From 468561eb0a9f4d8b4c5297d12706d7712b9920e2 Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Fri, 30 Jul 2021 15:58:12 +0100
Subject: [PATCH] wip

---
 verkle_trie_pedersen/_blst.so         | Bin 0 -> 306448 bytes
 verkle_trie_pedersen/blst.py          | 237 +++++++++
 verkle_trie_pedersen/compute_stats.sh |  15 +
 verkle_trie_pedersen/ipa.py           | 114 +++++
 verkle_trie_pedersen/pippenger.py     |  68 +++
 verkle_trie_pedersen/poly_utils.py    | 298 ++++++++++++
 verkle_trie_pedersen/verkle_trie.py   | 677 ++++++++++++++++++++++++++
 7 files changed, 1409 insertions(+)
 create mode 100755 verkle_trie_pedersen/_blst.so
 create mode 100644 verkle_trie_pedersen/blst.py
 create mode 100755 verkle_trie_pedersen/compute_stats.sh
 create mode 100644 verkle_trie_pedersen/ipa.py
 create mode 100644 verkle_trie_pedersen/pippenger.py
 create mode 100644 verkle_trie_pedersen/poly_utils.py
 create mode 100644 verkle_trie_pedersen/verkle_trie.py

diff --git a/verkle_trie_pedersen/_blst.so b/verkle_trie_pedersen/_blst.so
new file mode 100755
index 0000000000000000000000000000000000000000..3bc7193c520b6ffe9cf6d17a3e5692c880645c82
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diff --git a/verkle_trie_pedersen/blst.py b/verkle_trie_pedersen/blst.py
new file mode 100644
index 00000000..0dd6f080
--- /dev/null
+++ b/verkle_trie_pedersen/blst.py
@@ -0,0 +1,237 @@
+# This file was automatically generated by SWIG (http://www.swig.org).
+# Version 4.0.1
+#
+# Do not make changes to this file unless you know what you are doing--modify
+# the SWIG interface file instead.
+
+from sys import version_info as _swig_python_version_info
+if _swig_python_version_info < (2, 7, 0):
+    raise RuntimeError("Python 2.7 or later required")
+
+# Import the low-level C/C++ module
+if __package__ or "." in __name__:
+    from . import _blst
+else:
+    import _blst
+
+try:
+    import builtins as __builtin__
+except ImportError:
+    import __builtin__
+
+_swig_new_instance_method = _blst.SWIG_PyInstanceMethod_New
+_swig_new_static_method = _blst.SWIG_PyStaticMethod_New
+
+def _swig_repr(self):
+    try:
+        strthis = "proxy of " + self.this.__repr__()
+    except __builtin__.Exception:
+        strthis = ""
+    return "<%s.%s; %s >" % (self.__class__.__module__, self.__class__.__name__, strthis,)
+
+
+def _swig_setattr_nondynamic_instance_variable(set):
+    def set_instance_attr(self, name, value):
+        if name == "thisown":
+            self.this.own(value)
+        elif name == "this":
+            set(self, name, value)
+        elif hasattr(self, name) and isinstance(getattr(type(self), name), property):
+            set(self, name, value)
+        else:
+            raise AttributeError("You cannot add instance attributes to %s" % self)
+    return set_instance_attr
+
+
+def _swig_setattr_nondynamic_class_variable(set):
+    def set_class_attr(cls, name, value):
+        if hasattr(cls, name) and not isinstance(getattr(cls, name), property):
+            set(cls, name, value)
+        else:
+            raise AttributeError("You cannot add class attributes to %s" % cls)
+    return set_class_attr
+
+
+def _swig_add_metaclass(metaclass):
+    """Class decorator for adding a metaclass to a SWIG wrapped class - a slimmed down version of six.add_metaclass"""
+    def wrapper(cls):
+        return metaclass(cls.__name__, cls.__bases__, cls.__dict__.copy())
+    return wrapper
+
+
+class _SwigNonDynamicMeta(type):
+    """Meta class to enforce nondynamic attributes (no new attributes) for a class"""
+    __setattr__ = _swig_setattr_nondynamic_class_variable(type.__setattr__)
+
+
+BLST_SUCCESS = _blst.BLST_SUCCESS
+BLST_BAD_ENCODING = _blst.BLST_BAD_ENCODING
+BLST_POINT_NOT_ON_CURVE = _blst.BLST_POINT_NOT_ON_CURVE
+BLST_POINT_NOT_IN_GROUP = _blst.BLST_POINT_NOT_IN_GROUP
+BLST_AGGR_TYPE_MISMATCH = _blst.BLST_AGGR_TYPE_MISMATCH
+BLST_VERIFY_FAIL = _blst.BLST_VERIFY_FAIL
+BLST_PK_IS_INFINITY = _blst.BLST_PK_IS_INFINITY
+class SecretKey(object):
+    thisown = property(lambda x: x.this.own(), lambda x, v: x.this.own(v), doc="The membership flag")
+    __repr__ = _swig_repr
+    keygen = _swig_new_instance_method(_blst.SecretKey_keygen)
+    from_bendian = _swig_new_instance_method(_blst.SecretKey_from_bendian)
+    from_lendian = _swig_new_instance_method(_blst.SecretKey_from_lendian)
+    to_bendian = _swig_new_instance_method(_blst.SecretKey_to_bendian)
+    to_lendian = _swig_new_instance_method(_blst.SecretKey_to_lendian)
+
+    def __init__(self):
+        _blst.SecretKey_swiginit(self, _blst.new_SecretKey())
+    __swig_destroy__ = _blst.delete_SecretKey
+
+# Register SecretKey in _blst:
+_blst.SecretKey_swigregister(SecretKey)
+
+class P1_Affine(object):
+    thisown = property(lambda x: x.this.own(), lambda x, v: x.this.own(v), doc="The membership flag")
+    __repr__ = _swig_repr
+
+    def __init__(self, *args):
+        _blst.P1_Affine_swiginit(self, _blst.new_P1_Affine(*args))
+    dup = _swig_new_instance_method(_blst.P1_Affine_dup)
+    to_jacobian = _swig_new_instance_method(_blst.P1_Affine_to_jacobian)
+    serialize = _swig_new_instance_method(_blst.P1_Affine_serialize)
+    compress = _swig_new_instance_method(_blst.P1_Affine_compress)
+    on_curve = _swig_new_instance_method(_blst.P1_Affine_on_curve)
+    in_group = _swig_new_instance_method(_blst.P1_Affine_in_group)
+    is_inf = _swig_new_instance_method(_blst.P1_Affine_is_inf)
+    is_equal = _swig_new_instance_method(_blst.P1_Affine_is_equal)
+    core_verify = _swig_new_instance_method(_blst.P1_Affine_core_verify)
+    generator = _swig_new_static_method(_blst.P1_Affine_generator)
+    __swig_destroy__ = _blst.delete_P1_Affine
+
+# Register P1_Affine in _blst:
+_blst.P1_Affine_swigregister(P1_Affine)
+P1_Affine_generator = _blst.P1_Affine_generator
+
+class P1(object):
+    thisown = property(lambda x: x.this.own(), lambda x, v: x.this.own(v), doc="The membership flag")
+    __repr__ = _swig_repr
+
+    def __init__(self, *args):
+        _blst.P1_swiginit(self, _blst.new_P1(*args))
+    dup = _swig_new_instance_method(_blst.P1_dup)
+    to_affine = _swig_new_instance_method(_blst.P1_to_affine)
+    serialize = _swig_new_instance_method(_blst.P1_serialize)
+    compress = _swig_new_instance_method(_blst.P1_compress)
+    on_curve = _swig_new_instance_method(_blst.P1_on_curve)
+    in_group = _swig_new_instance_method(_blst.P1_in_group)
+    is_inf = _swig_new_instance_method(_blst.P1_is_inf)
+    is_equal = _swig_new_instance_method(_blst.P1_is_equal)
+    aggregate = _swig_new_instance_method(_blst.P1_aggregate)
+    sign_with = _swig_new_instance_method(_blst.P1_sign_with)
+    hash_to = _swig_new_instance_method(_blst.P1_hash_to)
+    encode_to = _swig_new_instance_method(_blst.P1_encode_to)
+    mult = _swig_new_instance_method(_blst.P1_mult)
+    cneg = _swig_new_instance_method(_blst.P1_cneg)
+    neg = _swig_new_instance_method(_blst.P1_neg)
+    add = _swig_new_instance_method(_blst.P1_add)
+    dbl = _swig_new_instance_method(_blst.P1_dbl)
+    generator = _swig_new_static_method(_blst.P1_generator)
+    __swig_destroy__ = _blst.delete_P1
+
+# Register P1 in _blst:
+_blst.P1_swigregister(P1)
+P1_generator = _blst.P1_generator
+
+class P2_Affine(object):
+    thisown = property(lambda x: x.this.own(), lambda x, v: x.this.own(v), doc="The membership flag")
+    __repr__ = _swig_repr
+
+    def __init__(self, *args):
+        _blst.P2_Affine_swiginit(self, _blst.new_P2_Affine(*args))
+    dup = _swig_new_instance_method(_blst.P2_Affine_dup)
+    to_jacobian = _swig_new_instance_method(_blst.P2_Affine_to_jacobian)
+    serialize = _swig_new_instance_method(_blst.P2_Affine_serialize)
+    compress = _swig_new_instance_method(_blst.P2_Affine_compress)
+    on_curve = _swig_new_instance_method(_blst.P2_Affine_on_curve)
+    in_group = _swig_new_instance_method(_blst.P2_Affine_in_group)
+    is_inf = _swig_new_instance_method(_blst.P2_Affine_is_inf)
+    is_equal = _swig_new_instance_method(_blst.P2_Affine_is_equal)
+    core_verify = _swig_new_instance_method(_blst.P2_Affine_core_verify)
+    generator = _swig_new_static_method(_blst.P2_Affine_generator)
+    __swig_destroy__ = _blst.delete_P2_Affine
+
+# Register P2_Affine in _blst:
+_blst.P2_Affine_swigregister(P2_Affine)
+P2_Affine_generator = _blst.P2_Affine_generator
+
+class P2(object):
+    thisown = property(lambda x: x.this.own(), lambda x, v: x.this.own(v), doc="The membership flag")
+    __repr__ = _swig_repr
+
+    def __init__(self, *args):
+        _blst.P2_swiginit(self, _blst.new_P2(*args))
+    dup = _swig_new_instance_method(_blst.P2_dup)
+    to_affine = _swig_new_instance_method(_blst.P2_to_affine)
+    serialize = _swig_new_instance_method(_blst.P2_serialize)
+    compress = _swig_new_instance_method(_blst.P2_compress)
+    on_curve = _swig_new_instance_method(_blst.P2_on_curve)
+    in_group = _swig_new_instance_method(_blst.P2_in_group)
+    is_inf = _swig_new_instance_method(_blst.P2_is_inf)
+    is_equal = _swig_new_instance_method(_blst.P2_is_equal)
+    aggregate = _swig_new_instance_method(_blst.P2_aggregate)
+    sign_with = _swig_new_instance_method(_blst.P2_sign_with)
+    hash_to = _swig_new_instance_method(_blst.P2_hash_to)
+    encode_to = _swig_new_instance_method(_blst.P2_encode_to)
+    mult = _swig_new_instance_method(_blst.P2_mult)
+    cneg = _swig_new_instance_method(_blst.P2_cneg)
+    neg = _swig_new_instance_method(_blst.P2_neg)
+    add = _swig_new_instance_method(_blst.P2_add)
+    dbl = _swig_new_instance_method(_blst.P2_dbl)
+    generator = _swig_new_static_method(_blst.P2_generator)
+    __swig_destroy__ = _blst.delete_P2
+
+# Register P2 in _blst:
+_blst.P2_swigregister(P2)
+P2_generator = _blst.P2_generator
+
+G1 = _blst.G1
+G2 = _blst.G2
+class PT(object):
+    thisown = property(lambda x: x.this.own(), lambda x, v: x.this.own(v), doc="The membership flag")
+    __repr__ = _swig_repr
+
+    def __init__(self, *args):
+        _blst.PT_swiginit(self, _blst.new_PT(*args))
+    dup = _swig_new_instance_method(_blst.PT_dup)
+    is_one = _swig_new_instance_method(_blst.PT_is_one)
+    is_equal = _swig_new_instance_method(_blst.PT_is_equal)
+    sqr = _swig_new_instance_method(_blst.PT_sqr)
+    mul = _swig_new_instance_method(_blst.PT_mul)
+    final_exp = _swig_new_instance_method(_blst.PT_final_exp)
+    __swig_destroy__ = _blst.delete_PT
+
+# Register PT in _blst:
+_blst.PT_swigregister(PT)
+
+class Pairing(object):
+    thisown = property(lambda x: x.this.own(), lambda x, v: x.this.own(v), doc="The membership flag")
+    __repr__ = _swig_repr
+
+    def __init__(self, *args):
+        _blst.Pairing_swiginit(self, _blst.new_Pairing(*args))
+    __swig_destroy__ = _blst.delete_Pairing
+    aggregate = _swig_new_instance_method(_blst.Pairing_aggregate)
+    mul_n_aggregate = _swig_new_instance_method(_blst.Pairing_mul_n_aggregate)
+    commit = _swig_new_instance_method(_blst.Pairing_commit)
+    merge = _swig_new_instance_method(_blst.Pairing_merge)
+    finalverify = _swig_new_instance_method(_blst.Pairing_finalverify)
+
+# Register Pairing in _blst:
+_blst.Pairing_swigregister(Pairing)
+
+cdata = _blst.cdata
+memmove = _blst.memmove
+
+cvar = _blst.cvar
+BLS12_381_G1 = cvar.BLS12_381_G1
+BLS12_381_NEG_G1 = cvar.BLS12_381_NEG_G1
+BLS12_381_G2 = cvar.BLS12_381_G2
+BLS12_381_NEG_G2 = cvar.BLS12_381_NEG_G2
+
diff --git a/verkle_trie_pedersen/compute_stats.sh b/verkle_trie_pedersen/compute_stats.sh
new file mode 100755
index 00000000..e1b16812
--- /dev/null
+++ b/verkle_trie_pedersen/compute_stats.sh
@@ -0,0 +1,15 @@
+echo -e "WIDTH_BITS\tWIDTH\tNUMBER_INITIAL_KEYS\tNUMBER_KEYS_PROOF\taverage_depth\tproof_size\tproof_time\tcheck_time" > stats.txt
+
+
+python verkle_trie.py 5 65536 500 >> stats.txt
+python verkle_trie.py 6 65536 500 >> stats.txt
+python verkle_trie.py 7 65536 500 >> stats.txt
+python verkle_trie.py 8 65536 500 >> stats.txt
+python verkle_trie.py 9 65536 500 >> stats.txt
+python verkle_trie.py 10 65536 500 >> stats.txt
+python verkle_trie.py 11 65536 500 >> stats.txt
+python verkle_trie.py 12 65536 500 >> stats.txt
+python verkle_trie.py 13 65536 500 >> stats.txt
+python verkle_trie.py 14 65536 500 >> stats.txt
+python verkle_trie.py 15 65536 500 >> stats.txt
+python verkle_trie.py 16 65536 500 >> stats.txt
diff --git a/verkle_trie_pedersen/ipa.py b/verkle_trie_pedersen/ipa.py
new file mode 100644
index 00000000..4997dc30
--- /dev/null
+++ b/verkle_trie_pedersen/ipa.py
@@ -0,0 +1,114 @@
+import blst
+import pippenger
+
+#
+# Utilities for dealing with polynomials in evaluation form
+#
+# A polynomial in evaluation for is defined by its values on DOMAIN,
+# where DOMAIN is [omega**0, omega**1, omega**2, ..., omega**(WIDTH-1)]
+# where omega is a WIDTH root of unity, i.e. omega**WIDTH % MODULUS == 1 
+#
+# Any polynomial of degree < WIDTH can be represented uniquely in this form,
+# and many operations (such as multiplication and exact division) are more
+# efficient.
+#
+# By precomputing the trusted setup in Lagrange basis, we can also easily
+# commit to a a polynomial in evaluation form.
+#
+
+class KzgUtils():
+
+    """
+    Class that defines helper function for Kate proofs in evaluation form (Lagrange basis)
+    """
+    def __init__(self, MODULUS, WIDTH, DOMAIN, SETUP, primefield):
+        self.MODULUS = MODULUS
+        self.WIDTH = WIDTH
+        self.DOMAIN = DOMAIN
+        self.SETUP = SETUP
+        self.primefield = primefield
+        # Precomputed inverses of 1 / (1 - DOMAIN[i])
+        self.inverses = [0] + [primefield.inv(1 - DOMAIN[i]) for i in range(1, WIDTH)]
+        self.inverse_width = primefield.inv(self.WIDTH)
+
+
+    def evaluate_polynomial_in_evaluation_form(self, f, z):
+        """
+        Takes a polynomial in evaluation form and evaluates it at one point outside the domain. 
+        Uses the barycentric formula:
+        f(z) = (1 - z**WIDTH) / WIDTH  *  sum_(i=0)^WIDTH  (f(DOMAIN[i]) * DOMAIN[i]) / (z - DOMAIN[i])
+        """
+        r = 0
+        for i in range(self.WIDTH):
+            r += self.primefield.div(f[i] * self.DOMAIN[i], (z - self.DOMAIN[i]) )
+        r = r * (pow(z, self.WIDTH, self.MODULUS) - 1) * self.inverse_width % self.MODULUS
+
+        return r
+
+
+    def compute_inner_quotient_in_evaluation_form(self, f, index):
+        """
+        Compute the quotient q(X) = (f(X) - f(DOMAIN[index])) / (X - DOMAIN[index]) in evaluation form.
+
+        Inner means that the value z = DOMAIN[index] is one of the points at which f is evaluated -- so unlike an outer
+        quotient (where z is not in DOMAIN), we need to do some extra work to compute q[index] where the formula above
+        is 0 / 0
+        """
+        q = [0] * self.WIDTH
+        y = f[index]
+        for i in range(self.WIDTH):
+            if i != index:
+                q[i] = (f[i] - y) * self.DOMAIN[-i] * self.inverses[index - i] % self.MODULUS
+                q[index] += - self.DOMAIN[(i - index) % self.WIDTH] * q[i] % self.MODULUS
+
+        return q
+
+
+    def compute_outer_quotient_in_evaluation_form(self, f, z, y):
+        """
+        Compute the quotient q(X) = (f(X) - y)) / (X - z) in evaluation form. Note that this only works if the quotient
+        is exact, i.e. f(z) = y, and otherwise returns garbage
+        """
+        q = [0] * self.WIDTH
+        for i in range(self.WIDTH):
+            q[i] = self.primefield.div(f[i] - y, self.DOMAIN[i] - z)
+
+        return q
+
+
+    def check_kzg_proof(self, C, z, y, pi):
+        """
+        Check the KZG proof 
+        e(C - [y], [1]) = e(pi, [s - z])
+        which is equivalent to
+        e(C - [y], [1]) * e(-pi, [s - z]) == 1
+        """
+        pairing = blst.PT(blst.G2().to_affine(), C.dup().add(blst.G1().mult(y).neg()).to_affine())
+        pairing.mul(blst.PT(self.SETUP["g2"][1].dup().add(blst.G2().mult(z).neg()).to_affine(), pi.dup().neg().to_affine()))
+
+        return pairing.final_exp().is_one()
+
+
+    def evaluate_and_compute_kzg_proof(self, f, z):
+        """
+        Evaluates a function f (given in evaluation form) at a point z (which can be in the DOMAIN or not)
+        and gives y = f(z) as well as a Kate proof that this is the correct result
+        """
+        if z in self.DOMAIN:
+            index = self.DOMAIN.index(z)
+            y = f[index]
+            q = self.compute_inner_quotient_in_evaluation_form(f, index)
+        else:
+            y = self.evaluate_polynomial_in_evaluation_form(f, z)
+            q = self.compute_outer_quotient_in_evaluation_form(f, z, y)
+
+        return y, pippenger.pippenger_simple(self.SETUP["g1_lagrange"], q)
+
+
+    def compute_commitment_lagrange(self, values):
+        """
+        Computes a commitment for a function given in evaluation form.
+        'values' is a dictionary and can have missing indices, which improves efficiency.
+        """
+        commitment = pippenger.pippenger_simple([self.SETUP["g1_lagrange"][i] for i in values.keys()], values.values())
+        return commitment
\ No newline at end of file
diff --git a/verkle_trie_pedersen/pippenger.py b/verkle_trie_pedersen/pippenger.py
new file mode 100644
index 00000000..51d72030
--- /dev/null
+++ b/verkle_trie_pedersen/pippenger.py
@@ -0,0 +1,68 @@
+import blst
+from itertools import zip_longest
+from collections import defaultdict
+from random import randint
+from time import time
+
+def integer_in_base(i, b):
+    r = []
+    while i > 0:
+        r.append(i % b)
+        i //= b
+    return r
+
+def pippenger_simple(group_elements, factors):
+    """
+    A naive implementation of a Pippenger-like multiexponentiation algorithm. Don't use this
+    in practice, a native implementation in the blst library will perform much better.
+    """
+    assert len(group_elements) == len(factors)
+    n = len(group_elements)
+    d = 1
+    while (d + 2) * 2**(d + 2) < n:
+        d += 1
+    b = 2**d
+    factors_decomposed = [integer_in_base(factor, b) for factor in factors]
+    result = blst.P1_generator().mult(0)
+    for bases in reversed(list(zip_longest(*factors_decomposed, fillvalue=0))):
+        total = blst.P1_generator().mult(0)
+        base_elements_dict = defaultdict(list)
+        for index, base in enumerate(bases):
+            if base > 0:
+                base_elements_dict[base].append(group_elements[index])
+        for base, base_elements in base_elements_dict.items():
+            if len(base_elements) > 0:
+                sum_base_elements = base_elements[0].dup()
+                for x in base_elements[1:]:
+                    sum_base_elements.add(x)
+                sum_base_elements.mult(base)
+                total.add(sum_base_elements)
+        result.mult(b).add(total)
+    return result
+
+def lincomb_naive(group_elements, factors):
+    """
+    Direct linear combination
+    """
+    assert len(group_elements) == len(factors)
+    result = blst.P1_generator().mult(0)
+    for g, f in zip(group_elements, factors):
+        result.add(g.dup().mult(f))
+    return result
+
+def test_pippenger(group_elements, factors):
+    """
+    Test and time pippenger_simple
+    """
+    time_a = time()
+    naive_result = lincomb_naive(group_elements, factors)
+    time_b = time()
+    print("n = {0} multiexp".format(len(group_elements)))
+    print("Naive linear combination: {0:.6f} s".format(time_b - time_a))
+    pippenger_result = pippenger_simple(group_elements, factors)
+    time_c = time()
+    print("Using simple Pippenger algorithm: {0:.6f} s".format(time_c - time_b))
+    assert naive_result.is_equal(pippenger_result)
+    
+if __name__ == "__main__":
+    test_pippenger([blst.P1_generator()]*16384, [randint(0, 2**255) for i in range(16384)])
\ No newline at end of file
diff --git a/verkle_trie_pedersen/poly_utils.py b/verkle_trie_pedersen/poly_utils.py
new file mode 100644
index 00000000..eb8972d3
--- /dev/null
+++ b/verkle_trie_pedersen/poly_utils.py
@@ -0,0 +1,298 @@
+# Creates an object that includes convenience operations for numbers
+# and polynomials in some prime field
+
+# Also added interpolation over an arbitrary DOMAIN (not roots of unity)
+#
+
+class PrimeField():
+    def __init__(self, MODULUS, WIDTH):
+        assert pow(2, MODULUS, MODULUS) == 2
+        self.WIDTH = WIDTH
+        self.MODULUS = MODULUS
+        self.DOMAIN = range(WIDTH)
+
+        self.A = self.zpoly(DOMAIN)
+        self.Aprime = self.formal_derivative(self.A)
+
+        # i-th Lagrange polynomial
+        self.lagrange_polys = []
+
+        # Aprime evaluated on the DOMAIN
+        self.Aprime_DOMAIN = []
+
+        # Aprime on the DOMAIN, inverted
+        self.Aprime_DOMAIN_inv = []
+        for i, x in enumerate(DOMAIN):
+            self.Aprime_DOMAIN.append(self.eval_poly_at(self.Aprime, x))
+            self.Aprime_DOMAIN_inv.append(self.inv(self.Aprime_DOMAIN[-1]))
+            self.lagrange_polys.append(self.mul_polys([self.Aprime_DOMAIN_inv[-1]], 
+                        self.div_polys(self.A, [-x, 1])))
+
+        # Inverses needed for quotients
+        self.INVERSES = [self.inv(x) for x in list(range(WIDTH)) + list(range(-WIDTH + 1, 0))]
+
+        
+    def formal_derivative(self, f):
+        return [(n + 1) * c % self.MODULUS for n, c in enumerate(f[1:])]
+
+
+    def evaluate_polynomial_in_evaluation_form(self, f, z):
+        """
+        Takes a polynomial in evaluation form and evaluates it at one point outside the DOMAIN. 
+        Uses the barycentric formula:
+        f(z) = A(z) *  sum_(i=0)^(WIDTH-1)  f(DOMAIN[i]) / A'(DOMAIN[i]) * 1 / (z - DOMAIN[i])
+        """
+        r = 0
+        for x, i in enumerate(self.DOMAIN):
+            r += self.div(f[i], self.Aprime_DOMAIN[i] * (z - x) )
+        r = r * self.eval_poly_at(self.A, z) % self.MODULUS
+
+        return r
+
+
+    def compute_inner_quotient_in_evaluation_form(self, f, index):
+        """
+        Compute the quotient q(X) = (f(X) - f(DOMAIN[index])) / (X - DOMAIN[index]) in evaluation form.
+
+        Inner means that the value z = DOMAIN[index] is one of the points at which f is evaluated -- so unlike an outer
+        quotient (where z is not in DOMAIN), we need to do some extra work to compute q[index] where the formula above
+        is 0 / 0
+        """
+        q = [0] * self.WIDTH
+        y = f[index]
+        for i in range(self.WIDTH):
+            if i != index:
+                q[i] = (f[i] - y) * self.inverses[index - i] * self.Aprime_DOMAIN[index] * self.Aprime_DOMAIN_inv[i] % self.MODULUS
+                q[index] += - self.DOMAIN[(i - index) % self.WIDTH] * q[i] % self.MODULUS
+
+        return q
+
+
+    def compute_outer_quotient_in_evaluation_form(self, f, z, y):
+        """
+        Compute the quotient q(X) = (f(X) - y)) / (X - z) in evaluation form. Note that this only works if the quotient
+        is exact, i.e. f(z) = y, and otherwise returns garbage
+        """
+        q = [0] * self.WIDTH
+        for i in range(self.WIDTH):
+            q[i] = self.primefield.div(f[i] - y, self.DOMAIN[i] - z)
+
+        return q
+
+    def add(self, x, y):
+        return (x+y) % self.MODULUS
+
+    def sub(self, x, y):
+        return (x-y) % self.MODULUS
+
+    def mul(self, x, y):
+        return (x*y) % self.MODULUS
+
+    def exp(self, x, p):
+        return pow(x, p, self.MODULUS)
+
+    # Modular inverse using the extended Euclidean algorithm
+    def inv(self, a):
+        if a == 0:
+            return 0
+        lm, hm = 1, 0
+        low, high = a % self.MODULUS, self.MODULUS
+        while low > 1:
+            r = high//low
+            nm, new = hm-lm*r, high-low*r
+            lm, low, hm, high = nm, new, lm, low
+        return lm % self.MODULUS
+
+    def multi_inv(self, values):
+        partials = [1]
+        for i in range(len(values)):
+            partials.append(self.mul(partials[-1], values[i] or 1))
+        inv = self.inv(partials[-1])
+        outputs = [0] * len(values)
+        for i in range(len(values), 0, -1):
+            outputs[i-1] = self.mul(partials[i-1], inv) if values[i-1] else 0
+            inv = self.mul(inv, values[i-1] or 1)
+        return outputs
+
+    def div(self, x, y):
+        return self.mul(x, self.inv(y))
+
+    # Evaluate a polynomial at a point
+    def eval_poly_at(self, p, x):
+        y = 0
+        power_of_x = 1
+        for i, p_coeff in enumerate(p):
+            y += power_of_x * p_coeff
+            power_of_x = (power_of_x * x) % self.MODULUS
+        return y % self.MODULUS
+        
+    # Arithmetic for polynomials
+    def add_polys(self, a, b):
+        return [((a[i] if i < len(a) else 0) + (b[i] if i < len(b) else 0))
+                % self.MODULUS for i in range(max(len(a), len(b)))]
+
+    def sub_polys(self, a, b):
+        return [((a[i] if i < len(a) else 0) - (b[i] if i < len(b) else 0))
+                % self.MODULUS for i in range(max(len(a), len(b)))]
+    
+    def mul_by_const(self, a, c):
+        return [(x*c) % self.MODULUS for x in a]
+    
+    def mul_polys(self, a, b):
+        o = [0] * (len(a) + len(b) - 1)
+        for i, aval in enumerate(a):
+            for j, bval in enumerate(b):
+                o[i+j] += a[i] * b[j]
+        return [x % self.MODULUS for x in o]
+    
+    def div_polys(self, a, b):
+        assert len(a) >= len(b)
+        a = [x for x in a]
+        o = []
+        apos = len(a) - 1
+        bpos = len(b) - 1
+        diff = apos - bpos
+        while diff >= 0:
+            quot = self.div(a[apos], b[bpos])
+            o.insert(0, quot)
+            for i in range(bpos, -1, -1):
+                a[diff+i] -= b[i] * quot
+            apos -= 1
+            diff -= 1
+        return [x % self.MODULUS for x in o]
+
+    def mod_polys(self, a, b):
+        return self.sub_polys(a, self.mul_polys(b, self.div_polys(a, b)))[:len(b)-1]
+
+    # Build a polynomial from a few coefficients
+    def sparse(self, coeff_dict):
+        o = [0] * (max(coeff_dict.keys()) + 1)
+        for k, v in coeff_dict.items():
+            o[k] = v % self.MODULUS
+        return o
+
+    # Build a polynomial that returns 0 at all specified xs
+    def zpoly(self, xs):
+        root = [1]
+        for x in xs:
+            root.insert(0, 0)
+            for j in range(len(root)-1):
+                root[j] -= root[j+1] * x
+        return [x % self.MODULUS for x in root]
+    
+    # Given p+1 y values and x values with no errors, recovers the original
+    # p+1 degree polynomial.
+    # Lagrange interpolation works roughly in the following way.
+    # 1. Suppose you have a set of points, eg. x = [1, 2, 3], y = [2, 5, 10]
+    # 2. For each x, generate a polynomial which equals its corresponding
+    #    y coordinate at that point and 0 at all other points provided.
+    # 3. Add these polynomials together.
+    
+    def lagrange_interp(self, xs, ys):
+        # Generate master numerator polynomial, eg. (x - x1) * (x - x2) * ... * (x - xn)
+        root = self.zpoly(xs)
+        assert len(root) == len(ys) + 1
+        # print(root)
+        # Generate per-value numerator polynomials, eg. for x=x2,
+        # (x - x1) * (x - x3) * ... * (x - xn), by dividing the master
+        # polynomial back by each x coordinate
+        nums = [self.div_polys(root, [-x, 1]) for x in xs]
+        # Generate denominators by evaluating numerator polys at each x
+        denoms = [self.eval_poly_at(nums[i], xs[i]) for i in range(len(xs))]
+        invdenoms = self.multi_inv(denoms)
+        # Generate output polynomial, which is the sum of the per-value numerator
+        # polynomials rescaled to have the right y values
+        b = [0 for y in ys]
+        for i in range(len(xs)):
+            yslice = self.mul(ys[i], invdenoms[i])
+            for j in range(len(ys)):
+                if nums[i][j] and ys[i]:
+                    b[j] += nums[i][j] * yslice
+        return [x % self.MODULUS for x in b]
+
+    # Optimized poly evaluation for degree 4
+    def eval_quartic(self, p, x):
+        xsq = x * x % self.MODULUS
+        xcb = xsq * x
+        return (p[0] + p[1] * x + p[2] * xsq + p[3] * xcb) % self.MODULUS
+
+    # Optimized version of the above restricted to deg-4 polynomials
+    def lagrange_interp_4(self, xs, ys):
+        x01, x02, x03, x12, x13, x23 = \
+            xs[0] * xs[1], xs[0] * xs[2], xs[0] * xs[3], xs[1] * xs[2], xs[1] * xs[3], xs[2] * xs[3]
+        m = self.MODULUS
+        eq0 = [-x12 * xs[3] % m, (x12 + x13 + x23), -xs[1]-xs[2]-xs[3], 1]
+        eq1 = [-x02 * xs[3] % m, (x02 + x03 + x23), -xs[0]-xs[2]-xs[3], 1]
+        eq2 = [-x01 * xs[3] % m, (x01 + x03 + x13), -xs[0]-xs[1]-xs[3], 1]
+        eq3 = [-x01 * xs[2] % m, (x01 + x02 + x12), -xs[0]-xs[1]-xs[2], 1]
+        e0 = self.eval_poly_at(eq0, xs[0])
+        e1 = self.eval_poly_at(eq1, xs[1])
+        e2 = self.eval_poly_at(eq2, xs[2])
+        e3 = self.eval_poly_at(eq3, xs[3])
+        e01 = e0 * e1
+        e23 = e2 * e3
+        invall = self.inv(e01 * e23)
+        inv_y0 = ys[0] * invall * e1 * e23 % m
+        inv_y1 = ys[1] * invall * e0 * e23 % m
+        inv_y2 = ys[2] * invall * e01 * e3 % m
+        inv_y3 = ys[3] * invall * e01 * e2 % m
+        return [(eq0[i] * inv_y0 + eq1[i] * inv_y1 + eq2[i] * inv_y2 + eq3[i] * inv_y3) % m for i in range(4)]
+    
+    # Optimized version of the above restricted to deg-2 polynomials
+    def lagrange_interp_2(self, xs, ys):
+        m = self.MODULUS
+        eq0 = [-xs[1] % m, 1]
+        eq1 = [-xs[0] % m, 1]
+        e0 = self.eval_poly_at(eq0, xs[0])
+        e1 = self.eval_poly_at(eq1, xs[1])
+        invall = self.inv(e0 * e1)
+        inv_y0 = ys[0] * invall * e1
+        inv_y1 = ys[1] * invall * e0
+        return [(eq0[i] * inv_y0 + eq1[i] * inv_y1) % m for i in range(2)]
+
+    # Optimized version of the above restricted to deg-4 polynomials
+    def multi_interp_4(self, xsets, ysets):
+        data = []
+        invtargets = []
+        for xs, ys in zip(xsets, ysets):
+            x01, x02, x03, x12, x13, x23 = \
+                xs[0] * xs[1], xs[0] * xs[2], xs[0] * xs[3], xs[1] * xs[2], xs[1] * xs[3], xs[2] * xs[3]
+            m = self.MODULUS
+            eq0 = [-x12 * xs[3] % m, (x12 + x13 + x23), -xs[1]-xs[2]-xs[3], 1]
+            eq1 = [-x02 * xs[3] % m, (x02 + x03 + x23), -xs[0]-xs[2]-xs[3], 1]
+            eq2 = [-x01 * xs[3] % m, (x01 + x03 + x13), -xs[0]-xs[1]-xs[3], 1]
+            eq3 = [-x01 * xs[2] % m, (x01 + x02 + x12), -xs[0]-xs[1]-xs[2], 1]
+            e0 = self.eval_quartic(eq0, xs[0])
+            e1 = self.eval_quartic(eq1, xs[1])
+            e2 = self.eval_quartic(eq2, xs[2])
+            e3 = self.eval_quartic(eq3, xs[3])
+            data.append([ys, eq0, eq1, eq2, eq3])
+            invtargets.extend([e0, e1, e2, e3])
+        invalls = self.multi_inv(invtargets)
+        o = []
+        for (i, (ys, eq0, eq1, eq2, eq3)) in enumerate(data):
+            invallz = invalls[i*4:i*4+4]
+            inv_y0 = ys[0] * invallz[0] % m
+            inv_y1 = ys[1] * invallz[1] % m
+            inv_y2 = ys[2] * invallz[2] % m
+            inv_y3 = ys[3] * invallz[3] % m
+            o.append([(eq0[i] * inv_y0 + eq1[i] * inv_y1 + eq2[i] * inv_y2 + eq3[i] * inv_y3) % m for i in range(4)])
+        # assert o == [self.lagrange_interp_4(xs, ys) for xs, ys in zip(xsets, ysets)]
+        return o
+
+
+if __name__ == "__main__":
+    primefield = PrimeField(11, [0,1,2,3])
+    for i, x in enumerate(primefield.DOMAIN):
+        assert primefield.eval_poly_at(primefield.lagrange_polys[i], x) == 1
+        for y in primefield.DOMAIN[:i] + primefield.DOMAIN[i+1:]:
+            assert primefield.eval_poly_at(primefield.lagrange_polys[i], y) == 0
+
+    poly = [3, 4, 3, 2]
+    poly_eval = [primefield.eval_poly_at(poly, x) for x in primefield.DOMAIN]
+
+    assert primefield.eval_poly_at(poly, 5) == primefield.evaluate_polynomial_in_evaluation_form(poly_eval, 5)
+
+    poly_eval_quotient = primefield.compute_inner_quotient_in_evaluation_form(poly_eval, 2)
+
+    poly_quotient = primefield.div_polys([poly[0] - poly_eval[2]] + poly[1:], [-3, 1])
\ No newline at end of file
diff --git a/verkle_trie_pedersen/verkle_trie.py b/verkle_trie_pedersen/verkle_trie.py
new file mode 100644
index 00000000..5d3158d1
--- /dev/null
+++ b/verkle_trie_pedersen/verkle_trie.py
@@ -0,0 +1,677 @@
+import pippenger
+import blst
+import hashlib
+from random import randint, shuffle
+from poly_utils import PrimeField
+from time import time
+from kzg_utils import KzgUtils
+from fft import fft
+import sys
+
+#
+# Proof of concept implementation for verkle tries
+#
+# All polynomials in this implementation are represented in evaluation form, i.e. by their values
+# on DOMAIN. See kzg_utils.py for more explanation
+#
+
+# BLS12_381 curve modulus
+MODULUS = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
+
+# Primitive root for the field
+PRIMITIVE_ROOT = 5
+
+assert pow(PRIMITIVE_ROOT, (MODULUS - 1) // 2, MODULUS) != 1
+assert pow(PRIMITIVE_ROOT, MODULUS - 1, MODULUS) == 1
+
+primefield = PrimeField(MODULUS)
+
+# Verkle trie parameters
+KEY_LENGTH = 256 # bits
+WIDTH_BITS = 10
+WIDTH = 2**WIDTH_BITS
+
+ROOT_OF_UNITY = pow(PRIMITIVE_ROOT, (MODULUS - 1) // WIDTH, MODULUS)
+DOMAIN = [pow(ROOT_OF_UNITY, i, MODULUS) for i in range(WIDTH)]
+
+# Number of key-value pairs to insert
+NUMBER_INITIAL_KEYS = 2**15
+
+# Number of keys to insert after computing initial tree
+NUMBER_ADDED_KEYS = 512
+
+# Number of keys to delete
+NUMBER_DELETED_KEYS = 512
+
+# Number of key/values pair in proof
+NUMBER_KEYS_PROOF = 5000
+
+def generate_setup(size, secret):
+    """
+    Generates a setup in the G1 group and G2 group, as well as the Lagrange polynomials in G1 (via FFT)
+    """
+    g1_setup = [blst.G1().mult(pow(secret, i, MODULUS)) for i in range(size)]
+    g2_setup = [blst.G2().mult(pow(secret, i, MODULUS)) for i in range(size)]
+    g1_lagrange = fft(g1_setup, MODULUS, ROOT_OF_UNITY, inv=True)
+    return {"g1": g1_setup, "g2": g2_setup, "g1_lagrange": g1_lagrange}
+
+
+def get_verkle_indices(key):
+    """
+    Generates the list of verkle indices for key
+    """
+    x = int.from_bytes(key, "big")
+    last_index_bits = KEY_LENGTH % WIDTH_BITS
+    index = (x % (2**last_index_bits)) << (WIDTH_BITS - last_index_bits)
+    x //= 2**last_index_bits
+    indices = [index]
+    for i in range((KEY_LENGTH - 1) // WIDTH_BITS):
+        index = x % WIDTH
+        x //= WIDTH
+        indices.append(index)
+    return tuple(reversed(indices))
+
+
+def hash(x):
+    if isinstance(x, bytes):
+        return hashlib.sha256(x).digest()
+    elif isinstance(x, blst.P1):
+        return hash(x.compress())
+    b = b""
+    for a in x:
+        if isinstance(a, bytes):
+            b += a
+        elif isinstance(a, int):
+            b += a.to_bytes(32, "little")
+        elif isinstance(a, blst.P1):
+            b += hash(a.compress())
+    return hash(b)
+
+
+def hash_to_int(x):
+    return int.from_bytes(hash(x), "little")
+
+
+def insert_verkle_node(root, key, value):
+    """
+    Insert node without updating hashes/commitments (useful for building a full trie)
+    """
+    current_node = root
+    indices = iter(get_verkle_indices(key))
+    index = None
+    while current_node["node_type"] == "inner":
+        previous_node = current_node
+        previous_index = index
+        index = next(indices)
+        if index in current_node:
+            current_node = current_node[index]
+        else:
+            current_node[index] = {"node_type": "leaf", "key": key, "value": value}
+            return
+    if current_node["key"] == key:
+        current_node["value"] = value
+    else:
+        previous_node[index] = {"node_type": "inner", "commitment": blst.G1().mult(0)}
+        insert_verkle_node(root, key, value)
+        insert_verkle_node(root, current_node["key"], current_node["value"])
+
+
+def update_verkle_node(root, key, value):
+    """
+    Update or insert node and update all commitments and hashes
+    """
+    current_node = root
+    indices = iter(get_verkle_indices(key))
+    index = None
+    path = []
+
+    new_node = {"node_type": "leaf", "key": key, "value": value}
+    add_node_hash(new_node)
+
+    while True:
+        index = next(indices)
+        path.append((index, current_node))
+        if index in current_node:
+            if current_node[index]["node_type"] == "leaf":
+                old_node = current_node[index]
+                if current_node[index]["key"] == key:
+                    current_node[index] = new_node
+                    value_change = (MODULUS + int.from_bytes(new_node["hash"], "little")
+                                    - int.from_bytes(old_node["hash"], "little")) % MODULUS
+                    break
+                else:
+                    new_inner_node = {"node_type": "inner"}
+                    new_index = next(indices)
+                    old_index = get_verkle_indices(old_node["key"])[len(path)]
+                    # TODO! Handle old_index == new_index
+                    assert old_index != new_index
+                    new_inner_node[new_index] = new_node
+                    new_inner_node[old_index] = old_node
+                    add_node_hash(new_inner_node)
+                    current_node[index] = new_inner_node
+                    value_change = (MODULUS + int.from_bytes(new_inner_node["hash"], "little")
+                                    - int.from_bytes(old_node["hash"], "little")) % MODULUS
+                    break
+            current_node = current_node[index]
+        else:
+            current_node[index] = new_node
+            value_change = int.from_bytes(new_node["hash"], "little") % MODULUS
+            break
+    
+    # Update all the parent commitments along 'path'
+    for index, node in reversed(path):
+        node["commitment"].add(SETUP["g1_lagrange"][index].dup().mult(value_change))
+        old_hash = node["hash"]
+        new_hash = hash(node["commitment"])
+        node["hash"] = new_hash
+        value_change = (MODULUS + int.from_bytes(new_hash, "little")
+                        - int.from_bytes(old_hash, "little")) % MODULUS
+
+
+def get_only_child(node):
+    """
+    Returns the only child of a node which has only one child. Returns 'None' if node has 0 or >1 children
+    """
+    child_count = 0
+    only_child = None
+    for key in node:
+        if isinstance(key, int):
+            child_count += 1
+            only_child = node[key]
+    return only_child if child_count == 1 else None
+
+
+def delete_verkle_node(root, key):
+    """
+    Delete node and update all commitments and hashes
+    """
+    current_node = root
+    indices = iter(get_verkle_indices(key))
+    index = None
+    path = []
+
+    while True:
+        index = next(indices)
+        path.append((index, current_node))
+        assert index in current_node, "Tried to delete non-existent key"
+        if current_node[index]["node_type"] == "leaf":
+            deleted_node = current_node[index]
+            assert deleted_node["key"] == key, "Tried to delete non-existent key"
+            del current_node[index]
+            value_change = (MODULUS - int.from_bytes(deleted_node["hash"], "little")) % MODULUS
+            break
+        current_node = current_node[index]
+    
+    # Update all the parent commitments along 'path'
+    replacement_node = None
+    for index, node in reversed(path):
+        if replacement_node != None:
+            node[index] = replacement_node
+            replacement_node = None
+        only_child = get_only_child(node)
+        if only_child != None and only_child["node_type"] == "leaf" and node != root:
+            replacement_node = only_child
+            value_change = (MODULUS + int.from_bytes(only_child["hash"], "little")
+                            - int.from_bytes(node["hash"], "little")) % MODULUS
+        else:            
+            node["commitment"].add(SETUP["g1_lagrange"][index].dup().mult(value_change))
+            old_hash = node["hash"]
+            new_hash = hash(node["commitment"])
+            node["hash"] = new_hash
+            value_change = (MODULUS + int.from_bytes(new_hash, "little")
+                            - int.from_bytes(old_hash, "little")) % MODULUS
+
+
+def add_node_hash(node):
+    """
+    Recursively adds all missing commitments and hashes to a verkle trie structure.
+    """
+    if node["node_type"] == "leaf":
+        node["hash"] = hash([node["key"], node["value"]])
+    if node["node_type"] == "inner":
+        lagrange_polynomials = []
+        values = {}
+        for i in range(WIDTH):
+            if i in node:
+                if "hash" not in node[i]:
+                    add_node_hash(node[i])
+                values[i] = int.from_bytes(node[i]["hash"], "little")
+        commitment = kzg_utils.compute_commitment_lagrange(values)
+        node["commitment"] = commitment
+        node["hash"] = hash(commitment.compress())
+
+
+def get_total_depth(root):
+    """
+    Computes the total depth (sum of the depth of all nodes) of a verkle trie
+    """
+    if root["node_type"] == "inner":
+        total_depth = 0
+        num_nodes = 0
+        for i in range(WIDTH):
+            if i in root:
+                depth, nodes = get_total_depth(root[i])
+                num_nodes += nodes
+                total_depth += nodes + depth
+        return total_depth, num_nodes
+    else:
+        return 0, 1
+
+
+def check_valid_tree(root, is_trie_root=True):
+    """
+    Checks that the tree is valid
+    """
+    if root["node_type"] == "inner":
+        if not is_trie_root:
+            only_child = get_only_child(root)
+            if only_child is not None:
+                assert only_child["node_type"] == "inner"
+    
+        lagrange_polynomials = []
+        values = {}
+        for i in range(WIDTH):
+            if i in root:
+                if "hash" not in root[i]:
+                    add_node_hash(node[i])
+                values[i] = int.from_bytes(root[i]["hash"], "little")
+        commitment = kzg_utils.compute_commitment_lagrange(values)
+        assert root["commitment"].is_equal(commitment)
+        assert root["hash"] == hash(commitment.compress())
+
+        for i in range(WIDTH):
+            if i in root:
+                check_valid_tree(root[i], False)
+    else:
+        assert root["hash"] == hash([root["key"], root["value"]])
+
+
+def get_average_depth(trie):
+    """
+    Get the average depth of nodes in a verkle trie
+    """
+    depth, nodes = get_total_depth(trie)
+    return depth / nodes
+
+
+def find_node(root, key):
+    """
+    Finds 'key' in verkle trie. Returns the full node (not just the value) or None if not present
+    """
+    current_node = root
+    indices = iter(get_verkle_indices(key))
+    while current_node["node_type"] == "inner":
+        index = next(indices)
+        if index in current_node:
+            current_node = current_node[index]
+        else:
+            return None
+    if current_node["key"] == key:
+        return current_node
+    return None
+
+
+def find_node_with_path(root, key):
+    """
+    As 'find_node', but returns the path of all nodes on the way to 'key' as well as their index
+    """
+    current_node = root
+    indices = iter(get_verkle_indices(key))
+    path = []
+    current_index_path = []
+    while current_node["node_type"] == "inner":
+        index = next(indices)
+        path.append((tuple(current_index_path), index, current_node))
+        current_index_path.append(index)
+        if index in current_node:
+            current_node = current_node[index]
+        else:
+            return path, None
+    if current_node["key"] == key:
+        return path, current_node
+    return path, None
+    
+
+def get_proof_size(proof):
+    depths, commitments_sorted_by_index_serialized, D_serialized, y, sigma_serialized = proof
+    size = len(depths) # assume 8 bit integer to represent the depth
+    size += 48 * len(commitments_sorted_by_index_serialized)
+    size += 48 + 32 + 48
+    return size
+
+lasttime = [0]
+
+
+def start_logging_time_if_eligible(string, eligible):
+    if eligible:
+        print(string, file=sys.stderr)
+        lasttime[0] = time()
+
+        
+def log_time_if_eligible(string, width, eligible):
+    if eligible:
+        print(string + ' ' * max(1, width - len(string)) + "{0:7.3f} s".format(time() - lasttime[0]), file=sys.stderr)
+        lasttime[0] = time()
+
+
+def make_kzg_multiproof(Cs, fs, indices, ys, display_times=True):
+    """
+    Computes a KZG multiproof according to the schema described here:
+    https://notes.ethereum.org/nrQqhVpQRi6acQckwm1Ryg
+
+    zs[i] = DOMAIN[indexes[i]]
+    """
+
+    # Step 1: Construct g(X) polynomial in evaluation form
+    r = hash_to_int([hash(C) for C in Cs] + indices + ys) % MODULUS
+
+    log_time_if_eligible("   Hashed to r", 30, display_times)
+
+    g = [0 for i in range(WIDTH)]
+    power_of_r = 1
+    for f, index in zip(fs, indices):
+        quotient = kzg_utils.compute_inner_quotient_in_evaluation_form(f, index)
+        for i in range(WIDTH):
+            g[i] += power_of_r * quotient[i]
+
+        power_of_r = power_of_r * r % MODULUS
+
+    log_time_if_eligible("   Computed g polynomial", 30, display_times)
+
+    D = kzg_utils.compute_commitment_lagrange({i: v for i, v in enumerate(g)})
+
+    log_time_if_eligible("   Computed commitment D", 30, display_times)
+
+    # Step 2: Compute f in evaluation form
+    
+    t = hash_to_int([r, D]) % MODULUS
+    
+    h = [0 for i in range(WIDTH)]
+    power_of_r = 1
+    
+    for f, index in zip(fs, indices):
+        denominator_inv = primefield.inv(t - DOMAIN[index])
+        for i in range(WIDTH):
+            h[i] += power_of_r * f[i] * denominator_inv % MODULUS
+            
+        power_of_r = power_of_r * r % MODULUS
+   
+    log_time_if_eligible("   Computed h polynomial", 30, display_times)
+
+    # Step 3: Evaluate and compute KZG proofs
+
+    y, pi = kzg_utils.evaluate_and_compute_kzg_proof(h, t)
+    w, rho = kzg_utils.evaluate_and_compute_kzg_proof(g, t)
+
+
+    # Compress both proofs into one
+
+    E = kzg_utils.compute_commitment_lagrange({i: v for i, v in enumerate(h)})
+    q = hash_to_int([E, D, y, w])
+    sigma = pi.dup().add(rho.dup().mult(q))
+
+    log_time_if_eligible("   Computed KZG proofs", 30, display_times)
+
+    return D.compress(), y, sigma.compress()
+
+
+def check_kzg_multiproof(Cs, indices, ys, proof, display_times=True):
+    """
+    Verifies a KZG multiproof according to the schema described here:
+    https://notes.ethereum.org/nrQqhVpQRi6acQckwm1Ryg
+    """
+
+    D_serialized, y, sigma_serialized = proof
+    D = blst.P1(D_serialized)
+    sigma = blst.P1(sigma_serialized)
+
+    # Step 1
+    r = hash_to_int([hash(C) for C in Cs] + indices + ys)
+
+    log_time_if_eligible("   Computed r hash", 30, display_times)
+    
+    # Step 2
+    t = hash_to_int([r, D])
+    E_coefficients = []
+    g_2_of_t = 0
+    power_of_r = 1
+
+    for index, y in zip(indices, ys):
+        E_coefficient = primefield.div(power_of_r, t - DOMAIN[index])
+        E_coefficients.append(E_coefficient)
+        g_2_of_t += E_coefficient * y % MODULUS
+            
+        power_of_r = power_of_r * r % MODULUS
+
+    log_time_if_eligible("   Computed g2 and e coeffs", 30, display_times)
+    
+    E = pippenger.pippenger_simple(Cs, E_coefficients)
+
+    log_time_if_eligible("   Computed E commitment", 30, display_times)
+
+    # Step 3 (Check KZG proofs)
+    w = (y - g_2_of_t) % MODULUS
+
+    q = hash_to_int([E, D, y, w])
+
+    if not kzg_utils.check_kzg_proof(E.dup().add(D.dup().mult(q)), t, y + q * w, sigma):
+        return False
+
+    log_time_if_eligible("   Checked KZG proofs", 30, display_times)
+
+    return True
+
+
+def make_verkle_proof(trie, keys, display_times=True):
+    """
+    Creates a proof for the 'keys' in the verkle trie given by 'trie'
+    """
+
+    start_logging_time_if_eligible("   Starting proof computation", display_times)
+
+    # Step 0: Find all keys in the trie
+    nodes_by_index = {}
+    nodes_by_index_and_subindex = {}
+    values = []
+    depths = []
+    for key in keys:
+        path, node = find_node_with_path(trie, key)
+        depths.append(len(path))
+        values.append(node["value"])
+        for index, subindex, node in path:
+            nodes_by_index[index] = node
+            nodes_by_index_and_subindex[(index, subindex)] = node
+
+    log_time_if_eligible("   Computed key paths", 30, display_times)
+    
+    # All commitments, but without any duplications. These are for sending over the wire as part of the proof
+    nodes_sorted_by_index = list(map(lambda x: x[1], sorted(nodes_by_index.items())))
+    
+    # Nodes sorted 
+    nodes_sorted_by_index_and_subindex = list(map(lambda x: x[1], sorted(nodes_by_index_and_subindex.items())))
+    
+    indices = list(map(lambda x: x[0][1], sorted(nodes_by_index_and_subindex.items())))
+    
+    ys = list(map(lambda x: int.from_bytes(x[1][x[0][1]]["hash"], "little"), sorted(nodes_by_index_and_subindex.items())))
+    
+    log_time_if_eligible("   Sorted all commitments", 30, display_times)
+
+    fs = []
+    Cs = [x["commitment"] for x in nodes_sorted_by_index_and_subindex]
+
+    for node in nodes_sorted_by_index_and_subindex:
+        fs.append([int.from_bytes(node[i]["hash"], "little") if i in node else 0 for i in range(WIDTH)])
+
+    D, y, sigma = make_kzg_multiproof(Cs, fs, indices, ys, display_times)
+
+    commitments_sorted_by_index_serialized = [x["commitment"].compress() for x in nodes_sorted_by_index[1:]]
+    
+    log_time_if_eligible("   Serialized commitments", 30, display_times)
+
+    return depths, commitments_sorted_by_index_serialized, D, y, sigma
+
+
+def check_verkle_proof(trie, keys, values, proof, display_times=True):
+    """
+    Checks Verkle tree proof according to
+    https://notes.ethereum.org/nrQqhVpQRi6acQckwm1Ryg?both
+    """
+
+    start_logging_time_if_eligible("   Starting proof check", display_times)
+
+    # Unpack the proof
+    depths, commitments_sorted_by_index_serialized, D_serialized, y, sigma_serialized = proof
+    commitments_sorted_by_index = [blst.P1(trie)] + [blst.P1(x) for x in commitments_sorted_by_index_serialized]
+
+    all_indices = set()
+    all_indices_and_subindices = set()
+    
+    leaf_values_by_index_and_subindex = {}
+
+    # Find all required indices
+    for key, value, depth in zip(keys, values, depths):
+        verkle_indices = get_verkle_indices(key)
+        for i in range(depth):
+            all_indices.add(verkle_indices[:i])
+            all_indices_and_subindices.add((verkle_indices[:i], verkle_indices[i]))
+        leaf_values_by_index_and_subindex[(verkle_indices[:depth - 1], verkle_indices[depth - 1])] = hash([key, value])
+    
+    all_indices = sorted(all_indices)
+    all_indices_and_subindices = sorted(all_indices_and_subindices)
+
+    log_time_if_eligible("   Computed indices", 30, display_times)
+
+    # Step 0: recreate the commitment list sorted by indices
+    commitments_by_index = {index: commitment for index, commitment in zip(all_indices, commitments_sorted_by_index)}
+    commitments_by_index_and_subindex = {index_and_subindex: commitments_by_index[index_and_subindex[0]]
+                                            for index_and_subindex in all_indices_and_subindices}
+
+    subhashes_by_index_and_subindex = {}
+    for index_and_subindex in all_indices_and_subindices:
+        full_subindex = index_and_subindex[0] + (index_and_subindex[1],)
+        if full_subindex in commitments_by_index:
+            subhashes_by_index_and_subindex[index_and_subindex] = hash(commitments_by_index[full_subindex])
+        else:
+            subhashes_by_index_and_subindex[index_and_subindex] = leaf_values_by_index_and_subindex[index_and_subindex]
+    
+    Cs = list(map(lambda x: x[1], sorted(commitments_by_index_and_subindex.items())))
+    
+    indices = list(map(lambda x: x[1], sorted(all_indices_and_subindices)))
+    
+    ys = list(map(lambda x: int.from_bytes(x[1], "little"), sorted(subhashes_by_index_and_subindex.items())))
+
+    log_time_if_eligible("   Recreated commitment lists", 30, display_times)
+
+    return check_kzg_multiproof(Cs, indices, ys, [D_serialized, y, sigma_serialized], display_times)
+
+
+if __name__ == "__main__":
+    if len(sys.argv) > 1:
+        WIDTH_BITS = int(sys.argv[1])
+        WIDTH = 2 ** WIDTH_BITS
+        ROOT_OF_UNITY = pow(PRIMITIVE_ROOT, (MODULUS - 1) // WIDTH, MODULUS)
+        DOMAIN = [pow(ROOT_OF_UNITY, i, MODULUS) for i in range(WIDTH)]
+
+        NUMBER_INITIAL_KEYS = int(sys.argv[2])
+
+        NUMBER_KEYS_PROOF = int(sys.argv[3])
+
+        NUMBER_DELETED_KEYS = 0
+        NUMBER_ADDED_KEYS = 0
+    
+    SETUP = generate_setup(WIDTH, 8927347823478352432985)
+    kzg_utils = KzgUtils(MODULUS, WIDTH, DOMAIN, SETUP, primefield)
+
+
+    # Build a random verkle trie
+    root = {"node_type": "inner", "commitment": blst.G1().mult(0)}
+
+    values = {}
+
+    for i in range(NUMBER_INITIAL_KEYS):
+        key = randint(0, 2**256-1).to_bytes(32, "little")
+        value = randint(0, 2**256-1).to_bytes(32, "little")
+        insert_verkle_node(root, key, value)
+        values[key] = value
+    
+    average_depth = get_average_depth(root)
+        
+    print("Inserted {0} elements for an average depth of {1:.3f}".format(NUMBER_INITIAL_KEYS, average_depth), file=sys.stderr)
+
+    time_a = time()
+    add_node_hash(root)
+    time_b = time()
+
+    print("Computed verkle root in {0:.3f} s".format(time_b - time_a), file=sys.stderr)
+
+    if NUMBER_ADDED_KEYS > 0:
+
+        time_a = time()
+        check_valid_tree(root)
+        time_b = time()
+        
+        print("[Checked tree valid: {0:.3f} s]".format(time_b - time_a), file=sys.stderr)
+
+        time_x = time()
+        for i in range(NUMBER_ADDED_KEYS):
+            key = randint(0, 2**256-1).to_bytes(32, "little")
+            value = randint(0, 2**256-1).to_bytes(32, "little")
+            update_verkle_node(root, key, value)
+            values[key] = value
+        time_y = time()
+            
+        print("Additionally inserted {0} elements in {1:.3f} s".format(NUMBER_ADDED_KEYS, time_y - time_x), file=sys.stderr)
+        print("Keys in tree now: {0}, average depth: {1:.3f}".format(get_total_depth(root)[1], get_average_depth(root)), file=sys.stderr)
+
+        time_a = time()
+        check_valid_tree(root)
+        time_b = time()
+        
+        print("[Checked tree valid: {0:.3f} s]".format(time_b - time_a), file=sys.stderr)
+    
+    if NUMBER_DELETED_KEYS > 0:
+
+        all_keys = list(values.keys())
+        shuffle(all_keys)
+
+        keys_to_delete = all_keys[:NUMBER_DELETED_KEYS]
+
+        time_a = time()
+        for key in keys_to_delete:
+            delete_verkle_node(root, key)
+            del values[key]
+        time_b = time()
+        
+        print("Deleted {0} elements in {1:.3f} s".format(NUMBER_DELETED_KEYS, time_b - time_a), file=sys.stderr)
+        print("Keys in tree now: {0}, average depth: {1:.3f}".format(get_total_depth(root)[1], get_average_depth(root)), file=sys.stderr)
+
+
+        time_a = time()
+        check_valid_tree(root)
+        time_b = time()
+        
+        print("[Checked tree valid: {0:.3f} s]".format(time_b - time_a), file=sys.stderr)
+
+
+    all_keys = list(values.keys())
+    shuffle(all_keys)
+
+    keys_in_proof = all_keys[:NUMBER_KEYS_PROOF]
+
+    time_a = time()
+    proof = make_verkle_proof(root, keys_in_proof)
+    time_b = time()
+    
+    proof_size = get_proof_size(proof)
+    proof_time = time_b - time_a
+    
+    print("Computed proof for {0} keys (size = {1} bytes) in {2:.3f} s".format(NUMBER_KEYS_PROOF, proof_size, time_b - time_a), file=sys.stderr)
+
+    time_a = time()
+    check_verkle_proof(root["commitment"].compress(), keys_in_proof, [values[key] for key in keys_in_proof], proof)
+    time_b = time()
+    check_time = time_b - time_a
+
+    print("Checked proof in {0:.3f} s".format(time_b - time_a), file=sys.stderr)
+
+    print("{0}\t{1}\t{2}\t{3}\t{4}\t{5}\t{6}\t{7}".format(WIDTH_BITS, WIDTH, NUMBER_INITIAL_KEYS, NUMBER_KEYS_PROOF, average_depth, proof_size, proof_time, check_time))
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