path: root/modexpng_fpga_model.py

#!/usr/bin/python3
#
#
# ModExpNG core math model.
#
#
# Copyright (c) 2019, NORDUnet A/S
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are
# met:
# - Redistributions of source code must retain the above copyright notice,
#   this list of conditions and the following disclaimer.
#
# - Redistributions in binary form must reproduce the above copyright
#   notice, this list of conditions and the following disclaimer in the
#   documentation and/or other materials provided with the distribution.
#
# - Neither the name of the NORDUnet nor the names of its contributors may
#   be used to endorse or promote products derived from this software
#   without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
# IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
# TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
# PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
# HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
# TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
# PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
# LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
#


# -------
# Imports
#--------

import sys
import importlib


# --------------
# Model Settings
# --------------

# length of public key
KEY_LENGTH = 1024

# how many parallel multipliers to use
NUM_MULTS  = 8


# ---------------
# Internal Values
# ---------------

# half of key length
_KEY_LENGTH_HALF = KEY_LENGTH // 2

# width of internal math pipeline
_WORD_WIDTH = 16

# folder with test vector scripts
_VECTOR_PATH = "/vector"

# name of test vector class
_VECTOR_CLASS = "Vector"


# ------------------
# Debugging Settings
# ------------------
DUMP_VECTORS = False
DUMP_INDICES = False
DUMP_MACS_CLEARING = False
DUMP_MACS_ACCUMULATION = True


#
# Multi-Precision Integer
#
class ModExpNG_Operand():

    def __init__(self, number, length, words = None):

        if words is None:

            # length must be divisible by word width
            if (length % _WORD_WIDTH) > 0:
                raise Exception("Bad number length!")

            self._init_from_number(number, length)

        else:

            # length must match words count
            if len(words) != length:
                raise Exception("Bad words count!")

            self._init_from_words(words, length)

    def format_verilog_concat(self, name):

        for i in range(len(self.words)):
            if i > 0:
                if (i % 4) == 0: print("")
                else:            print(" ", end='')
            print("%s[%2d] = 17'h%05x;" % (name, i, self.words[i]), end='')
        print("")

    def _init_from_words(self, words, count):

        for i in range(count):

            # word must not exceed 17 bits
            if words[i] >= (2 ** (_WORD_WIDTH + 1)):
                raise Exception("Word is too large!")

        self.words = words

    def _init_from_number(self, number, length):

        num_hexchars_per_word = _WORD_WIDTH // 4
        num_hexchars_total = length // num_hexchars_per_word

        value_hex = format(number, 'x')

        # value must not be larger than specified, but it can be smaller, so
        # we may need to prepend it with zeroes
        if len(value_hex) > num_hexchars_total:
            raise Exception("Number is too large!")
        else:
            while len(value_hex) < num_hexchars_total:
                value_hex = "0" + value_hex

        # create empty list
        self.words = list()

        # fill in words
        while len(value_hex) > 0:
            value_hex_part = value_hex[-num_hexchars_per_word:]
            value_hex = value_hex[:-num_hexchars_per_word]
            self.words.append(int(value_hex_part, 16))

    def number(self):
        ret = 0
        shift = 0
        for word in self.words:
            ret += word << shift
            shift += _WORD_WIDTH
        return ret


#
# Test Vector
#
class ModExpNG_TestVector():

    def __init__(self):

        # format target filename
        filename = "vector_" + str(KEY_LENGTH) + "_randomized"

        # add ./vector to import search path
        sys.path.insert(1, sys.path[0] + _VECTOR_PATH)

        # import from filename
        vector_module = importlib.import_module(filename)

        # get vector class
        vector_class = getattr(vector_module, _VECTOR_CLASS)

        # instantiate vector class
        vector_inst = vector_class()

        # obtain parts of vector
        self.m        = ModExpNG_Operand(vector_inst.m,         KEY_LENGTH)
        self.n        = ModExpNG_Operand(vector_inst.n,         KEY_LENGTH)
        self.d        = ModExpNG_Operand(vector_inst.d,         KEY_LENGTH)
        self.p        = ModExpNG_Operand(vector_inst.p,        _KEY_LENGTH_HALF)
        self.q        = ModExpNG_Operand(vector_inst.q,        _KEY_LENGTH_HALF)
        self.dp       = ModExpNG_Operand(vector_inst.dp,       _KEY_LENGTH_HALF)
        self.dq       = ModExpNG_Operand(vector_inst.dq,       _KEY_LENGTH_HALF)
        self.qinv     = ModExpNG_Operand(vector_inst.qinv,     _KEY_LENGTH_HALF)
        self.n_factor = ModExpNG_Operand(vector_inst.n_factor,  KEY_LENGTH)
        self.p_factor = ModExpNG_Operand(vector_inst.p_factor, _KEY_LENGTH_HALF)
        self.q_factor = ModExpNG_Operand(vector_inst.q_factor, _KEY_LENGTH_HALF)
        self.n_coeff  = ModExpNG_Operand(vector_inst.n_coeff,   KEY_LENGTH      + _WORD_WIDTH)
        self.p_coeff  = ModExpNG_Operand(vector_inst.p_coeff,  _KEY_LENGTH_HALF + _WORD_WIDTH)
        self.q_coeff  = ModExpNG_Operand(vector_inst.q_coeff,  _KEY_LENGTH_HALF + _WORD_WIDTH)
        self.x        = ModExpNG_Operand(vector_inst.x,         KEY_LENGTH)
        self.y        = ModExpNG_Operand(vector_inst.y,         KEY_LENGTH)


class ModExpNG_PartRecombinator():

    def _bit_select(self, x, msb, lsb):
        y = 0
        for pos in range(lsb, msb+1):
            y |= (x & (1 << pos)) >> lsb
        return y

    def _flush_pipeline(self):
        self.z0, self.y0, self.x0 = 0, 0, 0

    def _push_pipeline(self, part):

        # split next part into 16-bit words
        z = self._bit_select(part, 47, 32)
        y = self._bit_select(part, 31, 16)
        x = self._bit_select(part, 15,  0)

        # shift to the right
        z1 = z
        y1 = y + self.z0
        x1 = x + self.y0 + (self.x0 >> 16) # IMPORTANT: This carry can be up to two bits wide!!

        # save lower 16 bits of the rightmost cell
        t = self.x0 & 0xffff

        # update internal latches
        self.z0, self.y0, self.x0 = z1, y1, x1

        # done
        return t

    def recombine_square(self, parts, ab_num_words):

        # empty result so far
        words = list()

        # flush recombinator pipeline
        self._flush_pipeline()

        # the first tick produces null result, the last part produces
        # two words, so we need (2*n - 1) + 2 = 2*n + 1 ticks total
        # and should only save the result word during the last 2 * n ticks
        for i in range(2 * ab_num_words + 1):

            next_part = parts[i] if i < (2 * ab_num_words - 1) else 0
            next_word = self._push_pipeline(next_part)

            if i > 0:
                words.append(next_word)

        return words

    def recombine_triangle(self, parts, ab_num_words):

        # empty result so far
        words = list()

        # flush recombinator pipeline
        self._flush_pipeline()

        # the first tick produces null result, so we need n + 1 + 1 = n + 2
        # ticks total and should only save the result word during the last n ticks
        for i in range(ab_num_words + 2):

            next_part = parts[i] if i < (ab_num_words + 1) else 0
            next_word = self._push_pipeline(next_part)

            if i > 0:
                words.append(next_word)

        return words

    def recombine_rectangle(self, parts, ab_num_words):

        # empty result so far
        words = list()

        # flush recombinator pipeline
        self._flush_pipeline()

        # the first tick produces null result, the last part produces
        # two words, so we need 2 * n + 2 ticks total and should only save
        # the result word during the last 2 * n + 1 ticks
        for i in range(2 * ab_num_words + 2):

            next_part = parts[i] if i < (2 * ab_num_words) else 0
            next_word = self._push_pipeline(next_part)

            if i > 0:
                words.append(next_word)

        return words


class ModExpNG_WordMultiplier():

    def __init__(self):

        self._macs = list()
        self._indices = list()

        self._mac_aux = list()
        self._index_aux = list()

        for x in range(NUM_MULTS):
            self._macs.append(0)
            self._indices.append(0)

        self._mac_aux.append(0)
        self._index_aux.append(0)

    def _clear_all_macs(self):
        for x in range(NUM_MULTS):
            self._macs[x] = 0

    def _clear_one_mac(self, x):
        self._macs[x] = 0

    def _clear_mac_aux(self):
        self._mac_aux[0] = 0

    def _update_one_mac(self, x, value):
        self._macs[x] += value

    def _update_mac_aux(self, value):
        self._mac_aux[0] += value

    def _preset_indices(self, col):
        for x in range(len(self._indices)):
            self._indices[x] = col * len(self._indices) + x

    def _preset_index_aux(self, num_cols):
        self._index_aux[0] = num_cols * len(self._indices)

    def _rotate_indices(self, num_words):
        for x in range(len(self._indices)):
            if self._indices[x] > 0:
                self._indices[x] -= 1
            else:
                self._indices[x] = num_words - 1

    def _rotate_index_aux(self):
        self._index_aux[0] -= 1

    def multiply_square(self, a_wide, b_narrow, ab_num_words, dump=False):

        if dump: print("multiply_square()")

        num_cols = ab_num_words // NUM_MULTS

        parts = list()
        for i in range(2 * ab_num_words - 1):
            parts.append(0)

        for col in range(num_cols):

            for t in range(ab_num_words):

                if t == 0: self._preset_indices(col)    
                else:      self._rotate_indices(ab_num_words)

                if t == 0:
                    self._clear_all_macs()
                    if dump and DUMP_MACS_CLEARING:
                        print("t= 0, col=%2d > clear > all" % (col))
                else:
                    t1 = t - 1
                    if (t1 // 8) == col:
                        self._clear_one_mac(t1 % NUM_MULTS)
                        if dump and DUMP_MACS_CLEARING:
                            print("t=%2d, col=%2d > clear > x=%d:" % (t, col, t1 % NUM_MULTS))


                if dump and DUMP_INDICES:
                    print("t=%2d, col=%2d > indices:" % (t, col), end='')
                    for i in range(NUM_MULTS):
                        print(" %2d" % self._indices[i], end='')
                    print("")

                # current b-word
                bt = b_narrow.words[t]

                # multiply by a-words
                for x in range(NUM_MULTS):
                    ax = a_wide.words[self._indices[x]]
                    self._update_one_mac(x, ax * bt)

                    if t == (col * NUM_MULTS + x):
                        parts[t] = self._macs[x]

                if dump and DUMP_MACS_ACCUMULATION:
                    print("t=%2d, col=%2d > "% (t, col), end='')
                    for i in range(NUM_MULTS):
                        if i > 0: print(" | ", end='')
                        print("mac[%d]: 0x%012x" % (i, self._macs[i]), end='')
                    print("")

                # save the uppers part of product at end of column,
                # for the last column don't save the very last part
                if t == (ab_num_words - 1):
                    for x in range(NUM_MULTS):
                        if not (col == (num_cols - 1) and x == (NUM_MULTS - 1)):
                            parts[ab_num_words + col * NUM_MULTS + x] = self._macs[x]

        return parts

    def multiply_triangle(self, a_wide, b_narrow, ab_num_words):

        num_cols = ab_num_words // NUM_MULTS

        parts = list()
        for i in range(ab_num_words + 1):
            parts.append(0)

        for col in range(num_cols):

            last_col = col == (num_cols - 1)

            self._clear_all_macs()
            self._preset_indices(col)

            if last_col:
                self._clear_mac_aux()
                self._preset_index_aux(num_cols)

            for t in range(ab_num_words + 1):

                # current b-word
                bt = b_narrow.words[t]

                # multiply by a-words
                for x in range(NUM_MULTS):
                    ax = a_wide.words[self._indices[x]]
                    self._update_one_mac(x, ax * bt)

                    if t == (col * NUM_MULTS + x):
                        parts[t] = self._macs[x]

                # aux multiplier
                if last_col:
                    ax = a_wide.words[self._index_aux[0]]
                    self._update_mac_aux(ax * bt)

                    if t == ab_num_words:
                        parts[t] = self._mac_aux[0]

                # shortcut
                if not last_col:
                    if t == (NUM_MULTS * (col + 1) - 1): break

                # advance indices
                self._rotate_indices(ab_num_words)
                if last_col:
                    self._rotate_index_aux()

        return parts

    def multiply_rectangle(self, a_wide, b_narrow, ab_num_words):

        num_cols = ab_num_words // NUM_MULTS

        parts = list()
        for i in range(2 * ab_num_words):
            parts.append(0)

        for col in range(num_cols):

            self._clear_all_macs()
            self._preset_indices(col)

            for t in range(ab_num_words+1):

                # current b-word
                bt = b_narrow.words[t]

                # multiply by a-words
                for x in range(NUM_MULTS):
                    ax = a_wide.words[self._indices[x]]
                    self._update_one_mac(x, ax * bt)

                    # don't save one value for the very last time instant per column
                    if t < ab_num_words and t == (col * NUM_MULTS + x):
                        parts[t] = self._macs[x]
                        self._clear_one_mac(x)

                # save the uppers part of product at end of column
                if t == ab_num_words:
                    for x in range(NUM_MULTS):
                        parts[ab_num_words + col * NUM_MULTS + x] = self._macs[x]

                self._rotate_indices(ab_num_words)

        return parts


class ModExpNG_LowlevelOperator():

    def __init__(self):
        self._word_mask = 0
        for x in range(_WORD_WIDTH):
            self._word_mask |= (1 << x)

    def _check_word(self, a):
        if a < 0 or a >= (2 ** _WORD_WIDTH):
            raise Exception("Word out of range!")

    def _check_carry_borrow(self, cb):
        if cb < 0 or cb > 1:
            raise Exception("Carry or borrow out of range!")

    def add_words(self, a, b, c_in):

        self._check_word(a)
        self._check_word(b)
        self._check_carry_borrow(c_in)

        sum = a + b + c_in

        sum_s = sum & self._word_mask
        sum_c = (sum >> _WORD_WIDTH) & 1

        return (sum_c, sum_s)

    def sub_words(self, a, b, b_in):
        self._check_word(a)
        self._check_word(b)
        self._check_carry_borrow(b_in)

        dif = a - b - b_in

        if dif < 0:
            dif_b = 1
            dif_d = dif + 2 ** _WORD_WIDTH
        else:
            dif_b = 0
            dif_d = dif

        return (dif_b, dif_d)


class ModExpNG_Worker():

    def __init__(self):
        self.recombinator = ModExpNG_PartRecombinator()
        self.multiplier   = ModExpNG_WordMultiplier()
        self.lowlevel     = ModExpNG_LowlevelOperator()

    def exponentiate(self, iz, bz, e, n, n_factor, n_coeff, num_words):

        # working variables
        t1, t2 = iz, bz

        # length-1, length-2, length-3, ..., 1, 0 (left-to-right)
        for bit in range(_WORD_WIDTH * num_words - 1, -1, -1):

            if e.number() & (1 << bit):
                p1 = self.multiply(t1, t2, n, n_coeff, num_words)
                p2 = self.multiply(t2, t2, n, n_coeff, num_words)
            else:
                p1 = self.multiply(t1, t1, n, n_coeff, num_words)
                p2 = self.multiply(t2, t1, n, n_coeff, num_words)

            t1, t2 = p1, p2

            if (bit % 8) == 0:
                pct = float((_WORD_WIDTH * num_words - bit) / (_WORD_WIDTH * num_words)) * 100.0
                print("\rpct: %5.1f%%" % pct, end='')

        print("")

        return t1

    def subtract(self, a, b, n, ab_num_words):

        c_in = 0
        b_in = 0

        ab = list()
        ab_n = list()

        for x in range(ab_num_words):

            a_word = a.words[x]
            b_word = b.words[x]

            (b_out, d_out) = self.lowlevel.sub_words(a_word, b_word, b_in)
            (c_out, s_out) = self.lowlevel.add_words(d_out, n.words[x], c_in)

            ab.append(d_out)
            ab_n.append(s_out)

            (c_in, b_in) = (c_out, b_out)

        d = ab if not b_out else ab_n

        return ModExpNG_Operand(None, ab_num_words, d)

    def add(self, a, b, ab_num_words):

        c_in = 0

        ab = list()

        for x in range(2 * ab_num_words):

            a_word = a.words[x] if x < ab_num_words else 0
            b_word = b.words[x]

            (c_out, s_out) = self.lowlevel.add_words(a_word, b_word, c_in)

            ab.append(s_out)

            c_in = c_out

        return ModExpNG_Operand(None, 2*ab_num_words, ab)

    def multiply(self, a, b, n, n_coeff, ab_num_words, reduce_only=False, multiply_only=False, dump=False):

        if dump and DUMP_VECTORS:
            print("num_words = %d" % ab_num_words)
            a.format_verilog_concat("A")
            b.format_verilog_concat("B")
            n.format_verilog_concat("N")
            n_coeff.format_verilog_concat("N_COEFF")

        # 1.
        if reduce_only:
            ab = a
        else:
            ab_parts = self.multiplier.multiply_square(a, b, ab_num_words, dump)
            ab_words = self.recombinator.recombine_square(ab_parts, ab_num_words)
            ab = ModExpNG_Operand(None, 2 * ab_num_words, ab_words)

        if multiply_only:
            return ModExpNG_Operand(None, 2*ab_num_words, ab_words)

        # 2.
        q_parts = self.multiplier.multiply_triangle(ab, n_coeff, ab_num_words)
        q_words = self.recombinator.recombine_triangle(q_parts, ab_num_words)
        q = ModExpNG_Operand(None, ab_num_words + 1, q_words)

        # 3.
        m_parts = self.multiplier.multiply_rectangle(n, q, ab_num_words)
        m_words = self.recombinator.recombine_rectangle(m_parts, ab_num_words)
        m = ModExpNG_Operand(None, 2 * ab_num_words + 1, m_words)

        # 4.
        r_xwords = list()
        for i in range(2*ab_num_words):
            r_xwords.append(ab.words[i] + m.words[i])

        r_xwords.append(m.words[2 * ab_num_words])

        cy = 0
        for i in range(ab_num_words+1):
            s = r_xwords[i] + cy
            cy = s >> 16

        R = list()
        for i in range(ab_num_words):
            R.append(0)

        R[0] += cy # !!!

        for i in range(ab_num_words):
            R[i] += r_xwords[ab_num_words + i + 1]

        return ModExpNG_Operand(None, ab_num_words, R)

    def reduce(self, a):
        carry = 0
        for x in range(len(a.words)):
            a.words[x] += carry
            carry = (a.words[x] >> _WORD_WIDTH) & 1
            a.words[x] &= self.lowlevel._word_mask


if __name__ == "__main__":

    # load test vector
    # create worker
    # set numbers of words
    # obtain known good reference value with built-in math
    # create helper quantity
    # mutate blinding quantities with built-in math

    vector = ModExpNG_TestVector()
    worker = ModExpNG_Worker()

    n_num_words  = KEY_LENGTH  // _WORD_WIDTH
    pq_num_words = n_num_words // 2

    s_known  = pow(vector.m.number(), vector.d.number(), vector.n.number())

    i = ModExpNG_Operand(1, KEY_LENGTH)

    x_mutated_known = pow(vector.x.number(), 2, vector.n.number())
    y_mutated_known = pow(vector.y.number(), 2, vector.n.number())

    # bring one into Montgomery domain (glue 2**r to one)
    # bring blinding coefficients into Montgomery domain (glue 2**(2*r) to x and y)
    # blind message
    # convert message to non-redundant representation
    # first reduce message, this glues 2**-r to the message as a side effect
    # unglue 2**-r from message by gluing 2**r to it to compensate
    # bring message into Montgomery domain (glue 2**r to message)
    # do "easier" exponentiations
    # return "easier" parts from Montgomery domain (unglue 2**r from result)
    # do the "Garner's formula" part
    #  r = sp - sq mod p
    #  sr_qinv = sr * qinv mod p
    #  q_sr_qinv = q * sr_qinv
    #  s_crt = sq + q_sr_qinv
    # unblind s
    # mutate blinding factors
    ip_factor                    = worker.multiply(i,                            vector.p_factor,  vector.p, vector.p_coeff, pq_num_words)
    iq_factor                    = worker.multiply(i,                            vector.q_factor,  vector.q, vector.q_coeff, pq_num_words)

    x_factor                     = worker.multiply(vector.x,                     vector.n_factor,  vector.n, vector.n_coeff, n_num_words)
    y_factor                     = worker.multiply(vector.y,                     vector.n_factor,  vector.n, vector.n_coeff, n_num_words)

    m_blind                      = worker.multiply(vector.m,                     y_factor,         vector.n, vector.n_coeff, n_num_words)

    worker.reduce(m_blind)

    mp_blind_inverse_factor      = worker.multiply(m_blind,                      None,             vector.p, vector.p_coeff, pq_num_words, reduce_only=True)
    mq_blind_inverse_factor      = worker.multiply(m_blind,                      None,             vector.q, vector.q_coeff, pq_num_words, reduce_only=True)

    mp_blind                     = worker.multiply(mp_blind_inverse_factor,      vector.p_factor,  vector.p, vector.p_coeff, pq_num_words)
    mq_blind                     = worker.multiply(mq_blind_inverse_factor,      vector.q_factor,  vector.q, vector.q_coeff, pq_num_words)

    mp_blind_factor              = worker.multiply(mp_blind,                     vector.p_factor,  vector.p, vector.p_coeff, pq_num_words, dump=True)
    mq_blind_factor              = worker.multiply(mq_blind,                     vector.q_factor,  vector.q, vector.q_coeff, pq_num_words)

    sp_blind_factor              = worker.exponentiate(ip_factor, mp_blind_factor, vector.dp, vector.p, vector.p_factor, vector.p_coeff, pq_num_words)
    sq_blind_factor              = worker.exponentiate(iq_factor, mq_blind_factor, vector.dq, vector.q, vector.q_factor, vector.q_coeff, pq_num_words)

    sp_blind                     = worker.multiply(i,                            sp_blind_factor,  vector.p, vector.p_coeff, pq_num_words)
    sq_blind                     = worker.multiply(i,                            sq_blind_factor,  vector.q, vector.q_coeff, pq_num_words)

    sr_blind                     = worker.subtract(sp_blind, sq_blind, vector.p, pq_num_words)

    sr_qinv_blind_inverse_factor = worker.multiply(sr_blind,                     vector.qinv,      vector.p, vector.p_coeff, pq_num_words)
    sr_qinv_blind                = worker.multiply(sr_qinv_blind_inverse_factor, vector.p_factor,  vector.p, vector.p_coeff, pq_num_words)
    q_sr_qinv_blind              = worker.multiply(vector.q,                     sr_qinv_blind,    None,     None,           pq_num_words, multiply_only=True)

    s_crt_blinded                = worker.add(sq_blind, q_sr_qinv_blind, pq_num_words)

    s_crt_unblinded              = worker.multiply(s_crt_blinded,                x_factor,         vector.n, vector.n_coeff, n_num_words)

    x_mutated_factor             = worker.multiply(x_factor,                     x_factor,         vector.n, vector.n_coeff, n_num_words)
    y_mutated_factor             = worker.multiply(y_factor,                     y_factor,         vector.n, vector.n_coeff, n_num_words)

    x_mutated                    = worker.multiply(i,                            x_mutated_factor, vector.n, vector.n_coeff, n_num_words)
    y_mutated                    = worker.multiply(i,                            y_mutated_factor, vector.n, vector.n_coeff, n_num_words)

    worker.reduce(s_crt_unblinded)
    worker.reduce(x_mutated)
    worker.reduce(y_mutated)

    # check
    if s_crt_unblinded.number() != s_known:   print("ERROR: s_crt_unblinded != s_known!")
    else:                                     print("s is OK")

    if x_mutated.number() != x_mutated_known: print("ERROR: x_mutated != x_mutated_known!")
    else:                                     print("x_mutated is OK")

    if y_mutated.number() != y_mutated_known: print("ERROR: y_mutated != y_mutated_known!")
    else:                                     print("y_mutated is OK")


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