TECHNICAL FIELD
[0001] The present disclosure relates generally to cryptography, and in particular to blind decryption in public-key cryptosystems.
BACKGROUND
[0002] This section is intended to introduce the reader to various aspects of art, which may be related to various aspects of the present disclosure that are described and/or claimed below. This discussion is believed to be helpful in providing the reader with background information to facilitate a better understanding of the various aspects of the present disclosure. Accordingly, it should be understood that these statements are to be read in this light, and not as admissions of prior art.
[0003] In 1999 Pascal Paillier proposed a new public-key cryptosystem [
Pascal Paillier. Public-key cryptosystems based on composite degree residuosity classes. In Jacques Stern, editor, Advances in Cryptology - EUROCRYPT '99, volume 1592 of Lecture Notes in Computer Science, pages 223-238. Springer, 1999], which was later generalized by Damgård and Jurik [
Ivan Damgård and Mads Jurik. A generalisation, a simplification and some applications of Paillier's probabilistic public-key system. In Kwangjo Kim, editor, Public Key Cryptography, volume 1992 of Lecture Notes in Computer Science, pages 119-136. Springer, 2001] and which can be described as follows:
Let an integer s ≥ 1. Let also two (large) primes p and q and let N = pq. The public key is {N, s} and the private key is λ = lcm(p - 1, q - 1). The message space is
The encryption of a message
is given by
for some random element r drawn in
where
denotes the multiplicative group of the ring of integers modulo
N^{s+1},
[0004] The encryption is decrypted inductively using
λ from
where
and
The following relation is used
(1 +
N)
^{αNs-1} ≡ 1 +
αN^{s} (mod
N^{s+1}) for any
[0005] Hence, letting
C_{i} =
c^{λ} mod
N^{i+2}, for 0 ≤
i ≤ s - 1, and defining function
equation gives:
and expressed more generally
[0006] In a number of applications, it is required that the owner of the private decryption key
λ, called the decryptor, learns no information about a given plaintext.
[0007] In such a setting, a user wishing to get the decryption of a given ciphertext (encrypted under the decryptor's public-key) blinds the ciphertext. In more detail, if c denotes the ciphertext, the user chooses at random an element
and computes the blinded ciphertext
[0008] Upon receiving the blinded ciphertext
c*, the decryptor decrypts it and obtains a blinded message
m* = m +
µ (in
M). The decryptor sends
m* to the user. As the user knows the mask
µ (chosen by the user), the user can recover the plaintext corresponding to ciphertext
c as
m =
m* -
µ (mod
N^{s+1}).
[0009] It will be appreciated that the above protocol wastes bandwith. If ℓ(
N) is the bit-length of
N (typically 2048 or more), the communication between the user and the decryptor incurs an exchange of
bits.
[0010] It will be appreciated that it is desired to have a technique that allows to decrease this quantity. It would also be good to have a technique that is faster, in terms of computation, for both the user and the decryptor.
[0011] The present disclosure provides such a technique.
SUMMARY
[0012] In a first aspect, the disclosure is directed to a cryptographic device comprising: an interface configured to send a blinded Paillier ciphertext
c_{0} to a decryption device and to receive a return value
µ_{1} from the decryption device; and a processor configured to: obtain a Paillier ciphertext
c, the Paillier ciphertext c having been generated using an encryption method with a public key comprising a modulus
N being the product of at least two primes p,
q; calculate the blinded Paillier ciphertext
c_{0} by taking the Paillier ciphertext c modulo a value based on the modulus
N; calculate a first value
through a calculation involving an inverse of the blinded Paillier ciphertext
c_{0} modulo a value based on the modulus
N; generate a blinded plaintext
m* through a calculation involving a multiplication of the Paillier ciphertext c and the first value
; and generate a plaintext m through a calculation involving an addition of the blinded plaintext
m* and the return value
µ_{1} modulo a value based on the modulus
N.
[0013] In a second aspect, the disclosure is directed to a decryption device comprising: an interface configured to receive a blinded Paillier ciphertext
c_{0} from a cryptographic device and to send a return value
µ_{1} to the cryptographic device; and
a processor configured to: calculate a first key
λ_{0} through a calculation involving an inversion of a modulus
N modulo a value based on a private key
λ; calculate a second value
ρ_{0} through a calculation involving the blinded Paillier ciphertext
c_{0} to the power of the first key
λ_{0} modulo a value based on the modulus
N; calculate a third value
through a calculation involving the second value
ρ_{0} to the power of the modulus
N modulo a value based on the modulus N; and calculate the return value
µ_{1} through a calculation involving a multiplication of the third value
and the blinded Paillier ciphertext
c_{0}.
[0014] In a third aspect, the disclosure is directed to a cryptographic device comprising: an interface configured to send a blinded Paillier ciphertext
c_{0} to a decryption device and to receive at least one return value from the decryption device; and a processor configured to: obtain a Paillier ciphertext
c, the Paillier ciphertext
c having been generated using an encryption method with a public key comprising a modulus
N being the product of at least two primes
p, q; calculate the blinded Paillier ciphertext
c_{0} by taking the Paillier ciphertext
c modulo a value based on the modulus
N; calculate a first value
through a calculation involving an inverse of the blinded Paillier ciphertext
c_{0} modulo a value based on the modulus
N; obtain a third value
from the at least one return value; calculate an exponent value (1 +
N)
^{m} modulo a value based on the modulus
N through a calculation involving a multiplication between the Paillier ciphertext c and the third value
; and obtain a plaintext
m from the exponent value (1 +
N)
^{m} modulo a value based on the modulus
N using inductive decryption.
[0015] In a fourth aspect, the disclosure is directed to a decryption device comprising: an interface configured to receive a blinded Paillier ciphertext
c_{0} from a cryptographic device and to send at least one return value to the cryptographic device; and a processor configured to: calculate a first key
λ_{0} through a calculation involving an inversion of a modulus
N to the power of a value
s having been used to generate a Paillier ciphertext
c from which the blinded Paillier ciphertext
c_{0} was calculated, the inversion being taken modulo a value based on a private key
λ; calculate a second value
ρ_{0} through a calculation involving the blinded Paillier ciphertext
c_{0} to the power of the first key
λ_{0} modulo a value based on the modulus
N; calculate a third value
through a calculation involving the second value
ρ_{0} to the power of the modulus
N to the power of the value s modulo a value based on the modulus
N and the value s, the third value ; and obtain the at least one return value, the return value being equal to the third value
or a value based on the third value
minus a first component
and the modulus N, the first component
being equal to a value obtained by a calculation involving an inverse of the blinded Paillier ciphertext
c_{0} modulo a value based on the modulus
N.
[0016] In a fifth aspect, the disclosure is directed to a cryptographic method for generating a plaintext
m for a Paillier ciphertext
c, the method comprising, in a device comprising a processor: obtaining a Paillier ciphertext
c, the Paillier ciphertext
c having been generated using an encryption method with a public key comprising a modulus
N being the product of at least two primes
p,
q; calculating a blinded Paillier ciphertext
c_{0} by taking the Paillier ciphertext
c modulo a value based on the modulus
N; calculating a first value
through a calculation involving an inverse of the blinded Paillier ciphertext
c_{0} modulo a value based on the modulus
N; generating a blinded plaintext
m* through a calculation involving a multiplication of the Paillier ciphertext c and the first value
; and generating the plaintext m through a calculation involving an addition of the blinded plaintext
m* and the return value
µ_{1} modulo a value based on the modulus
N.
[0017] In a sixth aspect, the disclosure is directed to a cryptographic method for blind decryption of a blinded Paillier ciphertext
c_{0}, the method comprising, in a device comprising a processor: obtaining a first key
λ_{0}, the first key
λ_{0} having been generated through a calculation involving an inversion of a modulus N modulo a value based on a private key
λ; calculating a second value
ρ_{0} through a calculation involving the blinded Paillier ciphertext
c_{0} to the power of the first key
λ_{0} modulo a value based on the modulus
N; calculating a third value
through a calculation involving the second value
ρ_{0} to the power of the modulus N modulo a value based on the modulus
N; calculating a return value
µ_{1} through a calculation involving a multiplication of the third value
and the blinded Paillier ciphertext
c_{0}; and outputting the return value
µ_{1}.
[0018] In a seventh aspect, the disclosure is directed to a cryptographic method for generating a plaintext m for a Paillier ciphertext c, the method comprising, in a device comprising a processor: obtaining the Paillier ciphertext
c, the Paillier ciphertext c having been generated using an encryption method with a public key comprising a modulus
N being the product of at least two primes
p, q; calculating the blinded Paillier ciphertext
c_{0} by taking the Paillier ciphertext c modulo a value based on the modulus
N; calculating a first value
through a calculation involving an inverse of the blinded Paillier ciphertext
c_{0} modulo a value based on the modulus
N; obtaining a third value
from the at least one return value; calculating an exponent value (1 +
N)
^{m} modulo a value based on the modulus
N through a calculation involving a multiplication between the Paillier ciphertext
c and the third value
; and obtaining the plaintext
m from the exponent value (1 +
N)
^{m} modulo a value based on the modulus
N using inductive decryption.
[0019] In an eighth aspect, the disclosure is directed to a cryptographic method for blind decryption of a blinded Paillier ciphertext
c_{0}, the method comprising, in a device comprising a processor: obtaining a first key
λ_{0}, the first key
λ_{0} having been generated through a calculation involving an inversion of a modulus
N to the power of a value
s having been used to generate a Paillier ciphertext c from which the blinded Paillier ciphertext
c_{0} was calculated, the inversion being taken modulo a value based on a private key
λ; calculating a second value
ρ_{0} through a calculation involving the blinded Paillier ciphertext
c_{0} to the power of the first key
λ_{0} modulo a value based on the modulus
N; calculating a third value
through a calculation involving the second value
ρ_{0} to the power of the modulus
N to the power of the value
s modulo a value based on the modulus
N and the value
s; obtaining the at least one return value, the return value being equal to the third value
or a value based on the third value
minus a first component
and the modulus
N, the first component
being equal to a value obtained by a calculation involving an inverse of the blinded Paillier ciphertext
c_{0} modulo a value based on the modulus
N; and outputting the at least one return value.
[0020] In a ninth aspect, the disclosure is directed to a computer program product storing instructions that, when executed by a processor, perform the method of the fifth aspect.
[0021] In a tenth aspect, the disclosure is directed to a computer program product storing instructions that, when executed by a processor, perform the method of the sixth aspect.
[0022] In an eleventh aspect, the disclosure is directed to a computer program product storing instructions that, when executed by a processor, perform the method of the seventh aspect.
[0023] In a twelvth aspect, the disclosure is directed to a computer program product storing instructions that, when executed by a processor, perform the method of the eighth aspect.
BRIEF DESCRIPTION OF DRAWINGS
[0024] Preferred features of the present disclosure will now be described, by way of non-limiting example, with reference to the accompanying drawings, in which:
Figure 1 illustrates a blind Paillier decryption system and method according to a preferred embodiment; and
Figure 2 illustrates a system and a method for generalized Paillier decryption.
DESCRIPTION OF EMBODIMENTS
[0025] First it is observed that for any
[0026] Proof. For any integer
α, an application of the binomial identity immediately yields
[0027] As a result, the decryption of a ciphertext
c can be decrypted in two steps as:
- 1. ρ_{0} = c_{0}^{λ0} mod N, where λ_{0} = -N^{-s} mod λ and c_{0} = c mod N;
- 2. c ρ_{0}^{Ns} ≡ (1 + N)^{m} (mod N^{s+1}) from which m is obtained. This will be described further hereinafter as it depends on the embodiment.
[0028] Defining
and using equation (2), since
ρ_{0} ≡
r^{-1} mod N, it is possible to write
with
It should also be noted that
[0029] The Paillier cryptosystem corresponds to the case s = 1. Hence, letting
gives
whence
and thus
[0030] From the sole knowledge of
c, the user can therefore compute
[0031] Likewise, from the sole knowledge of
c_{0}, the decryptor can compute
[0032] It is worth noting that the value of
c_{0} leaks no information on message
m. The user thus ends up having
m* =
m -
µ_{1} mod
N and the decryptor having
µ_{1}. This is depicted in Figure 1 hereinafter.
[0033] Figure 1 illustrates a method for blind Paillier decryption according to the present disclosure. Figure 1 shows a system 100 comprising a user device 110 and a decryption device 120 ("decryptor"). Each device 110, 120 comprises an interface 111, 121 configured for communication with the other device, at least one processor ("CPU") 112, 122 and memory 113, 123. The devices also comprise other necessary hardware and software components such as internal connections, but these are not shown to simplify the illustration. Also shown are a first non-transitory computer program storage medium 114 and a second non-transitory computer program storage medium 116 that store instruction that, when executed by a processor, respectively perform the methods of the user device 110 and the decryptor 120 described hereinafter.
[0034] The user device 110 obtains S10 a first ciphertext
c for a message m from some external device that has calculated it using Paillier encryption:
c = (1 +
mN)
r^{N} mod
N^{2}, wherein
r is a random number and
N is a RSA-type modulus. The user device 110 then generates S11 a blinded ciphertext
c_{0} by calculating
c_{0} =
c mod
N and sends S12 the blinded ciphertext
c_{0} to the decryptor 120 over a connection 130.
[0035] The user device 110 then, advantageously while waiting for a response from the decryptor 120, generates S13 a first value by calculating
=
c_{0}^{-1} mod N and then generates a blinded plaintext by calculating
[0036] Upon reception of the blinded ciphertext
c_{0}, the decryptor 120 generates S15 a first key
λ_{0} from a private key
λ by calculating
λ_{0} = -
N^{-1} mod
λ, generates S16 a second value by calculating
ρ_{0} =
c_{0}^{λ0} mod N, generates S17 a third value by calculating
and, finally, generates S18 a return value
µ_{1} by calculating
The decryptor 120 returns S19 the return value
µ_{1} to the user device 110.
[0037] Upon reception of the return value
µ_{1}, the user device 110 calculates S20 the clear plaintext by calculating m =
m* +
µ_{1} mod
N. The clear plaintext m can then for example be output to a user or stored for later retrieval.
[0038] The technique can be generalized to the case
s ≥ 1; the expected relative gain then becomes (
s + 1)/
s. The technique is illustrated in Figure 2. The Figure illustrates a simplified version of the user device 210 and the decryptor 220, but it is to be understood that these devices comprise the necessary hardware and software components, essentially those illustrated for the devices in Figure 1. Figure 2 also illustrates a third non-transitory computer program storage medium 214 and a fourth non-transitory computer program storage medium 216 that store instruction that, when executed by a processor, respectively perform the methods of the user device 210 and the decryptor 220 described hereinafter.
[0039] The user device 210 obtains S30 a first ciphertext
c for a message m from some external device that has calculated it using Paillier encryption:
c = (1 +
mN)
r^{Ns} mod
N^{s+1}, wherein r is a random number and
N is an RSA-type modulus. The user device 210 then generates S31 a blinded ciphertext
c_{0} by calculating
c_{0} =
c mod
N as in Figure 1 and sends S32 the blinded ciphertext
c_{0} to the decryptor 220 over a connection 230.
[0040] The user device 210 then, advantageously while waiting for a response from the decryptor 220, generates S33 a first value by calculating
=
c_{0}^{-1} mod N.
[0041] Upon reception of the blinded ciphertext
c_{0}, the decryptor 220 generates S34 a first key
λ_{0} from a private key
λ by calculating
λ_{0} = -
N^{-s} mod
λ, generates S35 a second value by calculating
ρ_{0} =
c_{0}^{λ0} mod N, generates S36 a third value by calculating
generates S37 a fourth value by calculating
and, finally, generates S38 a return value by calculating
or equivalently
_{1}, ...,
_{s} and sends S39 this value or these values to the user device 210. It is noted that it is possible to view
as an integer in base
N, i.e.
=
[0042] The user device 210 recovers S40
from
_{1}, ...
_{s} and from
=
c_{0}^{-1} mod
N. The user device 210 then calculates S41
to obtain (1 +
N)
^{m} mod
N^{s+1}.
[0043] It then remains to obtain
m from
Y := (1 +
N)^{m} mod
N^{s+1}. Let
m =
with
Define
Y_{i} =
Y mod
N^{i+2}, for 0 ≤
i ≤ s - 1. Then:
[0044] The clear plaintext m can then for example be output to a user or stored for later retrieval.
[0045] In the generalization, with
s = 1 (i.e., based on the original Paillier cryptosystem), the user device 210 and the decryptor 220 exchange
c_{0} and
. Upon receiving
, the user device computes
and gets (1 +
N)
^{m} = 1 +
mN (mod
N^{2}) from which
m is obtained; namely
m =
While this embodiment is efficient when it comes to bandwidth as the blind Paillier decryption method illustrated in Figure 1, it is noted that the method illustrated in Figure 1 has the advantage that the calculations of the user device are simpler since in the on-line phase the user device has just to evaluate a mere addition modulo
N to obtain m.
[0046] In a variant of the methods illustrated in Figures 1 and 2,
c_{0} is obtained by reduction modulo a positive integer multiple (greater than 1) of the modulus N, for example 2N, which just requires the transmission of one more bit.
[0047] It will be appreciated that the blind-decryption protocol of the present disclosure is bandwidth optimal. This is particularly advantageous when a large number of Paillier ciphertexts are exchanged, which for example is the case in the privacy-preserving recommendation system described by Nikolaenko, Weinsberg, loannidis, Joye, Boneh and Taft [
Valeria Nikolaenko, Udi Weinsberg, Stratis loannidis, Marc Joye, Dan Boneh, and Nina Taft. Privacy-preserving ridge regression on hundreds of millions of records. In 34th IEEE Symposium on Security and Privacy (S&P 2013), pp. 334-348, IEEE Computer Society, 2013]
[0048] Each feature disclosed in the description and (where appropriate) the claims and drawings may be provided independently or in any appropriate combination. Features described as being implemented in hardware may also be implemented in software, and vice versa. Reference numerals appearing in the claims are by way of illustration only and shall have no limiting effect on the scope of the claims.
1. A cryptographic device (110) comprising:
an interface (111) configured to send a blinded Paillier ciphertext c_{0} to a decryption device (120) and to receive a return value µ_{1} from the decryption device (120); and
a processor (112) configured to:
- obtain a Paillier ciphertext c, the Paillier ciphertext c having been generated using an encryption method with a public key comprising a modulus N being the product of at least two primes p, q;
- calculate the blinded Paillier ciphertext c_{0} by taking the Paillier ciphertext c modulo a value based on the modulus N;
- calculate a first value
through a calculation involving an inverse of the blinded Paillier ciphertext c_{0} modulo a value based on the modulus N;
- generate a blinded plaintext m* through a calculation involving a multiplication of the Paillier ciphertext c and the first value
; and
- generate a plaintext m through a calculation involving an addition of the blinded plaintext m* and the return value µ_{1} modulo a value based on the modulus N.
2. A decryption device (120) comprising:
an interface (121) configured to receive a blinded Paillier ciphertext c_{0} from a cryptographic device (110) and to send a return value µ_{1} to the cryptographic device (110); and
a processor (122) configured to:
- calculate a first key λ_{0} through a calculation involving an inversion of a modulus N modulo a value based on a private key λ;
- calculate a second value ρ_{0} through a calculation involving the blinded Paillier ciphertext c_{0} to the power of the first key λ_{0} modulo a value based on the modulus N;
- calculate a third value
through a calculation involving the second value ρ_{0} to the power of the modulus N modulo a value based on the modulus N; and
- calculate the return value µ_{1} through a calculation involving a multiplication of the third value
and the blinded Paillier ciphertext c_{0}.
3. A cryptographic device (210) comprising:
an interface configured to send a blinded Paillier ciphertext c_{0} to a decryption device (120) and to receive at least one return value from the decryption device (120); and
a processor configured to:
- obtain a Paillier ciphertext c, the Paillier ciphertext c having been generated using an encryption method with a public key comprising a modulus N being the product of at least two primes p, q;
- calculate the blinded Paillier ciphertext c_{0} by taking the Paillier ciphertext c modulo a value based on the modulus N;
- calculate a first value
through a calculation involving an inverse of the blinded Paillier ciphertext c_{0} modulo a value based on the modulus N;
- obtain a third value
from the at least one return value;
- calculate an exponent value (1 + N)^{m} modulo a value based on the modulus N through a calculation involving a multiplication between the Paillier ciphertext c and the third value
; and
- obtain a plaintext m from the exponent value (1 + N)^{m} modulo a value based on the modulus N using inductive decryption.
4. A decryption device (220) comprising:
an interface configured to receive a blinded Paillier ciphertext c_{0} from a cryptographic device (110) and to send at least one return value to the cryptographic device (110); and
a processor configured to:
- calculate a first key λ_{0} through a calculation involving an inversion of a modulus N to the power of a value s having been used to generate a Paillier ciphertext c from which the blinded Paillier ciphertext c_{0} was calculated, the inversion being taken modulo a value based on a private key λ;
- calculate a second value ρ_{0} through a calculation involving the blinded Paillier ciphertext c_{0} to the power of the first key λ_{0} modulo a value based on the modulus N;
- calculate a third value
through a calculation involving the second value ρ_{0} to the power of the modulus N to the power of the value s modulo a value based on the modulus N and the value s, the third value ; and
- obtain the at least one return value, the return value being equal to the third value
or a value based on the third value
minus a first component
and the modulus N, the first component
being equal to a value obtained by a calculation involving an inverse of the blinded Paillier ciphertext c_{0} modulo a value based on the modulus N.
5. A cryptographic method for generating a plaintext
m for a Paillier ciphertext
c, the method comprising, in a device (110) comprising a processor (112):
- obtaining (S10) a Paillier ciphertext c, the Paillier ciphertext c having been generated using an encryption method with a public key comprising a modulus N being the product of at least two primes p, q;
- calculating (S11) a blinded Paillier ciphertext c_{0} by taking the Paillier ciphertext c modulo a value based on the modulus N;
- calculating (S13) a first value
through a calculation involving an inverse of the blinded Paillier ciphertext c_{0} modulo a value based on the modulus N;
- generating (S14) a blinded plaintext m* through a calculation involving a multiplication of the Paillier ciphertext c and the first value
; and
- generating (S20) the plaintext m through a calculation involving an addition of the blinded plaintext m* and a return value µ_{1} modulo a value based on the modulus N.
6. A cryptographic method for blind decryption of a blinded Paillier ciphertext
c_{0}, the method comprising, in a device (120) comprising a processor (122):
- obtaining (S15) a first key λ_{0}, the first key λ_{0} having been generated through a calculation involving an inversion of a modulus N modulo a value based on a private key λ;
- calculating (S16) a second value ρ_{0} through a calculation involving the blinded Paillier ciphertext c_{0} to the power of the first key λ_{0} modulo a value based on the modulus N;
- calculating (S17) a third value
through a calculation involving the second value ρ_{0} to the power of the modulus N modulo a value based on the modulus N;
- calculating (S18) a return value µ_{1} through a calculation involving a multiplication of the third value
and the blinded Paillier ciphertext c_{0}; and
- outputting (S19) the return value µ_{1}.
7. A cryptographic method for generating a plaintext m for a Paillier ciphertext
c, the method comprising, in a device (210) comprising a processor (212):
- obtaining (S30) the Paillier ciphertext c, the Paillier ciphertext c having been generated using an encryption method with a public key comprising a modulus N being the product of at least two primes p, q;
- calculating (S31) the blinded Paillier ciphertext c_{0} by taking the Paillier ciphertext c modulo a value based on the modulus N;
- calculating (S33) a first value
through a calculation involving an inverse of the blinded Paillier ciphertext c_{0} modulo a value based on the modulus N;
- obtaining (S40) a third value
from the at least one return value;
- calculating (S41) an exponent value (1 + N)^{m} modulo a value based on the modulus N through a calculation involving a multiplication between the Paillier ciphertext c and the third value
; and
- obtaining (S42) the plaintext m from the exponent value (1 + N)^{m} modulo a value based on the modulus N using inductive decryption.
8. A cryptographic method for blind decryption of a blinded Paillier ciphertext
c_{0}, the method comprising, in a device (220) comprising a processor (222):
- obtaining (S34) a first key λ_{0}, the first key λ_{0} having been generated through a calculation involving an inversion of a modulus N to the power of a value s having been used to generate a Paillier ciphertext c from which the blinded Paillier ciphertext c_{0} was calculated, the inversion being taken modulo a value based on a private key λ;
- calculating (S35) a second value ρ_{0} through a calculation involving the blinded Paillier ciphertext c_{0} to the power of the first key λ_{0} modulo a value based on the modulus N;
- calculating (S36) a third value
through a calculation involving the second value ρ_{0} to the power of the modulus N to the power of the value s modulo a value based on the modulus N and the value s;
- obtaining (S38) the at least one return value, the return value being equal to the third value
or a value based on the third value
minus a first component
and the modulus N, the first component
being equal to a value obtained by a calculation involving an inverse of the blinded Paillier ciphertext c_{0} modulo a value based on the modulus N; and
- outputting (S39) the at least one return value.
9. A computer program product (114) storing instructions that, when executed by a processor, perform the method of claim 5.
10. A computer program product (124) storing instructions that, when executed by a processor, perform the method of claim 6.
11. A computer program product (214) storing instructions that, when executed by a processor, perform the method of claim 7.
12. A computer program product (224) storing instructions that, when executed by a processor, perform the method of claim 8.