Fixed Point Arithmetic

In a paper (http://eprint.iacr.org/2016/250) to be presented at SAC (http://www.engr.mun.ca/~sac2016/), in St Johns Canada, in August HEAT researchers Ana Costache, Nigel P. Smart, Srinivas Vivek and Adrian Waller describe how to perform homomorphic operations on fixed point numbers. The work builds upon earlier work of Dowlin et al (http://research.microsoft.com/pubs/258435/ManualHEv2.pdf). In this blog post we present a quick overview of the technique.

To understand the technique you need to appreciate that all known efficient Somewhat Homomorphic Encryption (SHE) schemes are homomorphic over a given ring of polynomials. In particular the set of polymomials modulo both a prime $p$ (the so-called plaintext modulus) and a polynomial $F(X)$. Given such a plaintext space we can embed integers upto a bounded size within the plaintext space in a redundant manner as follows.

Suppose we have two integers, say $x=127$ and $y=78$, we write these as polynomials in {\em balanced base} $B$, for $B-3$. This means we write the integers in base $B$, but we allow plus or minus signs in the representation and the coefficients lie in $[0,\ldots,(B-1)/2]$. Thus we have
$$  127 \equiv 3^5-3^4 -3^3 -3^2+1 =X^5-X^4-X^3-X^2+1$$
and
$$  78 \equiv 3^4-3 = X^4-X$$.
When we add and multiply two such polynomials, and then substitute back in $X=3$, we will obtain the same result as multiplying and adding the original integers.  So for example
\begin{eqnarray*}  (X^5-X^4-X^3&-&X^2+1)
     \cdot
    (X^4-X)  \\
    &=& X^9-X^8-X^7-2 \cdot X^6+X^5+2 \cdot X^4+X^3-X.
\end{eqnarray*}
Notice how both the degree and the coefficient sizes have increased. If the resulting polynomial can still be represented in the plaintext space, i.e. the prime $p$ and the degree of $F(X)$ are large enough, then we can map the polynomial operation to the homomorphic operation. Thus assuming $p$ and $F(X)$ are {\em big enough} we can perform operations on integers; even though the homomorphic encryption scheme only supports operations on polynomials modulo $(p,F(X))$.

To encode a fixed point number we then simply encode it as the ratio of an integer numerator and a power of $B$ denominator; so for example $4.7037 \approx 127/3^3$. We can then apply operations on these fractional numbers, as long as we keep track of which denominator we are working with, by simply operating on the numberators. Thus each encrypted fixed point number is represented by an encrypted integer (the numerator) and a public power of three (the denominator).

Whilst this representation has been known before, and was given in the paper of Dowlin, a key issue was that one needed to know for a given homomorphic operation what the requisite sizes of $p$ and $F(X)$ should be to ensure correctness. It is this latter problem which our paper addresses; we show how to obtian lower bounds on the plaintext size which are needed to support a given homomorphic calculation.  In addition we show that a technique of Dowlin et al, which appears at first sight to be more efficient, turns out to be actually just the same as the above encoding method for fixed point numbers.

Fourier Transforms in the Encrypted Domain

Thales UK has identified Synthetic Aperture Radar (SAR) as a promising candidate for the application of Homomorphic Encryption (HE). It is proposed that the data collected from SAR can be protected throughout its processing up to the formation of image products. SAR data can be processed in many ways to produce different types of images; the simplest of which is the Single Look Complex Image. The main challenge of implementing the Single Look Complex Image algorithm in the encrypted domain is that of implementing the Fourier Transform.

Discrete Fourier Transform

Consider the Discrete Fourier Transform (DFT) for a sequence of complex numbers $x_0, x_1, \ldots, x_{N-1}$:

 

$X_k = \sum_{n=0}^{N-1} x_n \cdot e^{ -\frac{2\pi i k n} {N} } , \quad k=0,\ldots, N-1$

 

This is a linear transform; it requires only the summation of sequence samples and multiplication by scalar values $e^{-\frac{2\pi i k n}{N} }$ (sometimes called twiddle factors). We usually implement this via a Fast Fourier Transform (FFT) which reduces the computational complexity by the use of a few mathematical tricks.
 

Paillier Cryptosystem

The Paillier cryptosystem [1] encrypts plaintext messages $m\in \mathbb{Z}_n$, where $n$ is the product of two large prime numbers, to a ciphertext $c\in \mathbb{Z}^*_{n^2}$. Let $\text{Enc}$ and $\text{Dec}$ denote the encryption and decryption processes respectively. It can be shown that the Paillier cryptosystem satisfies the two following properties:

1)      For two plaintext messages $m_1, m_2\in \mathbb{Z}_n$
                  $\text{Dec}( \text{Enc}(m_1) \cdot \text{Enc}(m_2) \text{ mod } n^2)$ $ = m_1 + m_2 \text{ mod } n$.

2)      For a plaintext $m$ and any constant integer $k$

$\text{Dec} ( \text{Enc}(m)^k \text{ mod } n ) = km \text{ mod } n^2$.

Properties 1 and 2 allow us to add any two plaintexts and multiply any plaintext by an integer constant in the encrypted domain; these operations are sufficient to implement the Fourier Transform (DFT/FFT).
 

The Fourier Transform under Paillier Encryption

There are two major challenges in the implementation of a Fourier Transform under Paillier encryption:   

1)      Paillier encrypts and allows operations only on integer values modulo $n$. We must convert our sequence samples $x_0, x_1, \ldots, x_{N-1}$, and twiddle factors $e^{-\frac{2\pi ikn}{N} }$, into integers modulo $n$. We achieve this by multiplying by two appropriately chosen scale factors; one for the sequence samples and one for the twiddle factors. These scale factors must be dealt with throughout the Fourier Transform.

2)      The homomorphic addition and scalar multiplication operations result in plaintext values modulo $n$. That is, if we wish to sum two plaintext values $m_1, m_2$ in the encrypted domain we must ensure the resulting plaintext is no greater than $n-1$. Otherwise, we can say nothing about the magnitude of $m_1 + m_2$. Since we wish to perform many additions and scalar multiplications throughout the Fourier Transform, we require $n$ to be large enough to cope with the growth from all of these operations.

These two challenges have been discussed in [2]. Challenge 1 means that we need to deal with the quantisation scale factors when performing operations. The analysis of challenge 2 in [2] results in the following lower bound for the word length $n_p$ of the modulus $n$ after radix-2 FFT operations:

$n_p \geq v + (v-2) n_2 + n_1 + 3$

Where:

$v = \log_2 (N)$

$n_1 = b_1-1$, where $b_1$ is the number of bits of precision in the sequence samples

$n_2 = \left\lceil b_2 - \dfrac{v}{2} + \dfrac{1}{2} \right\rceil$ where $b_2$ is the hardware register size in an equivalent plaintext implementation of the Fourier Transform. Equivalently, $n_2$ is the number of bits of precision required in the twiddle factors to ensure the same precision in the result of the Fourier Transform as a standard (in clear) hardware implementation. 

Typical SAR applications perform multiple FFT operations, as well as other multiplications. For example, the Single Look Complex Image processing employs four 2-dimensional FFTs and 3 Hadamard products. Using the inequality above for one (1-dimensional) FFT operation, our analysis of the Single Look Complex Image processing results in a lower bound: 

 

$n_p \geq 8v + 8(v-2) n_2 + n_1 + 3(n_3-1) + 24$

 

for the modulus word length, where $n_3$ is the word length associated with each of the inputs to the Hadamard products. We expect values of the order $v=13$, $n_1 = 7$, $n_2=10$, $n_3=16$, giving a lower bound of $n_p\geq 1060$. It is likely that the modulus word length is required to be larger than this bound for security reasons so there is plenty of scope to perform Single Look Complex Image processing.

 

References

[1]  Paillier, "Public-Key Cryptosystems Based on Composite Degree Residuosity Classes", EUROCRYPT, Springer. pp. 223–238 (1999).

[2]  Bianchi, T., Piva, A., Barni, M, "On the Implementation of the Discrete Fourier Transform in the Encrypted Domain", IEE Transaction on Information and Security , 86-97 (2009).

Tightness of the error bound in Ring-LWE

About two months ago we have posted a short paper on eprint sharing some thoughts on the error size used in Ring-LWE, an increasingly important hard problem that was proposed by Lyubashevsky, Peikert and Regev [LPR] as a building block for cryptography. Besides resistance against quantum computers, one of the main features of Ring-LWE for HEAT is that it enables homomorphic encryption.

Very briefly, in Ring-LWE one fixes a degree $n$ number field $K$ with ring of integers $R$ and discriminant $\Delta$. One also fixes an integer modulus $q$ and an error parameter $r \in \mathbb{R}_{> 0}$, which we think of as depending on $n$ only. We write $R_q$ and $R_q^\vee$ for the reduction of $R$ and its codifferent (or `dual') modulo $q$. The Ring-LWE problem is about guessing a secret $\mathbf{s} \in R_q^\vee$ from a number of samples of the form $(\mathbf{a}, \mathbf{a} \cdot \mathbf{s} + \mathbf{e} )$, where $\mathbf{a} \in R_q$ is random and $\mathbf{e}$ is sampled from the distribution on $(K \otimes \mathbb{R})/q\mathbb{Z}$ that one obtains by pulling back a spherical Gaussian of width $r$ under the canonical embedding $K \hookrightarrow \mathbb{R}^n$.

In a recent series of papers Elias, Lauter, Ozman and Stange [ELOS], and Chen, Lauter and Stange [CLS1], [CLS2] investigated whether the existence of a homomorphism $\phi : R_q \rightarrow S$ to some small ring $S$ can be exploited to leak information about the secret $\mathbf{s}$. For instance if $R_q$ is of the form $\mathbb{Z}[x]/(q,f(x))$ for some $f(x) \in \mathbb{Z}[x]$ satisfying $f(1) = 0$ mod $q$, then $\phi$ could be evaluation at $1$: $$ \phi : R_q \rightarrow \mathbb{F}_q: \mathbf{a} \mapsto \mathbf{a}(1).$$ Although this idea is attractive, the concrete vulnerable instances proposed by [ELOS], [CLS1], [CLS2] do not follow the original set-up from [LPR]. Instead they

1. sample the secrets from the ring $R_q$, rather than its dual $R_q^\vee$,
2. use an error parameter of the form $|\Delta|^{1/2n}\cdot r$, rather than just $r$.

It is not entirely clear to us what the implications of 1. are, but at least one should scale up the error parameter in order to compensate for the relative sparseness of $R$, as is indeed done in 2. However the scalar $|\Delta|^{1/2n}$ is chosen from a Polynomial-LWE point of view and is not adapted to the way the Ring-LWE errors are generated. Indeed, the canonical embedding may be very skew, so even though `on average' things will be okay, there could exist a basis of $R$ with respect to which some of the coordinates of $\mathbf{e}$ are negligible. This in turn leads to exact linear equations in the secret $\mathbf{s}$, obtained by merely rounding, which allows for a recovery of the secret using elementary linear algebra.

In a previous paper (reported upon in a blogpost below) we showed that the families from [ELOS] indeed suffer from this skewness, and therefore were easy to start from. To some extent this also seems to apply to the examples from [CLS1] and [CLS2], as is discussed in the last section of our current paper, where we make a more systematic analysis of the skewness phenomenon. Our main contributions are as follows:

• We first give an informal motivation for why the most natural choice of scalar is $|\Delta|^{1/n}$, rather than $|\Delta|^{1/2n}$, if one wants to set up a non-dual version of Ring-LWE as in 1.
• Next we provide for any $\varepsilon > 0$ a family of number fields in which non-dual Ring-LWE can again be broken using elementary linear algebra as soon as the errors are scaled up by $|\Delta|^{(1 - \varepsilon)/n}$ only.
• Finally we show that also the actual (i.e., dual) Ring-LWE problem is easily broken for the same families, as soon as the errors proposed by [LPR] as scaled down by $|\Delta|^{\varepsilon/n}$.

So our conclusions strongly suggest that the problems with the examples from [ELOS] are related to 2., rather than 1. They also provide a tightness statement on the error size proposed by [LPR], at least from a discriminant point of view. Immediate research questions that arise are whether there exist weak instances of non-dual Ring-LWE, when set up with the scalar $|\Delta|^{1/n}$, and, more specifically, whether there exist attacks involving a homomorphism $\phi : R_q \rightarrow S$ that apply here? Or that apply to the actual Ring-LWE problem as introduced in [LPR]?

About a month ago Peikert posted a paper sharing similar (but more detailed) thoughts on the vulnerability of the Ring-LWE instantiations described in [ELOS], [CLS1] and [CLS2].

Wouter Castryck, Ilia Iliashenko, Frederik Vercauteren

CryptoExperts is hiring a Post-Doc Researcher

CryptoExperts (Paris, France) is looking for an excellent postdoctoral fellow to work on the design, the analysis, and the implementation of homomorphic encryption and lattice-based cryptography, who will be primarily involved in the HEAT project.

In this position, the candidate will also be involved in the French CryptoComp project on homomorphic encryption, which aims at developing a fully automated “source to source” compiler which convert any function into a “secure” version of this function able to run on encrypted data. 

In both projects, the challenging industrial use-cases will allow to ground the research in practical issues. The candidate will be able to carry research according to its research interests (theoretical and/or practical), and will be offered multiple travel opportunities and interactions with scientists all over Europe (including the HEAT partners). Participation to the open-source libraries on homomorphic encryption and proof-of-concept demonstrators, main outcomes of the HEAT project, will be expected.

Applicants should have a Ph.D. in a relevant subject or equivalent, have a strong publication record and interest in practical protocols. Implementation and coding experience are required.

The position is fully funded for two years (with a flexible starting date). Review of applications will begin immediately until the position has been filled. Knowledge of French is not mandatory. To apply, please send an e-mail to jobs (at) cryptoexperts.com with the following documents:

• Curriculum vitae, with publication list
• Reference letters

NIST Draft on Post Quantum Cryptography

On February 4th 2016, the National Institute of Standards and Technology (NIST) released a draft entitled Report on Post-Quantum Cryptography. This draft presents NIST's current understanding about the status of quantum computers and post-quantum cryptography, and its initial plan to move forward.

In particular, the NIST has decided that time has come to prepare for the transition to quantum-resistant cryptography. Some striking facts reported in the draft are:

• "researchers working on building a quantum computer have estimated that it is likely that a quantum computer capable of breaking RSA-2048 in a matter of hours could be built by 2030 for a budget of about a billion dollars" (p.6)
• "transitioning from 112 to 128 bits of security is perhaps less urgent than transitioning from existing cryptosystems to post-quantum cryptosystems." (p.6)

This engagement of the NIST to standardize post-quantum cryptography is a true opportunity for everyone to react before practical quantum computing is imminent. The NIST plan is as follows:

1. Feb. 3 2016 - Mar. 9 2016: public comment period for the draft Report on Post-Quantum Cryptography.
2. 2016: proposition of a draft on the evaluation criteria for quantum-resistant public key cryptography, and public comment period.
3. until late 2017: the NIST will accept proposals for quantum-resistant public key encryption, digital signatures, and key exchange algorithms.
4. 2018 + 3/5 years: public scrutiny of the candidates to be standardized.

The NIST emphasizes that this process should not be considered as a competition. The institute see its role as a manager of a transparent process engaging with the cryptographic community, that will eventually reach a consensus on cryptographic standards to be endorsed by industry and other standards organizations around the world.

The HEAT project focuses on bringing fully homomorphic encryption one step further, into products and standards. Obviously, this cannot be, and is not, orthogonal to the quantum-resistant cryptography process. In particular, most of the fully homomorphic encryption schemes belong to one of the main family of post-quantum cryptographic primitives: lattice-based cryptography. As a consequence, all the research that is, and will be, done on fully homomorphic encryption goes hand in hand with the process initiated by the NIST. 

In particular, the Security analysis and parameter recommendations outcome will directly feed the research on analysis of post-quantum primitives. Our open-source software and hardware libraries will include algorithms that will help better understand the performance potential of lattice-based cryptography. Also a member of the HEAT consortium is involved in the ISO standardization, and the main editor of the upcoming WG 2 standard on Homomorphic Encryption, that is, encryption that supports computing over encrypted data. A workshop on Cryptographic Standards and Evaluations  (AWACS 2016) will be organized by CryptoExperts, on behalf of the ECRYPT-CSA project, on May 8 2016, and will include talks and a panel discussion on the present and near-future trends in crypto standards (with an emphasis on the post-quantum rush). 

Last but not least, many members of the HEAT consortium are actively involved in the research on post-quantum primitives such as key-exchange or digital signatures.

Satellite Applications of Homomorphic Encryption

Thales UK has been investigating the use of Homomorphic Encryption (HE) in satellite scenarios. In particular, our assessment has focused on the data processing of imaging products; one of the many types of signal processing applications which could have been chosen.  

It is relatively simple to find signal processing satellite applications where HE could, in theory at least, provide real benefit. In particular, to provide end-to-end security of data from the sensor on the satellite to the end user, while allowing the potentially complex and specialised signal processing to be performed on shared (and less trusted from the end user’s point of view) infrastructure. The real challenge has been to find such applications that are potentially practical in the near term, using Somewhat Homomorphic Encryption (SHE).

Consider the Sentinel 2 Multi Spectral Imager which is a representative example of an earth observation satellite (in this case designed for environmental monitoring). In order to be suitable for the application of HE to provide sensor to end user security, all the signal processing algorithms would have to be amenable to HE. It turns out that many of them potentially are, in particular the deconvolution algorithms that use Fourier Transforms. However, other parts of the processing are more difficult to deal with in HE. For example, processing to deal with imperfections such as:

• saturated pixels
• no data pixels and partially corrected pixels (crosstalk correction)
• defective pixels

These are very simple algorithms, but involve a lot of conditional operators (<, >, etc. to test pixel values) that would be expensive when implemented on HE encrypted data.

A much more promising candidate is Synthetic Aperture Radar (SAR) (e.g. Envisat). SAR is used to provide high precision, and often 3D, imaging for applications such as sea ice and glacier monitoring, vegetation coverage analysis, disaster monitoring and traffic surveillance. Being a radio-based technique, it is not affected by weather or the lack of daylight. The raw data collected by the satellite consists of a series of radar echoes, and these must be processed to form an image. On examining the details of this processing, it can be seen that the core functionality consists of a series of:

• Fourier Transforms (and their inverses)
• Hadamard products of matrices

These can be implemented, in theory at least, with a low multiplicative depth, and without the need for conditional operators. This is therefore a good candidate application for SHE, but much work needs to be done to demonstrate its practicality. 

Provably Weak Instances of Ring-LWE Revisited

In Crypto 2015, Elias, Lauter, Ozman and Stange [ELOS15] proposed a successful attack on the decision version of the non-dual Ring-LWE problem over number fields $K$ for two specific families of defining polynomials. Both families have the common feature that the coefficients of the polynomials are dependent on the modulus $q$.

We extended this attack to the search version of Ring-LWE by using that the error distributions in such number fields are very skewed by moving away from the Minkowski space. Particularly, the largest errors either wrap around modulo $q$ (and thus make it impossible to decrypt the secret), or the smallest ones are so negligible that we obtain exact linear equations that reveal the secret. Our attack applies to every modulus up to $q$, and requires less samples.

Let $M$ be the $n \times n$ matrix of the canonical embedding $K \to \mathbb{C}^n$ with respect to the polynomial basis of $K$, and let $N$ be its inverse. The "skewness" of the errors can be nicely described by the singular value decomposition (SVD) of the matrix $N$ and the condition number $k(N) = s_1(N)/s_n(N)$, where $s_1$ and $s_n$ are the biggest and the smallest singular values correspondingly.

Recall that the SVD is given by $N = U \cdot \Sigma \cdot \overline{V}^t$, where $U$, $V$ are unitary matrices and $\Sigma$ is a diagonal matrix containing the singular values. The image of a unit sphere under $N$ will result in an ellipsoid whose axes are defined by the columns of $U$, with length equal to the corresponding singular value. Hence, the condition number captures how heavily this ellipsoid is deformed.

All vulnerable instances in [ELOS15] are defined by polynomials $f_{n, a, b} = x^n +ax + b$, where $a$ and $b$ are such that  $f(1) = 0 \bmod q$, which is the property that they exploit, but that our attack does not need. If we consider their first family of samples, where $a = 0$ and $b = q - 1$, the SVD of $N$ is of the form $I_{n \times n} \cdot \Sigma \cdot \overline{V}^t$. Thus a spherical error distribution on the Minkowski space with the parameter $\sigma$ induces an elliptical distribution on the number field, where the $i$th coordinate is distributed by a Gaussian with $\sigma_i = s_i(N) \cdot \sigma$. Moreover, the condition number is close to the modulus $q$, that is, the elliptical distribution is stretched along axes whose lengths range roughly from $\sigma_1$ to $q \cdot \sigma_1$. For the second family we did not find such an explicit form of the SVD. Nevertheless, the unitary matrix $U$ remains close to being diagonal (see the heat map below) and the singular values form a near-geometric series again leading to a condition number close to $q$.

It can be easily seen that for both families and for the choices of $\sigma$ made in [ELOS15], the Ring-LWE distribution generates negligible errors in the higher terms of the polynomial basis. More precisely each Ring-LWE sample $(a, b = a \cdot s + e)$ can be written as

\[ M_{a} \cdot (s_0, s_1, \dots, s_{n-1})^t = (b_{0}, b_{1}, \dots, b_{n-1})^t - (e_{0}, e_{1}, \dots, e_{n-1})^t , \]

where $M_{a}$ corresponds to multiplication by $a$. By rounding, the higher error terms are removed and the last equations become exact equations in the coefficients of the secret. If the highest $\left\lceil n/k \right\rceil$ error terms round to zero then we need only $k$ Ring-LWE samples to reveal the secret (for these concrete examples 10 samples amply suffice).

The corresponding paper will appear soon on the Eprint.

Wouter Castryck, Ilia Iliashenko and Fre Vercauteren.