def decrypt(item):
decryptor = simplefhe._decryptor
if simplefhe._private_key is None:
raise ValueError(
'Private key has not been set. Decryption not possible.')
if simplefhe._relin_keys is None:
raise ValueError(
'Relinearization keys have not been set. Decryption not possible.')
result = Plaintext()
decryptor.decrypt(item._ciphertext, result)
mode = item._mode
if mode['type'] == 'int':
if decryptor.invariant_noise_budget(item._ciphertext) == 0:
raise ValueError(
'The noise budget has been exhausted.' +
' Try calling `simplefhe.initialize` with a larger `poly_modulus_degree` or a smaller `max_int`.'
)
result = result.to_string()
result = int(result, 16)
if result > mode['modulus'] // 2:
result -= mode['modulus']
return result
else:
decoded = DoubleVector()
item._mode['encoder'].decode(result, decoded)
return float(decoded[0])
def encryption(value):
# IntegerEncoder with base 2
encoder = IntegerEncoder(context.plain_modulus())
# generate public/private keys
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
# encrypts public key
encryptor = Encryptor(context, public_key)
# perform computations on ciphertexts
evaluator = Evaluator(context)
# decrypts secret key
decryptor = Decryptor(context, secret_key)
# perform encryptions
plaintext = encoder.encode(value)
# convert into encrypted ciphertext
encrypt = Ciphertext()
encryptor.encrypt(plaintext, encrypt)
print("Encryption successful!")
print("Encrypted ciphertext: " + (str)(value) + " as " +
plaintext.to_string())
# noise budget of fresh encryptions
print("Noise budget: " + (str)(decryptor.invariant_noise_budget(encrypt)) +
" bits")
# decrypts result
result = Plaintext()
decryptor.decrypt(encrypt, result)
print("Decryption successful!")
print("Plaintext: " + result.to_string())
# decode for original integer
print("Original node: " + (str)(encoder.decode_int32(result)) + "\n")
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
encryptor = Encryptor(context, public_key)
evaluator = Evaluator(context)
decryptor = Decryptor(context, secret_key)
value=7
plain1 = encoder.encode(value1)
print("Encoded " + (str)(value) + " as polynomial " + plain1.to_string() + " (plain1)")
encrypted _data= Ciphertext()
encryptor.encrypt(plain, encrypted_data)
print("Noise budget in encrypted1: " + (str)(decryptor.invariant_noise_budget(encrypted_data)) + " bits")
# operations that can be performed --->
# result stored in encrypted1 data
evaluator.negate(encrypted1_data)
# result stored in encrypted1 data, encrpyted1 is modified
evaluator.add(encrypted1_data, encrypted2_data)
# result stored in encrypted1 data, encrpyted1 is modified
evaluator.multiply(encrypted1_data, encrypted2_data)
plain_result = Plaintext()
decryptor.decrypt(encrypted_data, plain_result)
print("Plaintext polynomial: " + plain_result.to_string())
print("Decoded integer: " + (str)(encoder.decode_int32(plain_result)))
def pickle_ciphertext():
parms = EncryptionParameters()
parms.set_poly_modulus("1x^2048 + 1")
parms.set_coeff_modulus(seal.coeff_modulus_128(2048))
parms.set_plain_modulus(1 << 8)
context = SEALContext(parms)
# Print the parameters that we have chosen
print_parameters(context);
encoder = IntegerEncoder(context.plain_modulus())
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
# To be able to encrypt, we need to construct an instance of Encryptor. Note that
# the Encryptor only requires the public key.
encryptor = Encryptor(context, public_key)
# Computations on the ciphertexts are performed with the Evaluator class.
evaluator = Evaluator(context)
# We will of course want to decrypt our results to verify that everything worked,
# so we need to also construct an instance of Decryptor. Note that the Decryptor
# requires the secret key.
decryptor = Decryptor(context, secret_key)
# We start by encoding two integers as plaintext polynomials.
value1 = 5;
plain1 = encoder.encode(value1);
print("Encoded " + (str)(value1) + " as polynomial " + plain1.to_string() + " (plain1)")
value2 = -7;
plain2 = encoder.encode(value2);
print("Encoded " + (str)(value2) + " as polynomial " + plain2.to_string() + " (plain2)")
# Encrypting the values is easy.
encrypted1 = Ciphertext()
encrypted2 = Ciphertext()
print("Encrypting plain1: ", encrypted1)
encryptor.encrypt(plain1, encrypted1)
print("Done (encrypted1)", encrypted1)
print("Encrypting plain2: ")
encryptor.encrypt(plain2, encrypted2)
print("Done (encrypted2)")
# output = open('ciphertest.pkl', 'wb')
# dill.dumps(encrypted_save, output)
# output.close()
# encrypted1 = dill.load(open('ciphertest.pkl', 'rb'))
output = open('session.pkl', 'wb')
dill.dump_session('session.pkl')
del encrypted1
sill.load_session('session.pkl')
# To illustrate the concept of noise budget, we print the budgets in the fresh
# encryptions.
print("Noise budget in encrypted1: " + (str)(decryptor.invariant_noise_budget(encrypted1)) + " bits")
print("Noise budget in encrypted2: " + (str)(decryptor.invariant_noise_budget(encrypted2)) + " bits")
# As a simple example, we compute (-encrypted1 + encrypted2) * encrypted2.
# Negation is a unary operation.
evaluator.negate(encrypted1)
# Negation does not consume any noise budget.
print("Noise budget in -encrypted1: " + (str)(decryptor.invariant_noise_budget(encrypted1)) + " bits")
# Addition can be done in-place (overwriting the first argument with the result,
# or alternatively a three-argument overload with a separate destination variable
# can be used. The in-place variants are always more efficient. Here we overwrite
# encrypted1 with the sum.
evaluator.add(encrypted1, encrypted2)
# It is instructive to think that addition sets the noise budget to the minimum
# of the input noise budgets. In this case both inputs had roughly the same
# budget going on, and the output (in encrypted1) has just slightly lower budget.
# Depending on probabilistic effects, the noise growth consumption may or may
# not be visible when measured in whole bits.
print("Noise budget in -encrypted1 + encrypted2: " + (str)(decryptor.invariant_noise_budget(encrypted1)) + " bits")
# Finally multiply with encrypted2. Again, we use the in-place version of the
# function, overwriting encrypted1 with the product.
evaluator.multiply(encrypted1, encrypted2)
# Multiplication consumes a lot of noise budget. This is clearly seen in the
# print-out. The user can change the plain_modulus to see its effect on the
# rate of noise budget consumption.
print("Noise budget in (-encrypted1 + encrypted2) * encrypted2: " + (str)(
decryptor.invariant_noise_budget(encrypted1)) + " bits")
# Now we decrypt and decode our result.
plain_result = Plaintext()
print("Decrypting result: ")
decryptor.decrypt(encrypted1, plain_result)
print("Done")
# Print the result plaintext polynomial.
print("Plaintext polynomial: " + plain_result.to_string())
# Decode to obtain an integer result.
print("Decoded integer: " + (str)(encoder.decode_int32(plain_result)))
def example_bfv_basics():
print("Example: BFV Basics")
#In this example, we demonstrate performing simple computations (a polynomial
#evaluation) on encrypted integers using the BFV encryption scheme.
#
#The first task is to set up an instance of the EncryptionParameters class.
#It is critical to understand how the different parameters behave, how they
#affect the encryption scheme, performance, and the security level. There are
#three encryption parameters that are necessary to set:
#
# - poly_modulus_degree (degree of polynomial modulus);
# - coeff_modulus ([ciphertext] coefficient modulus);
# - plain_modulus (plaintext modulus; only for the BFV scheme).
#
#The BFV scheme cannot perform arbitrary computations on encrypted data.
#Instead, each ciphertext has a specific quantity called the `invariant noise
#budget' -- or `noise budget' for short -- measured in bits. The noise budget
#in a freshly encrypted ciphertext (initial noise budget) is determined by
#the encryption parameters. Homomorphic operations consume the noise budget
#at a rate also determined by the encryption parameters. In BFV the two basic
#operations allowed on encrypted data are additions and multiplications, of
#which additions can generally be thought of as being nearly free in terms of
#noise budget consumption compared to multiplications. Since noise budget
#consumption compounds in sequential multiplications, the most significant
#factor in choosing appropriate encryption parameters is the multiplicative
#depth of the arithmetic circuit that the user wants to evaluate on encrypted
#data. Once the noise budget of a ciphertext reaches zero it becomes too
#corrupted to be decrypted. Thus, it is essential to choose the parameters to
#be large enough to support the desired computation; otherwise the result is
#impossible to make sense of even with the secret key.
parms = EncryptionParameters(scheme_type.BFV)
#The first parameter we set is the degree of the `polynomial modulus'. This
#must be a positive power of 2, representing the degree of a power-of-two
#cyclotomic polynomial; it is not necessary to understand what this means.
#
#Larger poly_modulus_degree makes ciphertext sizes larger and all operations
#slower, but enables more complicated encrypted computations. Recommended
#values are 1024, 2048, 4096, 8192, 16384, 32768, but it is also possible
#to go beyond this range.
#
#In this example we use a relatively small polynomial modulus. Anything
#smaller than this will enable only very restricted encrypted computations.
poly_modulus_degree = 4096
parms.set_poly_modulus_degree(poly_modulus_degree)
#Next we set the [ciphertext] `coefficient modulus' (coeff_modulus). This
#parameter is a large integer, which is a product of distinct prime numbers,
#each up to 60 bits in size. It is represented as a vector of these prime
#numbers, each represented by an instance of the SmallModulus class. The
#bit-length of coeff_modulus means the sum of the bit-lengths of its prime
#factors.
#
#A larger coeff_modulus implies a larger noise budget, hence more encrypted
#computation capabilities. However, an upper bound for the total bit-length
#of the coeff_modulus is determined by the poly_modulus_degree, as follows:
#
# +----------------------------------------------------+
# | poly_modulus_degree | max coeff_modulus bit-length |
# +---------------------+------------------------------+
# | 1024 | 27 |
# | 2048 | 54 |
# | 4096 | 109 |
# | 8192 | 218 |
# | 16384 | 438 |
# | 32768 | 881 |
# +---------------------+------------------------------+
#
#These numbers can also be found in native/src/seal/util/hestdparms.h encoded
#in the function SEAL_HE_STD_PARMS_128_TC, and can also be obtained from the
#function
#
# CoeffModulus::MaxBitCount(poly_modulus_degree).
#
#For example, if poly_modulus_degree is 4096, the coeff_modulus could consist
#of three 36-bit primes (108 bits).
#
#Microsoft SEAL comes with helper functions for selecting the coeff_modulus.
#For new users the easiest way is to simply use
#
# CoeffModulus::BFVDefault(poly_modulus_degree),
#
#which returns std::vector<SmallModulus> consisting of a generally good choice
#for the given poly_modulus_degree.
parms.set_coeff_modulus(CoeffModulus.BFVDefault(poly_modulus_degree))
#The plaintext modulus can be any positive integer, even though here we take
#it to be a power of two. In fact, in many cases one might instead want it
#to be a prime number; we will see this in later examples. The plaintext
#modulus determines the size of the plaintext data type and the consumption
#of noise budget in multiplications. Thus, it is essential to try to keep the
#plaintext data type as small as possible for best performance. The noise
#budget in a freshly encrypted ciphertext is
#
# ~ log2(coeff_modulus/plain_modulus) (bits)
#
#and the noise budget consumption in a homomorphic multiplication is of the
#form log2(plain_modulus) + (other terms).
#
#The plaintext modulus is specific to the BFV scheme, and cannot be set when
#using the CKKS scheme.
parms.set_plain_modulus(1024)
#Now that all parameters are set, we are ready to construct a SEALContext
#object. This is a heavy class that checks the validity and properties of the
#parameters we just set.
context = SEALContext.Create(parms)
#Print the parameters that we have chosen.
print("Set encryption parameters and print")
print_parameters(context)
print("~~~~~~ A naive way to calculate 4(x^2+1)(x+1)^2. ~~~~~~")
#The encryption schemes in Microsoft SEAL are public key encryption schemes.
#For users unfamiliar with this terminology, a public key encryption scheme
#has a separate public key for encrypting data, and a separate secret key for
#decrypting data. This way multiple parties can encrypt data using the same
#shared public key, but only the proper recipient of the data can decrypt it
#with the secret key.
#
#We are now ready to generate the secret and public keys. For this purpose
#we need an instance of the KeyGenerator class. Constructing a KeyGenerator
#automatically generates the public and secret key, which can immediately be
#read to local variables.
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
#To be able to encrypt we need to construct an instance of Encryptor. Note
#that the Encryptor only requires the public key, as expected.
encryptor = Encryptor(context, public_key)
#Computations on the ciphertexts are performed with the Evaluator class. In
#a real use-case the Evaluator would not be constructed by the same party
#that holds the secret key.
evaluator = Evaluator(context)
#We will of course want to decrypt our results to verify that everything worked,
#so we need to also construct an instance of Decryptor. Note that the Decryptor
#requires the secret key.
decryptor = Decryptor(context, secret_key)
#As an example, we evaluate the degree 4 polynomial
#
# 4x^4 + 8x^3 + 8x^2 + 8x + 4
#
#over an encrypted x = 6. The coefficients of the polynomial can be considered
#as plaintext inputs, as we will see below. The computation is done modulo the
#plain_modulus 1024.
#
#While this examples is simple and easy to understand, it does not have much
#practical value. In later examples we will demonstrate how to compute more
#efficiently on encrypted integers and real or complex numbers.
#
#Plaintexts in the BFV scheme are polynomials of degree less than the degree
#of the polynomial modulus, and coefficients integers modulo the plaintext
#modulus. For readers with background in ring theory, the plaintext space is
#the polynomial quotient ring Z_T[X]/(X^N + 1), where N is poly_modulus_degree
#and T is plain_modulus.
#
#To get started, we create a plaintext containing the constant 6. For the
#plaintext element we use a constructor that takes the desired polynomial as
#a string with coefficients represented as hexadecimal numbers.
x = 6
x_plain = Plaintext(str(x))
print("Express x = {} as a plaintext polynomial 0x{}.".format(
x, x_plain.to_string()))
#We then encrypt the plaintext, producing a ciphertext.
x_encrypted = Ciphertext()
print("Encrypt x_plain to x_encrypted.")
encryptor.encrypt(x_plain, x_encrypted)
#In Microsoft SEAL, a valid ciphertext consists of two or more polynomials
#whose coefficients are integers modulo the product of the primes in the
#coeff_modulus. The number of polynomials in a ciphertext is called its `size'
#and is given by Ciphertext::size(). A freshly encrypted ciphertext always
#has size 2.
print(" + size of freshly encrypted x: {}".format(x_encrypted.size()))
#There is plenty of noise budget left in this freshly encrypted ciphertext.
print(" + noise budget in freshly encrypted x: {} bits".format(
decryptor.invariant_noise_budget(x_encrypted)))
#We decrypt the ciphertext and print the resulting plaintext in order to
#demonstrate correctness of the encryption.
x_decrypted = Plaintext()
decryptor.decrypt(x_encrypted, x_decrypted)
print(" + decryption of x_encrypted: 0x{} ...... Correct.".format(
x_decrypted.to_string()))
#When using Microsoft SEAL, it is typically advantageous to compute in a way
#that minimizes the longest chain of sequential multiplications. In other
#words, encrypted computations are best evaluated in a way that minimizes
#the multiplicative depth of the computation, because the total noise budget
#consumption is proportional to the multiplicative depth. For example, for
#our example computation it is advantageous to factorize the polynomial as
#
# 4x^4 + 8x^3 + 8x^2 + 8x + 4 = 4(x + 1)^2 * (x^2 + 1)
#
#to obtain a simple depth 2 representation. Thus, we compute (x + 1)^2 and
#(x^2 + 1) separately, before multiplying them, and multiplying by 4.
#
#First, we compute x^2 and add a plaintext "1". We can clearly see from the
#print-out that multiplication has consumed a lot of noise budget. The user
#can vary the plain_modulus parameter to see its effect on the rate of noise
#budget consumption.
print("Compute x_sq_plus_one (x^2+1).")
x_sq_plus_one = Ciphertext()
evaluator.square(x_encrypted, x_sq_plus_one)
plain_one = Plaintext("1")
evaluator.add_plain_inplace(x_sq_plus_one, plain_one)
#Encrypted multiplication results in the output ciphertext growing in size.
#More precisely, if the input ciphertexts have size M and N, then the output
#ciphertext after homomorphic multiplication will have size M+N-1. In this
#case we perform a squaring, and observe both size growth and noise budget
#consumption.
print(" + size of x_sq_plus_one: {}".format(x_sq_plus_one.size()))
print(" + noise budget in x_sq_plus_one: {} bits".format(
decryptor.invariant_noise_budget(x_sq_plus_one)))
#Even though the size has grown, decryption works as usual as long as noise
#budget has not reached 0.
decrypted_result = Plaintext()
decryptor.decrypt(x_sq_plus_one, decrypted_result)
print(" + decryption of x_sq_plus_one: 0x{} ...... Correct.".format(
decrypted_result.to_string()))
#Next, we compute (x + 1)^2.
print("Compute x_plus_one_sq ((x+1)^2).")
x_plus_one_sq = Ciphertext()
evaluator.add_plain(x_encrypted, plain_one, x_plus_one_sq)
evaluator.square_inplace(x_plus_one_sq)
print(" + size of x_plus_one_sq: {}".format(x_plus_one_sq.size()))
print(" + noise budget in x_plus_one_sq: {} bits".format(
decryptor.invariant_noise_budget(x_plus_one_sq)))
decryptor.decrypt(x_plus_one_sq, decrypted_result)
print(" + decryption of x_plus_one_sq: 0x{} ...... Correct.".format(
decrypted_result.to_string()))
#Finally, we multiply (x^2 + 1) * (x + 1)^2 * 4.
print("Compute encrypted_result (4(x^2+1)(x+1)^2).")
encrypted_result = Ciphertext()
plain_four = Plaintext("4")
evaluator.multiply_plain_inplace(x_sq_plus_one, plain_four)
evaluator.multiply(x_sq_plus_one, x_plus_one_sq, encrypted_result)
print(" + size of encrypted_result: {}".format(encrypted_result.size()))
print(" + noise budget in encrypted_result: {} bits".format(
decryptor.invariant_noise_budget(encrypted_result)))
print("NOTE: Decryption can be incorrect if noise budget is zero.")
print("~~~~~~ A better way to calculate 4(x^2+1)(x+1)^2. ~~~~~~")
#Noise budget has reached 0, which means that decryption cannot be expected
#to give the correct result. This is because both ciphertexts x_sq_plus_one
#and x_plus_one_sq consist of 3 polynomials due to the previous squaring
#operations, and homomorphic operations on large ciphertexts consume much more
#noise budget than computations on small ciphertexts. Computing on smaller
#ciphertexts is also computationally significantly cheaper.
#`Relinearization' is an operation that reduces the size of a ciphertext after
#multiplication back to the initial size, 2. Thus, relinearizing one or both
#input ciphertexts before the next multiplication can have a huge positive
#impact on both noise growth and performance, even though relinearization has
#a significant computational cost itself. It is only possible to relinearize
#size 3 ciphertexts down to size 2, so often the user would want to relinearize
#after each multiplication to keep the ciphertext sizes at 2.
#Relinearization requires special `relinearization keys', which can be thought
#of as a kind of public key. Relinearization keys can easily be created with
#the KeyGenerator.
#Relinearization is used similarly in both the BFV and the CKKS schemes, but
#in this example we continue using BFV. We repeat our computation from before,
#but this time relinearize after every multiplication.
#We use KeyGenerator::relin_keys() to create relinearization keys.
print("Generate relinearization keys.")
relin_keys = keygen.relin_keys()
#We now repeat the computation relinearizing after each multiplication.
print("Compute and relinearize x_squared (x^2),")
print("then compute x_sq_plus_one (x^2+1)")
x_squared = Ciphertext()
evaluator.square(x_encrypted, x_squared)
print(" + size of x_squared: {}".format(x_squared.size()))
evaluator.relinearize_inplace(x_squared, relin_keys)
print(" + size of x_squared (after relinearization): {}".format(
x_squared.size()))
evaluator.add_plain(x_squared, plain_one, x_sq_plus_one)
print(" + noise budget in x_sq_plus_one: {} bits".format(
decryptor.invariant_noise_budget(x_sq_plus_one)))
decryptor.decrypt(x_sq_plus_one, decrypted_result)
print(" + decryption of x_sq_plus_one: 0x{} ...... Correct.".format(
decrypted_result.to_string()))
x_plus_one = Ciphertext()
print("Compute x_plus_one (x+1),")
print("then compute and relinearize x_plus_one_sq ((x+1)^2).")
evaluator.add_plain(x_encrypted, plain_one, x_plus_one)
evaluator.square(x_plus_one, x_plus_one_sq)
print(" + size of x_plus_one_sq: {}".format(x_plus_one_sq.size()))
evaluator.relinearize_inplace(x_plus_one_sq, relin_keys)
print(" + noise budget in x_plus_one_sq: {} bits".format(
decryptor.invariant_noise_budget(x_plus_one_sq)))
decryptor.decrypt(x_plus_one_sq, decrypted_result)
print(" + decryption of x_plus_one_sq: 0x{} ...... Correct.".format(
decrypted_result.to_string()))
print("Compute and relinearize encrypted_result (4(x^2+1)(x+1)^2).")
evaluator.multiply_plain_inplace(x_sq_plus_one, plain_four)
evaluator.multiply(x_sq_plus_one, x_plus_one_sq, encrypted_result)
print(" + size of encrypted_result: {}".format(encrypted_result.size()))
evaluator.relinearize_inplace(encrypted_result, relin_keys)
print(" + size of encrypted_result (after relinearization): {}".format(
encrypted_result.size()))
print(" + noise budget in encrypted_result: {} bits".format(
decryptor.invariant_noise_budget(encrypted_result)))
print("NOTE: Notice the increase in remaining noise budget.")
#Relinearization clearly improved our noise consumption. We have still plenty
#of noise budget left, so we can expect the correct answer when decrypting.
print("Decrypt encrypted_result (4(x^2+1)(x+1)^2).")
decryptor.decrypt(encrypted_result, decrypted_result)
print(" + decryption of 4(x^2+1)(x+1)^2 = 0x{} ...... Correct.".format(
decrypted_result.to_string()))
def example_levels():
print("Example: Levels")
#In this examples we describe the concept of `levels' in BFV and CKKS and the
#related objects that represent them in Microsoft SEAL.
#
#In Microsoft SEAL a set of encryption parameters (excluding the random number
#generator) is identified uniquely by a 256-bit hash of the parameters. This
#hash is called the `parms_id' and can be easily accessed and printed at any
#time. The hash will change as soon as any of the parameters is changed.
#
#When a SEALContext is created from a given EncryptionParameters instance,
#Microsoft SEAL automatically creates a so-called `modulus switching chain',
#which is a chain of other encryption parameters derived from the original set.
#The parameters in the modulus switching chain are the same as the original
#parameters with the exception that size of the coefficient modulus is
#decreasing going down the chain. More precisely, each parameter set in the
#chain attempts to remove the last coefficient modulus prime from the
#previous set; this continues until the parameter set is no longer valid
#(e.g., plain_modulus is larger than the remaining coeff_modulus). It is easy
#to walk through the chain and access all the parameter sets. Additionally,
#each parameter set in the chain has a `chain index' that indicates its
#position in the chain so that the last set has index 0. We say that a set
#of encryption parameters, or an object carrying those encryption parameters,
#is at a higher level in the chain than another set of parameters if its the
#chain index is bigger, i.e., it is earlier in the chain.
#
#Each set of parameters in the chain involves unique pre-computations performed
#when the SEALContext is created, and stored in a SEALContext::ContextData
#object. The chain is basically a linked list of SEALContext::ContextData
#objects, and can easily be accessed through the SEALContext at any time. Each
#node can be identified by the parms_id of its specific encryption parameters
#(poly_modulus_degree remains the same but coeff_modulus varies).
parms = EncryptionParameters(scheme_type.BFV)
poly_modulus_degree = 8192
parms.set_poly_modulus_degree(poly_modulus_degree)
#In this example we use a custom coeff_modulus, consisting of 5 primes of
#sizes 50, 30, 30, 50, and 50 bits. Note that this is still OK according to
#the explanation in `1_bfv_basics.cpp'. Indeed,
#
# CoeffModulus::MaxBitCount(poly_modulus_degree)
#
#returns 218 (greater than 50+30+30+50+50=210).
#
#Due to the modulus switching chain, the order of the 5 primes is significant.
#The last prime has a special meaning and we call it the `special prime'. Thus,
#the first parameter set in the modulus switching chain is the only one that
#involves the special prime. All key objects, such as SecretKey, are created
#at this highest level. All data objects, such as Ciphertext, can be only at
#lower levels. The special prime should be as large as the largest of the
#other primes in the coeff_modulus, although this is not a strict requirement.
#
# special prime +---------+
# |
# v
#coeff_modulus: { 50, 30, 30, 50, 50 } +---+ Level 4 (all keys; `key level')
# |
# |
# coeff_modulus: { 50, 30, 30, 50 } +---+ Level 3 (highest `data level')
# |
# |
# coeff_modulus: { 50, 30, 30 } +---+ Level 2
# |
# |
# coeff_modulus: { 50, 30 } +---+ Level 1
# |
# |
# coeff_modulus: { 50 } +---+ Level 0 (lowest level)
parms.set_coeff_modulus(
CoeffModulus.Create(poly_modulus_degree,
IntVector([50, 30, 30, 50, 50])))
#In this example the plain_modulus does not play much of a role; we choose
#some reasonable value.
parms.set_plain_modulus(PlainModulus.Batching(poly_modulus_degree, 20))
context = SEALContext.Create(parms)
print_parameters(context)
#There are convenience method for accessing the SEALContext::ContextData for
#some of the most important levels:
#
# SEALContext::key_context_data(): access to key level ContextData
# SEALContext::first_context_data(): access to highest data level ContextData
# SEALContext::last_context_data(): access to lowest level ContextData
#
#We iterate over the chain and print the parms_id for each set of parameters.
print("Print the modulus switching chain.")
#First print the key level parameter information.
context_data = context.key_context_data()
print("----> Level (chain index): {} ...... key_context_data()".format(
context_data.chain_index()))
print(" parms_id: {}".format(list2hex(context_data.parms_id())))
print("coeff_modulus primes:", end=' ')
for prime in context_data.parms().coeff_modulus():
print(hex(prime.value()), end=' ')
print("")
print("\\")
print(" \\-->")
#Next iterate over the remaining (data) levels.
context_data = context.first_context_data()
while (context_data):
print(" Level (chain index): {} ".format(context_data.chain_index()),
end='')
if context_data.parms_id() == context.first_parms_id():
print(" ...... first_context_data()")
elif context_data.parms_id() == context.last_parms_id():
print(" ...... last_context_data()")
else:
print("")
print(" parms_id: {}".format(list2hex(context_data.parms_id())))
print("coeff_modulus primes:", end=' ')
for prime in context_data.parms().coeff_modulus():
print(hex(prime.value()), end=' ')
print("")
print("\\")
print(" \\-->")
#Step forward in the chain.
context_data = context_data.next_context_data()
print(" End of chain reached")
#We create some keys and check that indeed they appear at the highest level.
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
relin_keys = keygen.relin_keys()
galois_keys = keygen.galois_keys()
print("Print the parameter IDs of generated elements.")
print(" + public_key: {}".format(list2hex(public_key.parms_id())))
print(" + secret_key: {}".format(list2hex(secret_key.parms_id())))
print(" + relin_keys: {}".format(list2hex(relin_keys.parms_id())))
print(" + galois_keys: {}".format(list2hex(galois_keys.parms_id())))
encryptor = Encryptor(context, public_key)
evaluator = Evaluator(context)
decryptor = Decryptor(context, secret_key)
#In the BFV scheme plaintexts do not carry a parms_id, but ciphertexts do. Note
#how the freshly encrypted ciphertext is at the highest data level.
plain = Plaintext("1x^3 + 2x^2 + 3x^1 + 4")
encrypted = Ciphertext()
encryptor.encrypt(plain, encrypted)
print(" + plain: {} (not set in BFV)".format(
list2hex(plain.parms_id())))
print(" + encrypted: {}".format(list2hex(encrypted.parms_id())))
#`Modulus switching' is a technique of changing the ciphertext parameters down
#in the chain. The function Evaluator::mod_switch_to_next always switches to
#the next level down the chain, whereas Evaluator::mod_switch_to switches to
#a parameter set down the chain corresponding to a given parms_id. However, it
#is impossible to switch up in the chain.
print("Perform modulus switching on encrypted and print.")
context_data = context.first_context_data()
print("---->")
while (context_data.next_context_data()):
print(" Level (chain index): {} ".format(context_data.chain_index()))
print(" parms_id of encrypted: {}".format(
list2hex(encrypted.parms_id())))
print(" Noise budget at this level: {} bits".format(
decryptor.invariant_noise_budget(encrypted)))
print("\\")
print(" \\-->")
evaluator.mod_switch_to_next_inplace(encrypted)
context_data = context_data.next_context_data()
print(" Level (chain index): {}".format(context_data.chain_index()))
print(" parms_id of encrypted: {}".format(encrypted.parms_id()))
print(" Noise budget at this level: {} bits".format(
decryptor.invariant_noise_budget(encrypted)))
print("\\")
print(" \\-->")
print(" End of chain reached")
#At this point it is hard to see any benefit in doing this: we lost a huge
#amount of noise budget (i.e., computational power) at each switch and seemed
#to get nothing in return. Decryption still works.
print("Decrypt still works after modulus switching.")
decryptor.decrypt(encrypted, plain)
print(" + Decryption of encrypted: {} ...... Correct.".format(
plain.to_string()))
#However, there is a hidden benefit: the size of the ciphertext depends
#linearly on the number of primes in the coefficient modulus. Thus, if there
#is no need or intention to perform any further computations on a given
#ciphertext, we might as well switch it down to the smallest (last) set of
#parameters in the chain before sending it back to the secret key holder for
#decryption.
#
#Also the lost noise budget is actually not an issue at all, if we do things
#right, as we will see below.
#
#First we recreate the original ciphertext and perform some computations.
print("Computation is more efficient with modulus switching.")
print("Compute the 8th power.")
encryptor.encrypt(plain, encrypted)
print(" + Noise budget fresh: {} bits".format(
decryptor.invariant_noise_budget(encrypted)))
evaluator.square_inplace(encrypted)
evaluator.relinearize_inplace(encrypted, relin_keys)
print(" + Noise budget of the 2nd power: {} bits".format(
decryptor.invariant_noise_budget(encrypted)))
evaluator.square_inplace(encrypted)
evaluator.relinearize_inplace(encrypted, relin_keys)
print(" + Noise budget of the 4th power: {} bits".format(
decryptor.invariant_noise_budget(encrypted)))
#Surprisingly, in this case modulus switching has no effect at all on the
#noise budget.
evaluator.mod_switch_to_next_inplace(encrypted)
print(" + Noise budget after modulus switching: {} bits".format(
decryptor.invariant_noise_budget(encrypted)))
#This means that there is no harm at all in dropping some of the coefficient
#modulus after doing enough computations. In some cases one might want to
#switch to a lower level slightly earlier, actually sacrificing some of the
#noise budget in the process, to gain computational performance from having
#smaller parameters. We see from the print-out that the next modulus switch
#should be done ideally when the noise budget is down to around 25 bits.
evaluator.square_inplace(encrypted)
evaluator.relinearize_inplace(encrypted, relin_keys)
print(" + Noise budget of the 8th power: {} bits".format(
decryptor.invariant_noise_budget(encrypted)))
evaluator.mod_switch_to_next_inplace(encrypted)
print(" + Noise budget after modulus switching: {} bits".format(
decryptor.invariant_noise_budget(encrypted)))
#At this point the ciphertext still decrypts correctly, has very small size,
#and the computation was as efficient as possible. Note that the decryptor
#can be used to decrypt a ciphertext at any level in the modulus switching
#chain.
decryptor.decrypt(encrypted, plain)
print(" + Decryption of the 8th power (hexadecimal) ...... Correct.")
print(" {}".format(plain.to_string()))
#In BFV modulus switching is not necessary and in some cases the user might
#not want to create the modulus switching chain, except for the highest two
#levels. This can be done by passing a bool `false' to SEALContext::Create.
context = SEALContext.Create(parms, False)
#We can check that indeed the modulus switching chain has been created only
#for the highest two levels (key level and highest data level). The following
#loop should execute only once.
print("Optionally disable modulus switching chain expansion.")
print("Print the modulus switching chain.")
print("---->")
context_data = context.key_context_data()
while (context_data):
print(" Level (chain index): {}".format(context_data.chain_index()))
print(" parms_id: {}".format(list2hex(context_data.parms_id())))
print("coeff_modulus primes:", end=' ')
for prime in context_data.parms().coeff_modulus():
print(hex(prime.value()), end=' ')
print("")
print("\\")
print(" \\-->")
context_data = context_data.next_context_data()
print(" End of chain reached")
def example_integer_encoder():
print("Example: Encoders / Integer Encoder")
#[IntegerEncoder] (For BFV scheme only)
#
#The IntegerEncoder encodes integers to BFV plaintext polynomials as follows.
#First, a binary expansion of the integer is computed. Next, a polynomial is
#created with the bits as coefficients. For example, the integer
#
# 26 = 2^4 + 2^3 + 2^1
#
#is encoded as the polynomial 1x^4 + 1x^3 + 1x^1. Conversely, plaintext
#polynomials are decoded by evaluating them at x=2. For negative numbers the
#IntegerEncoder simply stores all coefficients as either 0 or -1, where -1 is
#represented by the unsigned integer plain_modulus - 1 in memory.
#
#Since encrypted computations operate on the polynomials rather than on the
#encoded integers themselves, the polynomial coefficients will grow in the
#course of such computations. For example, computing the sum of the encrypted
#encoded integer 26 with itself will result in an encrypted polynomial with
#larger coefficients: 2x^4 + 2x^3 + 2x^1. Squaring the encrypted encoded
#integer 26 results also in increased coefficients due to cross-terms, namely,
#
# (2x^4 + 2x^3 + 2x^1)^2 = 1x^8 + 2x^7 + 1x^6 + 2x^5 + 2x^4 + 1x^2;
#
#further computations will quickly increase the coefficients much more.
#Decoding will still work correctly in this case (evaluating the polynomial
#at x=2), but since the coefficients of plaintext polynomials are really
#integers modulo plain_modulus, implicit reduction modulo plain_modulus may
#yield unexpected results. For example, adding 1x^4 + 1x^3 + 1x^1 to itself
#plain_modulus many times will result in the constant polynomial 0, which is
#clearly not equal to 26 * plain_modulus. It can be difficult to predict when
#such overflow will take place especially when computing several sequential
#multiplications.
#
#The IntegerEncoder is easy to understand and use for simple computations,
#and can be a good tool to experiment with for users new to Microsoft SEAL.
#However, advanced users will probably prefer more efficient approaches,
#such as the BatchEncoder or the CKKSEncoder.
parms = EncryptionParameters(scheme_type.BFV)
poly_modulus_degree = 4096
parms.set_poly_modulus_degree(poly_modulus_degree)
parms.set_coeff_modulus(CoeffModulus.BFVDefault(poly_modulus_degree))
#There is no hidden logic behind our choice of the plain_modulus. The only
#thing that matters is that the plaintext polynomial coefficients will not
#exceed this value at any point during our computation; otherwise the result
#will be incorrect.
parms.set_plain_modulus(512)
context = SEALContext.Create(parms)
print_parameters(context)
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
encryptor = Encryptor(context, public_key)
evaluator = Evaluator(context)
decryptor = Decryptor(context, secret_key)
#We create an IntegerEncoder.
encoder = IntegerEncoder(context)
#First, we encode two integers as plaintext polynomials. Note that encoding
#is not encryption: at this point nothing is encrypted.
value1 = 5
plain1 = encoder.encode(value1)
print("Encode {} as polynomial {} (plain1), ".format(
value1, plain1.to_string()))
value2 = -7
plain2 = encoder.encode(value2)
print(" encode {} as polynomial {} (plain2)".format(
value2, plain2.to_string()))
#Now we can encrypt the plaintext polynomials.
encrypted1 = Ciphertext()
encrypted2 = Ciphertext()
print("Encrypt plain1 to encrypted1 and plain2 to encrypted2.")
encryptor.encrypt(plain1, encrypted1)
encryptor.encrypt(plain2, encrypted2)
print(" + Noise budget in encrypted1: {} bits".format(
decryptor.invariant_noise_budget(encrypted1)))
print(" + Noise budget in encrypted2: {} bits".format(
decryptor.invariant_noise_budget(encrypted2)))
#As a simple example, we compute (-encrypted1 + encrypted2) * encrypted2.
encryptor.encrypt(plain2, encrypted2)
encrypted_result = Ciphertext()
print(
"Compute encrypted_result = (-encrypted1 + encrypted2) * encrypted2.")
evaluator.negate(encrypted1, encrypted_result)
evaluator.add_inplace(encrypted_result, encrypted2)
evaluator.multiply_inplace(encrypted_result, encrypted2)
print(" + Noise budget in encrypted_result: {} bits".format(
decryptor.invariant_noise_budget(encrypted_result)))
plain_result = Plaintext()
print("Decrypt encrypted_result to plain_result.")
decryptor.decrypt(encrypted_result, plain_result)
#Print the result plaintext polynomial. The coefficients are not even close
#to exceeding our plain_modulus, 512.
print(" + Plaintext polynomial: {}".format(plain_result.to_string()))
#Decode to obtain an integer result.
print("Decode plain_result.")
print(" + Decoded integer: {} ...... Correct.".format(
encoder.decode_int32(plain_result)))
def example_basics_ii():
print_example_banner("Example: Basics II")
# In this example we explain what relinearization is, how to use it, and how
# it affects noise budget consumption.
# First we set the parameters, create a SEALContext, and generate the public
# and secret keys. We use slightly larger parameters than be fore to be able
# to do more homomorphic multiplications.
parms = EncryptionParameters()
parms.set_poly_modulus("1x^8192 + 1")
# The default coefficient modulus consists of the following primes:
# 0x7fffffffba0001,
# 0x7fffffffaa0001,
# 0x7fffffff7e0001,
# 0x3fffffffd60001.
# The total size is 219 bits.
parms.set_coeff_modulus(seal.coeff_modulus_128(8192))
parms.set_plain_modulus(1 << 10)
context = SEALContext(parms)
print_parameters(context)
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
# We also set up an Encryptor, Evaluator, and Decryptor here. We will
# encrypt polynomials directly in this example, so there is no need for
# an encoder.
encryptor = Encryptor(context, public_key)
evaluator = Evaluator(context)
decryptor = Decryptor(context, secret_key)
# There are actually two more types of keys in SEAL: `evaluation keys' and
# `Galois keys'. Here we will discuss evaluation keys, and Galois keys will
# be discussed later in example_batching().
# In SEAL, a valid ciphertext consists of two or more polynomials with
# coefficients integers modulo the product of the primes in coeff_modulus.
# The current size of a ciphertext can be found using Ciphertext::size().
# A freshly encrypted ciphertext always has size 2.
#plain1 = Plaintext("1x^2 + 2x^1 + 3")
plain1 = Plaintext("1x^2 + 2x^1 + 3")
encrypted = Ciphertext()
print("")
print("Encrypting " + plain1.to_string() + ": ")
encryptor.encrypt(plain1, encrypted)
print("Done")
print("Size of a fresh encryption: " + (str)(encrypted.size()))
print("Noise budget in fresh encryption: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
# Homomorphic multiplication results in the output ciphertext growing in size.
# More precisely, if the input ciphertexts have size M and N, then the output
# ciphertext after homomorphic multiplication will have size M+N-1. In this
# case we square encrypted twice to observe this growth (also observe noise
# budget consumption).
evaluator.square(encrypted)
print("Size after squaring: " + (str)(encrypted.size()))
print("Noise budget after squaring: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
plain2 = Plaintext()
decryptor.decrypt(encrypted, plain2)
print("Second power: " + plain2.to_string())
evaluator.square(encrypted)
print("Size after squaring: " + (str)(encrypted.size()))
print("Noise budget after squaring: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
# It does not matter that the size has grown -- decryption works as usual.
# Observe from the print-out that the coefficients in the plaintext have
# grown quite large. One more squaring would cause some of them to wrap
# around plain_modulus (0x400), and as a result we would no longer obtain
# the expected result as an integer-coefficient polynomial. We can fix this
# problem to some extent by increasing plain_modulus. This would make sense,
# since we still have plenty of noise budget left.
plain2 = Plaintext()
decryptor.decrypt(encrypted, plain2)
print("Fourth power: " + plain2.to_string())
# The problem here is that homomorphic operations on large ciphertexts are
# computationally much more costly than on small ciphertexts. Specifically,
# homomorphic multiplication on input ciphertexts of size M and N will require
# O(M*N) polynomial multiplications to be performed, and an addition will
# require O(M+N) additions. Relinearization reduces the size of the ciphertexts
# after multiplication back to the initial size (2). Thus, relinearizing one
# or both inputs before the next multiplication, or e.g. before serializing the
# ciphertexts, can have a huge positive impact on performance.
# Another problem is that the noise budget consumption in multiplication is
# bigger when the input ciphertexts sizes are bigger. In a complicated
# computation the contribution of the sizes to the noise budget consumption
# can actually become the dominant term. We will point this out again below
# once we get to our example.
# Relinearization itself has both a computational cost and a noise budget cost.
# These both depend on a parameter called `decomposition bit count', which can
# be any integer at least 1 [dbc_min()] and at most 60 [dbc_max()]. A large
# decomposition bit count makes relinearization fast, but consumes more noise
# budget. A small decomposition bit count can make relinearization slower, but
# might not change the noise budget by any observable amount.
# Relinearization requires a special type of key called `evaluation keys'.
# These can be created by the KeyGenerator for any decomposition bit count.
# To relinearize a ciphertext of size M >= 2 back to size 2, we actually need
# M-2 evaluation keys. Attempting to relinearize a too large ciphertext with
# too few evaluation keys will result in an exception being thrown.
# We repeat our computation, but this time relinearize after both squarings.
# Since our ciphertext never grows past size 3 (we relinearize after every
# multiplication), it suffices to generate only one evaluation key.
# First, we need to create evaluation keys. We use a decomposition bit count
# of 16 here, which can be thought of as quite small.
ev_keys16 = EvaluationKeys()
# This function generates one single evaluation key. Another overload takes
# the number of keys to be generated as an argument, but one is all we need
# in this example (see above).
keygen.generate_evaluation_keys(16, ev_keys16)
print("")
print("Encrypting " + plain1.to_string() + ": ")
encryptor.encrypt(plain1, encrypted)
print("Done")
print("Size of a fresh encryption: " + (str)(encrypted.size()))
print("Noise budget in fresh encryption: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
evaluator.square(encrypted)
print("Size after squaring: " + (str)(encrypted.size()))
print("Noise budget after squaring: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
evaluator.relinearize(encrypted, ev_keys16)
print("Size after relinearization: " + (str)(encrypted.size()))
print("Noise budget after relinearizing (dbs = " +
(str)(ev_keys16.decomposition_bit_count()) + "): " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
evaluator.square(encrypted)
print("Size after second squaring: " + (str)(encrypted.size()) + " bits")
print("Noise budget after second squaring: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
evaluator.relinearize(encrypted, ev_keys16)
print("Size after relinearization: " + (str)(encrypted.size()))
print("Noise budget after relinearizing (dbs = " +
(str)(ev_keys16.decomposition_bit_count()) + "): " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
decryptor.decrypt(encrypted, plain2)
print("Fourth power: " + plain2.to_string())
# Of course the result is still the same, but this time we actually
# used less of our noise budget. This is not surprising for two reasons:
# - We used a very small decomposition bit count, which is why
# relinearization itself did not consume the noise budget by any
# observable amount;
# - Since our ciphertext sizes remain small throughout the two
# squarings, the noise budget consumption rate in multiplication
# remains as small as possible. Recall from above that operations
# on larger ciphertexts actually cause more noise growth.
# To make matters even more clear, we repeat the computation a third time,
# now using the largest possible decomposition bit count (60). We are not
# measuring the time here, but relinearization with these evaluation keys
# is significantly faster than with ev_keys16.
ev_keys60 = EvaluationKeys()
keygen.generate_evaluation_keys(seal.dbc_max(), ev_keys60)
print("")
print("Encrypting " + plain1.to_string() + ": ")
encryptor.encrypt(plain1, encrypted)
print("Done")
print("Size of a fresh encryption: " + (str)(encrypted.size()))
print("Noise budget in fresh encryption: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
evaluator.square(encrypted)
print("Size after squaring: " + (str)(encrypted.size()))
print("Noise budget after squaring: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
evaluator.relinearize(encrypted, ev_keys60)
print("Size after relinearization: " + (str)(encrypted.size()))
print("Noise budget after relinearizing (dbc = " +
(str)(ev_keys60.decomposition_bit_count()) + "): " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
evaluator.square(encrypted)
print("Size after second squaring: " + (str)(encrypted.size()))
print("Noise budget after second squaring: " +
(str)(decryptor.invariant_noise_budget) + " bits")
evaluator.relinearize(encrypted, ev_keys60)
print("Size after relinearization: " + (str)(encrypted.size()))
print("Noise budget after relinearizing (dbc = " +
(str)(ev_keys60.decomposition_bit_count()) + "): " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
decryptor.decrypt(encrypted, plain2)
print("Fourth power: " + plain2.to_string())
# Observe from the print-out that we have now used significantly more of our
# noise budget than in the two previous runs. This is again not surprising,
# since the first relinearization chops off a huge part of the noise budget.
# However, note that the second relinearization does not change the noise
# budget by any observable amount. This is very important to understand when
# optimal performance is desired: relinearization always drops the noise
# budget from the maximum (freshly encrypted ciphertext) down to a fixed
# amount depending on the encryption parameters and the decomposition bit
# count. On the other hand, homomorphic multiplication always consumes the
# noise budget from its current level. This is why the second relinearization
# does not change the noise budget anymore: it is already consumed past the
# fixed amount determinted by the decomposition bit count and the encryption
# parameters.
# We now perform a third squaring and observe an even further compounded
# decrease in the noise budget. Again, relinearization does not consume the
# noise budget at this point by any observable amount, even with the largest
# possible decomposition bit count.
evaluator.square(encrypted)
print("Size after third squaring " + (str)(encrypted.size()))
print("Noise budget after third squaring: " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
evaluator.relinearize(encrypted, ev_keys60)
print("Size after relinearization: " + (str)(encrypted.size()))
print("Noise budget after relinearizing (dbc = " +
(str)(ev_keys60.decomposition_bit_count()) + "): " +
(str)(decryptor.invariant_noise_budget(encrypted)) + " bits")
decryptor.decrypt(encrypted, plain2)
print("Eighth power: " + plain2.to_string())
def example_basics_i():
print_example_banner("Example: Basics I")
# In this example we demonstrate setting up encryption parameters and other
# relevant objects for performing simple computations on encrypted integers.
# SEAL uses the Fan-Vercauteren (FV) homomorphic encryption scheme. We refer to
# https://eprint.iacr.org/2012/144 for full details on how the FV scheme works.
# For better performance, SEAL implements the "FullRNS" optimization of FV, as
# described in https://eprint.iacr.org/2016/510.
# The first task is to set up an instance of the EncryptionParameters class.
# It is critical to understand how these different parameters behave, how they
# affect the encryption scheme, performance, and the security level. There are
# three encryption parameters that are necessary to set:
# - poly_modulus (polynomial modulus);
# - coeff_modulus ([ciphertext] coefficient modulus);
# - plain_modulus (plaintext modulus).
# A fourth parameter -- noise_standard_deviation -- has a default value of 3.19
# and should not be necessary to modify unless the user has a specific reason
# to and knows what they are doing.
# The encryption scheme implemented in SEAL cannot perform arbitrary computations
# on encrypted data. Instead, each ciphertext has a specific quantity called the
# `invariant noise budget' -- or `noise budget' for short -- measured in bits.
# The noise budget of a freshly encrypted ciphertext (initial noise budget) is
# determined by the encryption parameters. Homomorphic operations consume the
# noise budget at a rate also determined by the encryption parameters. In SEAL
# the two basic homomorphic operations are additions and multiplications, of
# which additions can generally be thought of as being nearly free in terms of
# noise budget consumption compared to multiplications. Since noise budget
# consumption is compounding in sequential multiplications, the most significant
# factor in choosing appropriate encryption parameters is the multiplicative
# depth of the arithmetic circuit that needs to be evaluated. Once the noise
# budget in a ciphertext reaches zero, it becomes too corrupted to be decrypted.
# Thus, it is essential to choose the parameters to be large enough to support
# the desired computation; otherwise the result is impossible to make sense of
# even with the secret key.
parms = EncryptionParameters()
# We first set the polynomial modulus. This must be a power-of-2 cyclotomic
# polynomial, i.e. a polynomial of the form "1x^(power-of-2) + 1". The polynomial
# modulus should be thought of mainly affecting the security level of the scheme;
# larger polynomial modulus makes the scheme more secure. At the same time, it
# makes ciphertext sizes larger, and consequently all operations slower.
# Recommended degrees for poly_modulus are 1024, 2048, 4096, 8192, 16384, 32768,
# but it is also possible to go beyond this. Since we perform only a very small
# computation in this example, it suffices to use a very small polynomial modulus
parms.set_poly_modulus("1x^2048 + 1")
# Next we choose the [ciphertext] coefficient modulus (coeff_modulus). The size
# of the coefficient modulus should be thought of as the most significant factor
# in determining the noise budget in a freshly encrypted ciphertext: bigger means
# more noise budget. Unfortunately, a larger coefficient modulus also lowers the
# security level of the scheme. Thus, if a large noise budget is required for
# complicated computations, a large coefficient modulus needs to be used, and the
# reduction in the security level must be countered by simultaneously increasing
# the polynomial modulus.
# To make parameter selection easier for the user, we have constructed sets of
# largest allowed coefficient moduli for 128-bit and 192-bit security levels
# for different choices of the polynomial modulus. These recommended parameters
# follow the Security white paper at http://HomomorphicEncryption.org. However,
# due to the complexity of this topic, we highly recommend the user to directly
# consult an expert in homomorphic encryption and RLWE-based encryption schemes
# to determine the security of their parameter choices.
# Our recommended values for the coefficient modulus can be easily accessed
# through the functions
# coeff_modulus_128bit(int)
# coeff_modulus_192bit(int)
# for 128-bit and 192-bit security levels. The integer parameter is the degree
# of the polynomial modulus.
# In SEAL the coefficient modulus is a positive composite number -- a product
# of distinct primes of size up to 60 bits. When we talk about the size of the
# coefficient modulus we mean the bit length of the product of the small primes.
# The small primes are represented by instances of the SmallModulus class; for
# example coeff_modulus_128bit(int) returns a vector of SmallModulus instances.
# It is possible for the user to select their own small primes. Since SEAL uses
# the Number Theoretic Transform (NTT) for polynomial multiplications modulo the
# factors of the coefficient modulus, the factors need to be prime numbers
# congruent to 1 modulo 2*degree(poly_modulus). We have generated a list of such
# prime numbers of various sizes, that the user can easily access through the
# functions
# small_mods_60bit(int)
# small_mods_50bit(int)
# small_mods_40bit(int)
# small_mods_30bit(int)
# each of which gives access to an array of primes of the denoted size. These
# primes are located in the source file util/globals.cpp.
# Performance is mainly affected by the size of the polynomial modulus, and the
# number of prime factors in the coefficient modulus. Thus, it is important to
# use as few factors in the coefficient modulus as possible.
# In this example we use the default coefficient modulus for a 128-bit security
# level. Concretely, this coefficient modulus consists of only one 56-bit prime
# factor: 0xfffffffff00001.
parms.set_coeff_modulus(seal.coeff_modulus_128(2048))
# The plaintext modulus can be any positive integer, even though here we take
# it to be a power of two. In fact, in many cases one might instead want it to
# be a prime number; we will see this in example_batching(). The plaintext
# modulus determines the size of the plaintext data type, but it also affects
# the noise budget in a freshly encrypted ciphertext, and the consumption of
# the noise budget in homomorphic multiplication. Thus, it is essential to try
# to keep the plaintext data type as small as possible for good performance.
# The noise budget in a freshly encrypted ciphertext is
# ~ log2(coeff_modulus/plain_modulus) (bits)
# and the noise budget consumption in a homomorphic multiplication is of the
# form log2(plain_modulus) + (other terms).
parms.set_plain_modulus(1 << 8)
# Now that all parameters are set, we are ready to construct a SEALContext
# object. This is a heavy class that checks the validity and properties of
# the parameters we just set, and performs and stores several important
# pre-computations.
context = SEALContext(parms)
# Print the parameters that we have chosen
print_parameters(context)
# Plaintexts in the FV scheme are polynomials with coefficients integers modulo
# plain_modulus. To encrypt for example integers instead, one can use an
# `encoding scheme' to represent the integers as such polynomials. SEAL comes
# with a few basic encoders:
# [IntegerEncoder]
# Given an integer base b, encodes integers as plaintext polynomials as follows.
# First, a base-b expansion of the integer is computed. This expansion uses
# a `balanced' set of representatives of integers modulo b as the coefficients.
# Namely, when b is odd the coefficients are integers between -(b-1)/2 and
# (b-1)/2. When b is even, the integers are between -b/2 and (b-1)/2, except
# when b is two and the usual binary expansion is used (coefficients 0 and 1).
# Decoding amounts to evaluating the polynomial at x=b. For example, if b=2,
# the integer
# 26 = 2^4 + 2^3 + 2^1
# is encoded as the polynomial 1x^4 + 1x^3 + 1x^1. When b=3,
# 26 = 3^3 - 3^0
# is encoded as the polynomial 1x^3 - 1. In memory polynomial coefficients are
# always stored as unsigned integers by storing their smallest non-negative
# representatives modulo plain_modulus. To create a base-b integer encoder,
# use the constructor IntegerEncoder(plain_modulus, b). If no b is given, b=2
# is used.
# [FractionalEncoder]
# The FractionalEncoder encodes fixed-precision rational numbers as follows.
# It expands the number in a given base b, possibly truncating an infinite
# fractional part to finite precision, e.g.
# 26.75 = 2^4 + 2^3 + 2^1 + 2^(-1) + 2^(-2)
# when b=2. For the sake of the example, suppose poly_modulus is 1x^1024 + 1.
# It then represents the integer part of the number in the same way as in
# IntegerEncoder (with b=2 here), and moves the fractional part instead to the
# highest degree part of the polynomial, but with signs of the coefficients
# changed. In this example we would represent 26.75 as the polynomial
# -1x^1023 - 1x^1022 + 1x^4 + 1x^3 + 1x^1.
# In memory the negative coefficients of the polynomial will be represented as
# their negatives modulo plain_modulus.
# [PolyCRTBuilder]
# If plain_modulus is a prime congruent to 1 modulo 2*degree(poly_modulus), the
# plaintext elements can be viewed as 2-by-(degree(poly_modulus) / 2) matrices
# with elements integers modulo plain_modulus. When a desired computation can be
# vectorized, using PolyCRTBuilder can result in massive performance improvements
# over naively encrypting and operating on each input number separately. Thus,
# in more complicated computations this is likely to be by far the most important
# and useful encoder. In example_batching() we show how to use and operate on
# encrypted matrix plaintexts.
# For performance reasons, in homomorphic encryption one typically wants to keep
# the plaintext data types as small as possible, which can make it challenging to
# prevent data type overflow in more complicated computations, especially when
# operating on rational numbers that have been scaled to integers. When using
# PolyCRTBuilder estimating whether an overflow occurs is a fairly standard task,
# as the matrix slots are integers modulo plain_modulus, and each slot is operated
# on independently of the others. When using IntegerEncoder or FractionalEncoder
# it is substantially more difficult to estimate when an overflow occurs in the
# plaintext, and choosing the plaintext modulus very carefully to be large enough
# is critical to avoid unexpected results. Specifically, one needs to estimate how
# large the largest coefficient in the polynomial view of all of the plaintext
# elements becomes, and choose the plaintext modulus to be larger than this value.
# SEAL comes with an automatic parameter selection tool that can help with this
# task, as is demonstrated in example_parameter_selection().
# Here we choose to create an IntegerEncoder with base b=2.
encoder = IntegerEncoder(context.plain_modulus())
# We are now ready to generate the secret and public keys. For this purpose we need
# an instance of the KeyGenerator class. Constructing a KeyGenerator automatically
# generates the public and secret key, which can then be read to local variables.
# To create a fresh pair of keys one can call KeyGenerator::generate() at any time.
keygen = KeyGenerator(context)
public_key = keygen.public_key()
secret_key = keygen.secret_key()
# To be able to encrypt, we need to construct an instance of Encryptor. Note that
# the Encryptor only requires the public key.
encryptor = Encryptor(context, public_key)
# Computations on the ciphertexts are performed with the Evaluator class.
evaluator = Evaluator(context)
# We will of course want to decrypt our results to verify that everything worked,
# so we need to also construct an instance of Decryptor. Note that the Decryptor
# requires the secret key.
decryptor = Decryptor(context, secret_key)
# We start by encoding two integers as plaintext polynomials.
value1 = 5
plain1 = encoder.encode(value1)
print("Encoded " + (str)(value1) + " as polynomial " + plain1.to_string() +
" (plain1)")
value2 = -7
plain2 = encoder.encode(value2)
print("Encoded " + (str)(value2) + " as polynomial " + plain2.to_string() +
" (plain2)")
# Encrypting the values is easy.
encrypted1 = Ciphertext()
encrypted2 = Ciphertext()
print("Encrypting plain1: ")
encryptor.encrypt(plain1, encrypted1)
print("Done (encrypted1)")
print("Encrypting plain2: ")
encryptor.encrypt(plain2, encrypted2)
print("Done (encrypted2)")
# To illustrate the concept of noise budget, we print the budgets in the fresh
# encryptions.
print("Noise budget in encrypted1: " +
(str)(decryptor.invariant_noise_budget(encrypted1)) + " bits")
print("Noise budget in encrypted2: " +
(str)(decryptor.invariant_noise_budget(encrypted2)) + " bits")
# As a simple example, we compute (-encrypted1 + encrypted2) * encrypted2.
# Negation is a unary operation.
evaluator.negate(encrypted1)
# Negation does not consume any noise budget.
print("Noise budget in -encrypted1: " +
(str)(decryptor.invariant_noise_budget(encrypted1)) + " bits")
# Addition can be done in-place (overwriting the first argument with the result,
# or alternatively a three-argument overload with a separate destination variable
# can be used. The in-place variants are always more efficient. Here we overwrite
# encrypted1 with the sum.
evaluator.add(encrypted1, encrypted2)
# It is instructive to think that addition sets the noise budget to the minimum
# of the input noise budgets. In this case both inputs had roughly the same
# budget going on, and the output (in encrypted1) has just slightly lower budget.
# Depending on probabilistic effects, the noise growth consumption may or may
# not be visible when measured in whole bits.
print("Noise budget in -encrypted1 + encrypted2: " +
(str)(decryptor.invariant_noise_budget(encrypted1)) + " bits")
# Finally multiply with encrypted2. Again, we use the in-place version of the
# function, overwriting encrypted1 with the product.
evaluator.multiply(encrypted1, encrypted2)
# Multiplication consumes a lot of noise budget. This is clearly seen in the
# print-out. The user can change the plain_modulus to see its effect on the
# rate of noise budget consumption.
print("Noise budget in (-encrypted1 + encrypted2) * encrypted2: " +
(str)(decryptor.invariant_noise_budget(encrypted1)) + " bits")
# Now we decrypt and decode our result.
plain_result = Plaintext()
print("Decrypting result: ")
decryptor.decrypt(encrypted1, plain_result)
print("Done")
# Print the result plaintext polynomial.
print("Plaintext polynomial: " + plain_result.to_string())
# Decode to obtain an integer result.
print("Decoded integer: " + (str)(encoder.decode_int32(plain_result)))
