def example_ckks_basics():
print("Example: CKKS Basics");
#In this example we demonstrate evaluating a polynomial function
#
# PI*x^3 + 0.4*x + 1
#
#on encrypted floating-point input data x for a set of 4096 equidistant points
#in the interval [0, 1]. This example demonstrates many of the main features
#of the CKKS scheme, but also the challenges in using it.
#
# We start by setting up the CKKS scheme.
parms = EncryptionParameters(scheme_type.CKKS)
#We saw in `2_encoders.cpp' that multiplication in CKKS causes scales
#in ciphertexts to grow. The scale of any ciphertext must not get too close
#to the total size of coeff_modulus, or else the ciphertext simply runs out of
#room to store the scaled-up plaintext. The CKKS scheme provides a `rescale'
#functionality that can reduce the scale, and stabilize the scale expansion.
#
#Rescaling is a kind of modulus switch operation (recall `3_levels.cpp').
#As modulus switching, it removes the last of the primes from coeff_modulus,
#but as a side-effect it scales down the ciphertext by the removed prime.
#Usually we want to have perfect control over how the scales are changed,
#which is why for the CKKS scheme it is more common to use carefully selected
#primes for the coeff_modulus.
#
#More precisely, suppose that the scale in a CKKS ciphertext is S, and the
#last prime in the current coeff_modulus (for the ciphertext) is P. Rescaling
#to the next level changes the scale to S/P, and removes the prime P from the
#coeff_modulus, as usual in modulus switching. The number of primes limits
#how many rescalings can be done, and thus limits the multiplicative depth of
#the computation.
#
#It is possible to choose the initial scale freely. One good strategy can be
#to is to set the initial scale S and primes P_i in the coeff_modulus to be
#very close to each other. If ciphertexts have scale S before multiplication,
#they have scale S^2 after multiplication, and S^2/P_i after rescaling. If all
#P_i are close to S, then S^2/P_i is close to S again. This way we stabilize the
#scales to be close to S throughout the computation. Generally, for a circuit
#of depth D, we need to rescale D times, i.e., we need to be able to remove D
#primes from the coefficient modulus. Once we have only one prime left in the
#coeff_modulus, the remaining prime must be larger than S by a few bits to
#preserve the pre-decimal-point value of the plaintext.
#
#Therefore, a generally good strategy is to choose parameters for the CKKS
#scheme as follows:
#
# (1) Choose a 60-bit prime as the first prime in coeff_modulus. This will
# give the highest precision when decrypting;
# (2) Choose another 60-bit prime as the last element of coeff_modulus, as
# this will be used as the special prime and should be as large as the
# largest of the other primes;
# (3) Choose the intermediate primes to be close to each other.
#
#We use CoeffModulus::Create to generate primes of the appropriate size. Note
#that our coeff_modulus is 200 bits total, which is below the bound for our
#poly_modulus_degree: CoeffModulus::MaxBitCount(8192) returns 218.
poly_modulus_degree = 8192
parms.set_poly_modulus_degree(poly_modulus_degree)
parms.set_coeff_modulus(CoeffModulus.Create(
poly_modulus_degree, IntVector([60, 40, 40, 60])))
#We choose the initial scale to be 2^40. At the last level, this leaves us
#60-40=20 bits of precision before the decimal point, and enough (roughly
#10-20 bits) of precision after the decimal point. Since our intermediate
#primes are 40 bits (in fact, they are very close to 2^40), we can achieve
#scale stabilization as described above.
scale = 2.0**40
context = SEALContext.Create(parms)
print_parameters(context)
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
relin_keys = keygen.relin_keys()
encryptor = Encryptor(context, public_key)
evaluator = Evaluator(context)
decryptor = Decryptor(context, secret_key)
encoder = CKKSEncoder(context)
slot_count = encoder.slot_count()
print("Number of slots: {}".format(slot_count))
step_size = 1.0 / (slot_count - 1)
input = DoubleVector(list(map(lambda x: x*step_size, range(0, slot_count))))
print("Input vector: ")
print_vector(input)
print("Evaluating polynomial PI*x^3 + 0.4x + 1 ...")
#We create plaintexts for PI, 0.4, and 1 using an overload of CKKSEncoder::encode
#that encodes the given floating-point value to every slot in the vector.
plain_coeff3 = Plaintext()
plain_coeff1 = Plaintext()
plain_coeff0 = Plaintext()
encoder.encode(3.14159265, scale, plain_coeff3)
encoder.encode(0.4, scale, plain_coeff1)
encoder.encode(1.0, scale, plain_coeff0)
x_plain = Plaintext()
print("Encode input vectors.")
encoder.encode(input, scale, x_plain)
x1_encrypted = Ciphertext()
encryptor.encrypt(x_plain, x1_encrypted)
#To compute x^3 we first compute x^2 and relinearize. However, the scale has
#now grown to 2^80.
x3_encrypted = Ciphertext()
print("Compute x^2 and relinearize:")
evaluator.square(x1_encrypted, x3_encrypted)
evaluator.relinearize_inplace(x3_encrypted, relin_keys)
print(" + Scale of x^2 before rescale: {} bits".format(log2(x3_encrypted.scale())))
#Now rescale; in addition to a modulus switch, the scale is reduced down by
#a factor equal to the prime that was switched away (40-bit prime). Hence, the
#new scale should be close to 2^40. Note, however, that the scale is not equal
#to 2^40: this is because the 40-bit prime is only close to 2^40.
print("Rescale x^2.")
evaluator.rescale_to_next_inplace(x3_encrypted)
print(" + Scale of x^2 after rescale: {} bits".format(log2(x3_encrypted.scale())))
#Now x3_encrypted is at a different level than x1_encrypted, which prevents us
#from multiplying them to compute x^3. We could simply switch x1_encrypted to
#the next parameters in the modulus switching chain. However, since we still
#need to multiply the x^3 term with PI (plain_coeff3), we instead compute PI*x
#first and multiply that with x^2 to obtain PI*x^3. To this end, we compute
#PI*x and rescale it back from scale 2^80 to something close to 2^40.
print("Compute and rescale PI*x.")
x1_encrypted_coeff3 = Ciphertext()
evaluator.multiply_plain(x1_encrypted, plain_coeff3, x1_encrypted_coeff3)
print(" + Scale of PI*x before rescale: {} bits".format(log2(x1_encrypted_coeff3.scale())))
evaluator.rescale_to_next_inplace(x1_encrypted_coeff3)
print(" + Scale of PI*x after rescale: {} bits".format(log2(x1_encrypted_coeff3.scale())))
#Since x3_encrypted and x1_encrypted_coeff3 have the same exact scale and use
#the same encryption parameters, we can multiply them together. We write the
#result to x3_encrypted, relinearize, and rescale. Note that again the scale
#is something close to 2^40, but not exactly 2^40 due to yet another scaling
#by a prime. We are down to the last level in the modulus switching chain.
print("Compute, relinearize, and rescale (PI*x)*x^2.")
evaluator.multiply_inplace(x3_encrypted, x1_encrypted_coeff3)
evaluator.relinearize_inplace(x3_encrypted, relin_keys)
print(" + Scale of PI*x^3 before rescale: {} bits".format(log2(x3_encrypted.scale())))
evaluator.rescale_to_next_inplace(x3_encrypted)
print(" + Scale of PI*x^3 after rescale: {} bits".format(log2(x3_encrypted.scale())))
#Next we compute the degree one term. All this requires is one multiply_plain
#with plain_coeff1. We overwrite x1_encrypted with the result.
print("Compute and rescale 0.4*x.")
evaluator.multiply_plain_inplace(x1_encrypted, plain_coeff1)
print(" + Scale of 0.4*x before rescale: {} bits".format(log2(x1_encrypted.scale())))
evaluator.rescale_to_next_inplace(x1_encrypted)
print(" + Scale of 0.4*x after rescale: {} bits".format(log2(x1_encrypted.scale())))
#Now we would hope to compute the sum of all three terms. However, there is
#a serious problem: the encryption parameters used by all three terms are
#different due to modulus switching from rescaling.
#
#Encrypted addition and subtraction require that the scales of the inputs are
#the same, and also that the encryption parameters (parms_id) match. If there
#is a mismatch, Evaluator will throw an exception.
print("Parameters used by all three terms are different.")
print(" + Modulus chain index for x3_encrypted: {}".format(
context.get_context_data(x3_encrypted.parms_id()).chain_index()))
print(" + Modulus chain index for x1_encrypted: {}".format(
context.get_context_data(x1_encrypted.parms_id()).chain_index()))
print(" + Modulus chain index for plain_coeff0: {}".format(
context.get_context_data(plain_coeff0.parms_id()).chain_index()))
#Let us carefully consider what the scales are at this point. We denote the
#primes in coeff_modulus as P_0, P_1, P_2, P_3, in this order. P_3 is used as
#the special modulus and is not involved in rescalings. After the computations
#above the scales in ciphertexts are:
#
# - Product x^2 has scale 2^80 and is at level 2;
# - Product PI*x has scale 2^80 and is at level 2;
# - We rescaled both down to scale 2^80/P_2 and level 1;
# - Product PI*x^3 has scale (2^80/P_2)^2;
# - We rescaled it down to scale (2^80/P_2)^2/P_1 and level 0;
# - Product 0.4*x has scale 2^80;
# - We rescaled it down to scale 2^80/P_2 and level 1;
# - The contant term 1 has scale 2^40 and is at level 2.
#
#Although the scales of all three terms are approximately 2^40, their exact
#values are different, hence they cannot be added together.
print("The exact scales of all three terms are different:")
print(" + Exact scale in PI*x^3: {0:0.10f}".format(x3_encrypted.scale()))
print(" + Exact scale in 0.4*x: {0:0.10f}".format(x1_encrypted.scale()))
print(" + Exact scale in 1: {0:0.10f}".format(plain_coeff0.scale()))
#There are many ways to fix this problem. Since P_2 and P_1 are really close
#to 2^40, we can simply "lie" to Microsoft SEAL and set the scales to be the
#same. For example, changing the scale of PI*x^3 to 2^40 simply means that we
#scale the value of PI*x^3 by 2^120/(P_2^2*P_1), which is very close to 1.
#This should not result in any noticeable error.
#
#Another option would be to encode 1 with scale 2^80/P_2, do a multiply_plain
#with 0.4*x, and finally rescale. In this case we would need to additionally
#make sure to encode 1 with appropriate encryption parameters (parms_id).
#
#In this example we will use the first (simplest) approach and simply change
#the scale of PI*x^3 and 0.4*x to 2^40.
print("Normalize scales to 2^40.")
x3_encrypted.set_scale(2.0**40)
x1_encrypted.set_scale(2.0**40)
#We still have a problem with mismatching encryption parameters. This is easy
#to fix by using traditional modulus switching (no rescaling). CKKS supports
#modulus switching just like the BFV scheme, allowing us to switch away parts
#of the coefficient modulus when it is simply not needed.
print("Normalize encryption parameters to the lowest level.")
last_parms_id = x3_encrypted.parms_id()
evaluator.mod_switch_to_inplace(x1_encrypted, last_parms_id)
evaluator.mod_switch_to_inplace(plain_coeff0, last_parms_id)
#All three ciphertexts are now compatible and can be added.
print("Compute PI*x^3 + 0.4*x + 1.")
encrypted_result = Ciphertext()
evaluator.add(x3_encrypted, x1_encrypted, encrypted_result)
evaluator.add_plain_inplace(encrypted_result, plain_coeff0)
#First print the true result.
plain_result = Plaintext()
print("Decrypt and decode PI*x^3 + 0.4x + 1.")
print(" + Expected result:")
true_result = DoubleVector(list(map(lambda x: (3.14159265 * x * x + 0.4)* x + 1, input)))
print_vector(true_result)
#Decrypt, decode, and print the result.
decryptor.decrypt(encrypted_result, plain_result)
result = DoubleVector()
encoder.decode(plain_result, result)
print(" + Computed result ...... Correct.")
print_vector(result)
class SealOps:
@classmethod
def with_env(cls):
parms = EncryptionParameters(scheme_type.CKKS)
parms.set_poly_modulus_degree(POLY_MODULUS_DEGREE)
parms.set_coeff_modulus(
CoeffModulus.Create(POLY_MODULUS_DEGREE, PRIME_SIZE_LIST))
context = SEALContext.Create(parms)
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
relin_keys = keygen.relin_keys()
galois_keys = keygen.galois_keys()
return cls(context=context,
public_key=public_key,
secret_key=secret_key,
relin_keys=relin_keys,
galois_keys=galois_keys,
poly_modulus_degree=POLY_MODULUS_DEGREE,
scale=SCALE)
def __init__(self,
context: SEALContext,
scale: float,
poly_modulus_degree: int,
public_key: PublicKey = None,
secret_key: SecretKey = None,
relin_keys: RelinKeys = None,
galois_keys: GaloisKeys = None):
self.scale = scale
self.context = context
self.encoder = CKKSEncoder(context)
self.evaluator = Evaluator(context)
self.encryptor = Encryptor(context, public_key)
self.decryptor = Decryptor(context, secret_key)
self.relin_keys = relin_keys
self.galois_keys = galois_keys
self.poly_modulus_degree_log = np.log2(poly_modulus_degree)
def encrypt(self, matrix: np.array):
matrix = Matrix.from_numpy_array(array=matrix)
cipher_matrix = CipherMatrix(rows=matrix.rows, cols=matrix.cols)
for i in range(matrix.rows):
encoded_row = Plaintext()
self.encoder.encode(matrix[i], self.scale, encoded_row)
self.encryptor.encrypt(encoded_row, cipher_matrix[i])
return cipher_matrix
def decrypt(self, cipher_matrix: CipherMatrix) -> Matrix:
matrix = Matrix(rows=cipher_matrix.rows, cols=cipher_matrix.cols)
for i in range(matrix.rows):
row = Vector()
encoded_row = Plaintext()
self.decryptor.decrypt(cipher_matrix[i], encoded_row)
self.encoder.decode(encoded_row, row)
matrix[i] = row
return matrix
def add(self, matrix_a: CipherMatrix,
matrix_b: CipherMatrix) -> CipherMatrix:
self.validate_same_dimension(matrix_a, matrix_b)
result_matrix = CipherMatrix(rows=matrix_a.rows, cols=matrix_a.cols)
for i in range(matrix_a.rows):
a_tag, b_tag = self.get_matched_scale_vectors(
matrix_a[i], matrix_b[i])
self.evaluator.add(a_tag, b_tag, result_matrix[i])
return result_matrix
def add_plain(self, matrix_a: CipherMatrix,
matrix_b: np.array) -> CipherMatrix:
matrix_b = Matrix.from_numpy_array(matrix_b)
self.validate_same_dimension(matrix_a, matrix_b)
result_matrix = CipherMatrix(rows=matrix_a.rows, cols=matrix_a.cols)
for i in range(matrix_a.rows):
row = matrix_b[i]
encoded_row = Plaintext()
self.encoder.encode(row, self.scale, encoded_row)
self.evaluator.mod_switch_to_inplace(encoded_row,
matrix_a[i].parms_id())
self.evaluator.add_plain(matrix_a[i], encoded_row,
result_matrix[i])
return result_matrix
def multiply_plain(self, matrix_a: CipherMatrix,
matrix_b: np.array) -> CipherMatrix:
matrix_b = Matrix.from_numpy_array(matrix_b)
self.validate_same_dimension(matrix_a, matrix_b)
result_matrix = CipherMatrix(rows=matrix_a.rows, cols=matrix_a.cols)
for i in range(matrix_a.rows):
row = matrix_b[i]
encoded_row = Plaintext()
self.encoder.encode(row, self.scale, encoded_row)
self.evaluator.mod_switch_to_inplace(encoded_row,
matrix_a[i].parms_id())
self.evaluator.multiply_plain(matrix_a[i], encoded_row,
result_matrix[i])
return result_matrix
def dot_vector(self, a: Ciphertext, b: Ciphertext) -> Ciphertext:
result = Ciphertext()
self.evaluator.multiply(a, b, result)
self.evaluator.relinearize_inplace(result, self.relin_keys)
self.vector_sum_inplace(result)
self.get_vector_first_element(result)
self.evaluator.rescale_to_next_inplace(result)
return result
def dot_vector_with_plain(self, a: Ciphertext,
b: DoubleVector) -> Ciphertext:
result = Ciphertext()
b_plain = Plaintext()
self.encoder.encode(b, self.scale, b_plain)
self.evaluator.multiply_plain(a, b_plain, result)
self.vector_sum_inplace(result)
self.get_vector_first_element(result)
self.evaluator.rescale_to_next_inplace(result)
return result
def get_vector_range(self, vector_a: Ciphertext, i: int,
j: int) -> Ciphertext:
cipher_range = Ciphertext()
one_and_zeros = DoubleVector([0.0 if x < i else 1.0 for x in range(j)])
plain = Plaintext()
self.encoder.encode(one_and_zeros, self.scale, plain)
self.evaluator.mod_switch_to_inplace(plain, vector_a.parms_id())
self.evaluator.multiply_plain(vector_a, plain, cipher_range)
return cipher_range
def dot_matrix_with_matrix_transpose(self, matrix_a: CipherMatrix,
matrix_b: CipherMatrix):
result_matrix = CipherMatrix(rows=matrix_a.rows, cols=matrix_a.cols)
rows_a = matrix_a.rows
cols_b = matrix_b.rows
for i in range(rows_a):
vector_dot_products = []
zeros = Plaintext()
for j in range(cols_b):
vector_dot_products += [
self.dot_vector(matrix_a[i], matrix_b[j])
]
if j == 0:
zero = DoubleVector()
self.encoder.encode(zero, vector_dot_products[j].scale(),
zeros)
self.evaluator.mod_switch_to_inplace(
zeros, vector_dot_products[j].parms_id())
self.evaluator.add_plain(vector_dot_products[j], zeros,
result_matrix[i])
else:
self.evaluator.rotate_vector_inplace(
vector_dot_products[j], -j, self.galois_keys)
self.evaluator.add_inplace(result_matrix[i],
vector_dot_products[j])
for vec in result_matrix:
self.evaluator.relinearize_inplace(vec, self.relin_keys)
self.evaluator.rescale_to_next_inplace(vec)
return result_matrix
def dot_matrix_with_plain_matrix_transpose(self, matrix_a: CipherMatrix,
matrix_b: np.array):
matrix_b = Matrix.from_numpy_array(matrix_b)
result_matrix = CipherMatrix(rows=matrix_a.rows, cols=matrix_a.cols)
rows_a = matrix_a.rows
cols_b = matrix_b.rows
for i in range(rows_a):
vector_dot_products = []
zeros = Plaintext()
for j in range(cols_b):
vector_dot_products += [
self.dot_vector_with_plain(matrix_a[i], matrix_b[j])
]
if j == 0:
zero = DoubleVector()
self.encoder.encode(zero, vector_dot_products[j].scale(),
zeros)
self.evaluator.mod_switch_to_inplace(
zeros, vector_dot_products[j].parms_id())
self.evaluator.add_plain(vector_dot_products[j], zeros,
result_matrix[i])
else:
self.evaluator.rotate_vector_inplace(
vector_dot_products[j], -j, self.galois_keys)
self.evaluator.add_inplace(result_matrix[i],
vector_dot_products[j])
for vec in result_matrix:
self.evaluator.relinearize_inplace(vec, self.relin_keys)
self.evaluator.rescale_to_next_inplace(vec)
return result_matrix
@staticmethod
def validate_same_dimension(matrix_a, matrix_b):
if matrix_a.rows != matrix_b.rows or matrix_a.cols != matrix_b.cols:
raise ArithmeticError("Matrices aren't of the same dimension")
def vector_sum_inplace(self, cipher: Ciphertext):
rotated = Ciphertext()
for i in range(int(self.poly_modulus_degree_log - 1)):
self.evaluator.rotate_vector(cipher, pow(2, i), self.galois_keys,
rotated)
self.evaluator.add_inplace(cipher, rotated)
def get_vector_first_element(self, cipher: Ciphertext):
one_and_zeros = DoubleVector([1.0])
plain = Plaintext()
self.encoder.encode(one_and_zeros, self.scale, plain)
self.evaluator.multiply_plain_inplace(cipher, plain)
def get_matched_scale_vectors(self, a: Ciphertext,
b: Ciphertext) -> (Ciphertext, Ciphertext):
a_tag = Ciphertext(a)
b_tag = Ciphertext(b)
a_index = self.context.get_context_data(a.parms_id()).chain_index()
b_index = self.context.get_context_data(b.parms_id()).chain_index()
# Changing the mod if required, else just setting the scale
if a_index < b_index:
self.evaluator.mod_switch_to_inplace(b_tag, a.parms_id())
elif a_index > b_index:
self.evaluator.mod_switch_to_inplace(a_tag, b.parms_id())
a_tag.set_scale(self.scale)
b_tag.set_scale(self.scale)
return a_tag, b_tag