index_html,
+        "--constraint", constraints_txt,
+        "--requirement", requirements_txt,
+        expect_error=True,
+    )
+
+    assert (
+        "THESE PACKAGES DO NOT MATCH THE HASHES FROM THE REQUIREMENTS FILE."
+    ) in result.stderr, str(result)
+
+
+def test_new_resolver_hash_intersect_empty_from_constraint(script):
+    find_links = _create_find_links(script)
+
+    constraints_txt = script.scratch_path / "constraints.txt"
+    constraints_txt.write_text(
+        """
+        base==0.1.0 --hash=sha256:{sdist_hash}
+        base==0.1.0 --hash=sha256:{wheel_hash}
+        """.format(
+            sdist_hash=find_links.sdist_hash,
+            wheel_hash=find_links.wheel_hash,
+        ),
+    )
+
+    result = script.pip(
+        "install",
+        "--use-feature=2020-resolver",
+        "--no-cache-dir",
+        "--no-deps",
+        "--no-index",
+        "--find-links", find_links.index_html,
+        "--constraint", constraints_txt,
+        "base==0.1.0",
+        expect_error=True,
+    )
+
+    message = (
+        "Hashes are required in --require-hashes mode, but they are missing "
+        "from some requirements."
+    )
+    assert message in result.stderr, str(result)
diff --git a/tests/unit/resolution_resolvelib/test_requirement.py b/tests/unit/resolution_resolvelib/test_requirement.py
index 21de3df4a..a03edb6f7 100644
--- a/tests/unit/resolution_resolvelib/test_requirement.py
+++ b/tests/unit/resolution_resolvelib/test_requirement.py
@@ -1,8 +1,7 @@
 import pytest
-from pip._vendor.packaging.specifiers import SpecifierSet
 from pip._vendor.resolvelib import BaseReporter, Resolver
 
-from pip._internal.resolution.resolvelib.base import Candidate
+from pip._internal.resolution.resolvelib.base import Candidate, Constraint
 from pip._internal.utils.urls import path_to_url
 
 # NOTE: All tests are prefixed `test_rlr` (for "test resolvelib resolver").
@@ -59,7 +58,7 @@ def test_new_resolver_correct_number_of_matches(test_cases, factory):
     """Requirements should return the correct number of candidates"""
     for spec, _, match_count in test_cases:
         req = factory.make_requirement_from_spec(spec, comes_from=None)
-        matches = factory.find_candidates([req], SpecifierSet())
+        matches = factory.find_candidates([req], Constraint.empty())
         assert len(list(matches)) == match_count
 
 
@@ -68,7 +67,7 @@ def test_new_resolver_candidates_match_requirement(test_cases, factory):
     """
     for spec, _, _ in test_cases:
         req = factory.make_requirement_from_spec(spec, comes_from=None)
-        for c in factory.find_candidates([req], SpecifierSet()):
+        for c in factory.find_candidates([req], Constraint.empty()):
             assert isinstance(c, Candidate)
             assert req.is_satisfied_by(c)