Lattice based cryptography

Lattice based public key encryption (and its cousins known as knapsack and coding based encryption) have almost as long a history as discrete logarithm and factoring based schemes. Already in 1976, right after the Diffie-Hellman key exchange was discovered (and before RSA), Ralph Merkle was working on building public key encryption from the NP hard knapsack problem (see Diffie’s recollection). This can be thought of as the task of solving a linear equation of the form \(Ax = y\) (where \(A\) is a given matrix, \(y\) is a given vector, and the unknown are \(x\)) over the real numbers but with the additional constraint that \(x\) must be either \(0\) or \(1\). His proposal evolved into the Merkle-Hellman system proposed in 1978 (which was broken in 1984).

McEliece proposed in 1978 a system based on the difficulty of the decoding problem for general linear codes. This is the task of solving noisy linear equations where one is given \(A\) and \(y\) such that \(y=Ax+e\) for a “small” error vector \(e\), and needs to recover \(x\). Crucially, here we work in a finite field, such as working modulo \(q\) for some prime \(q\) (that can even be \(2\)) rather than over the reals or rationals. There are special matrices \(A^*\) for which we know how to solve this problem efficiently: these are known as efficiently decodable error correcting codes. McEliece suggested a scheme where the key generator lets \(A\) be a “scrambled” version of a special \(A^*\) (based on the Goppa algebraic geometric code). So, someone that knows the scrambling could solve the problem, but (hopefully) someone that doesn’t know it wouldn’t. McEliece’s system has so far not been broken.

In a 1996 breakthrough, Ajtai showed a private key scheme based on integer lattices that had a very curious property- its security could be based on the assumption that certain problems were only hard in the worst case, and moreover variants of these problems were known to be NP hard. This re-ignited the hope that we could perhaps realize the old dream of basing crypto on the mere assumption that \(P\neq \ensuremath{\mathit{NP}}\). Alas, we now understand that there are fundamental barriers to this approach.

Nevertheless, Ajtai’s work attracted significant interest, and within a year both Ajtai and Dwork, as well as Goldreich, Goldwasser and Halevi came up with lattice based constructions for public key encryption (the former based also on worst case assumptions). At about the same time, Hoffstein, Pipher, and Silverman came up with their NTRU public key system which is based on stronger assumptions but offers better performance, and they started a company around it together with Daniel Lieman.

You may note that I haven’t yet said what lattices are; we will do so later, but for now if you simply think of questions involving linear equations modulo some prime \(q\), you will get enough of the intuition that you need. (The lattice viewpoint is more geometric, and we’ll discuss it more below; it was first used to attack cryptosystems and in particular break the Merkle-Hellman knapsack scheme and many of its variants.)

Lattice based cryptography has captured a lot of attention recently from both theory and practice. In the theory side, many cool new constructions are now based on lattice based cryptography, and chief among them fully homomorphic encryption, as well as indistinguishability obfuscation (though the latter’s security’s foundations are still far less solid). On the applied side, the steady advances in the technology of quantum computers have finally gotten practitioners worried about RSA, Diffie Hellman and Elliptic Curves. While current constructions for quantum computers are nowhere near being able to, say, factor larger numbers that can be done classically (or even than can be done by hand), given that it takes many years to develop new standards and get them deployed, many believe the effort to transition away from these factoring/dlog based schemes should start today (or perhaps should have started several years ago). Based on this, the National Institute of Standards and Technology has started a process to identify “post quantum” public key encryption scheme. All the finalist for public-key encryption are based on lattices/codes.

Cryptography has the peculiar/unfortunate feature that if a machine is built that can factor large integers in 20 years, it can still be used to break the communication we transmit today, provided this communication was recorded. So, if you have some data that you expect you’d want still kept secret in 20 years (as many government and commercial entities do), you might have reasons to worry. Currently lattice based cryptography is the only real “game in town” for potentially quantum-resistant public key encryption schemes.

Lattice based cryptography is a huge area, and in this lecture and this course we only touch on few aspects of it. I highly recommend Chris Peikert’s Survey for a much more in depth treatment of this area.

Quick linear algebra recap

A field \(\mathbb{F}\) is a set that supports the operations \(+,\cdot\) and contains the numbers \(0\) and \(1\) (more formally the additive identity and multiplicative identity) with the usual properties that the real numbers have. (That is associative, commutative, and distributive law, the fact that for every \(x \in \mathbb{F}\) there is an element \(-x\) such that \(x + (-x) = 0\) and that if \(x \neq 0\) there is an element \(x^{-1}\) such that \(x \cdot x^{-1} = 1\).) Apart from the real numbers, the main field we will be interested in this section is the field \(\Z_q\) of the numbers \(\{0,1,\ldots,q-1\}\) with addition and multiplication done modulo \(q\), where \(q\) is a prime number.¹

You should be comfortable with the following notions (these are covered in a number of sources, including the appendix of Katz-Lindell and Shoup’s online-available book):

• A vector \(v \in \mathbb{F}^n\) and a matrix \(M \in \mathbb{F}^{m \times n}\). An \(m\times n\) matrix has \(m\) rows and \(n\) columns. We think of vectors as column vectors and so we can think of a vector \(v \in \mathbb{F}^n\) as an \(n\times 1\) matrix. We write the \(i\)-th coordinate of \(v\) as \(v_i\) and the \((i,j)\)-th coordinate of \(M\) as \(M_{i,j}\) (i.e. the coordinate in the \(i\)-th row and the \(j\)-th column.) We often write a vector \(v\) as \((v_1,\ldots,v_n)\) but we still mean that it’s a column vector unless we say otherwise.

• If \(\alpha \in \mathbb{F}\) is a scalar (i.e., a number) and \(v \in \mathbb{F}^n\) is a vector then \(\alpha v\) is the vector \((\alpha v_1 ,\ldots, \alpha v_n)\). If \(u,v\) are \(n\) dimensional vectors then \(u+v\) is the vector \((u_1+v_1,\ldots,u_n+v_n)\).

• A linear subspace \(V \subseteq \mathbb{F}^n\) is a non-empty set of vectors such that for every vectors \(u,v \in V\) and \(\alpha,\beta \in \mathbb{F}\), \(\alpha u + \beta v \in V\). In particular this means that \(V\) contains the all zero vector \(0^n\) (can you see why?). A subset \(A \subseteq V\) is linearly independent if there is no collection \(a_1,\ldots,a_k \in A\) and scalars \(\alpha_1,\ldots,\alpha_k\) such that \(\sum \alpha_i a_i = 0^n\). It is known (and not hard to prove) that if \(A\) is linearly independent then \(|A| \leq n\). It is known that for every such linear subspace there is a linearly independent set \(B = \{ b_1,\ldots,b_d \}\) of vectors, with \(d \leq n\), such that for every \(u \in V\) there exist \(\alpha_1,\ldots,\alpha_d\) such that \(v = \sum \alpha_i b_i\). Such a set is known as a basis for \(V\). A subspace \(V\) has many bases, but all of them have the same size \(d\) which is known as the dimension of \(V\). An affine subspace is a set \(U\) of the form \(\{ u_0 + v : v\in V \}\) where \(V\) is a linear subspace. We can also write \(U\) as \(u_0 + V\). We denote the dimension of \(U\) as the dimension of \(V\) in such a case.

• The inner product (also known as “dot product”) \(\langle u,v \rangle\) between two vectors of the same dimension \(n\) is defined as \(\sum u_i v_i\) (addition done in the field \(\mathbb{F}\)).²

• The matrix product \(\ensuremath{\mathit{AB}}\) of an \(m \times k\) and a \(k\times n\) matrix results in an \(m\times n\) matrix. If we think of the rows of \(A\) as the vectors \(A_1,\ldots,A_m \in \mathbb{F}^k\) and the columns of \(B\) as \(B_1,\ldots,B_n \in \mathbb{F}^k\), then the \((i,j)\)-th coordinate of \(\ensuremath{\mathit{AB}}\) is \(\langle A_i , B_j \rangle\). Matrix product is associative and satisfies the distributive law but is not commutative: there are pairs of square matrices \(A,B\) such that \(\ensuremath{\mathit{AB}} \neq \ensuremath{\mathit{BA}}\).

• The transpose of an \(n\times m\) matrix \(A\) is the \(m\times n\) matrix \(A^\top\) such that \((A^\top)_{i,j} = A_{j,i}\).

• The inverse of a square \(n\times n\) matrix \(A\) is the matrix \(A^{-1}\) such that \(\ensuremath{\mathit{AA}}^{-1} = I\) where \(I\) is the \(n\times n\) identity matrix such that \(I_{i,j}=1\) if \(i=j\) and \(I_{i,j}=0\) otherwise.

• The rank of an \(m\times n\) matrix \(A\) is the minimum number \(r\) such that we can write \(A\) as \(\sum_{i=1}^r u_i(v_i)^\top\) where \(u_i \in \mathbb{F}^m\) and \(v_i \in \mathbb{F}^n\). We can think of the \(u_i\)’s as the columns of an \(m\times r\) matrix \(U\) and the \(v_i\)’s as the rows of an \(r\times n\) matrix \(V\), and hence the rank of \(A\) is the minimum \(r\) such that \(A=\ensuremath{\mathit{UV}}\) where \(U\) is \(m\times r\) and \(V\) is \(r\times n\). It can be shown that an \(n\times n\) matrix is full rank if and only if it has an inverse.

• Solving linear equations can be thought of as the task of given an \(m \times n\) matrix \(A\) and \(m\)-dimensional vector \(y\), finding the \(n\)-dimensional vector \(x\) such that \(Ax = y\). If the rank of \(A\) is at least \(n\) (which in particular means that \(m \geq n\)) then by dropping \(m-n\) rows of \(A\) and coordinates of \(y\) we can obtain the equation \(A'x = y'\) where \(A'\) is an \(n\times n\) matrix that has an inverse. In this case a solution (if it exists) will be equal to \((A')^{-1}y\). If for a set of equations we have \(m>n\) and we can find two such matrices \(A',A''\) such that \((A')^{-1}y \neq (A'')^{-1}y\) then we say it is over determined and in such a case it has no solutions. If a set of equations has more variables \(n\) than equations \(m\) we say it’s under-determined. In such a case it either has no solutions or the solutions form an affine subspace of dimension at least \(n-m\).

• The gaussian elimination algorithm can be used to obtain, given a set of equations \(Ax = y\) a solution to \(x\) if such exists or a certification that no solution exists. It can be executed in time polynomial in the dimensions and the bit complexity of the numbers involved. This algorithm can also be used to obtain an inverse of a given matrix \(A\), if such an inverse exists.

A world without Gaussian elimination

The general approach people use to get a public key encryption is to obtain a hard computational problem with some mathematical structure. We’ve seen this in the discrete logarithm problem, where the task is to invert the map \(a \mapsto g^a \pmod{p}\), and the integer factoring problem, where the task is to invert the map \(a,b \mapsto a\cdot b\). Perhaps the simplest structure to consider is the task of solving linear equations.

Pretend that we didn’t know of Gaussian elimination,³ and that if we picked a “generic” matrix \(A\) then the map \(x \mapsto Ax\) would be hard to invert. (Here and elsewhere, our default interpretation of a vector \(x\) is as a column vector, and hence if \(x\) is \(n\) dimensional and \(A\) is \(m\times n\) then \(Ax\) is \(m\) dimensional. We use \(x^\top\) to denote the row vector obtained by transposing \(x\).) Could we use that to get a public key encryption scheme?

Here is a concrete approach. Let us fix some prime \(q\) (think of it as polynomial size, e.g., \(q\) is smaller than \(1024\) or so, though people can and sometimes do consider \(q\) of exponential size), and all computation below will be done modulo \(q\). The secret key is a vector \(x\in\Z_q^n\), and the public key is \((A,y)\) where \(A\) is a random \(m\times n\) matrix with entries in \(\Z_q\) and \(y=Ax\). Under our assumption, it is hard to recover the secret key from the public key, but how do we use the public key to encrypt?

The crucial observation is that even if we don’t know how to solve linear equations, we can still combine several equations to get new ones. To keep things simple, let’s consider the case of encrypting a single bit.

If \(a_1,\ldots,a_m\) are the rows of \(A\), we can think of the public key as the set of equations \(\langle a_1,x \rangle=y_1,\ldots, \langle a_m,x \rangle=y_m\) in the unknown variables \(x\). The idea is that to encrypt the value \(0\) we will generate a new correct equation on \(x\), while to encrypt the value \(1\) we will generate an incorrect equation. To decrypt a ciphertext \((a,\sigma)\in \Z_q^{n+1}\), we think of it as an equation of the form \(\langle a,x \rangle=\sigma\) and output \(1\) if and only if the equation is incorrect.

How does the encrypting algorithm, that does not know \(x\), get a correct or incorrect equation on demand? One way would be to simply take two equations \(\langle a_i,x \rangle=y_i\) and \(\langle a_j,x \rangle=y_j\) and add them together to get the equation \(\langle a_i+a_j,x \rangle=y_i+y_j\). This equation is correct and so one can use it to encrypt \(0\), while to encrypt \(1\) we simply add some fixed nonzero number \(\alpha\in\Z_q\) to the right hand side to get the incorrect equation \(\langle a_i+a_j,x \rangle= y_i+y_j + \alpha\). However, even if it’s hard to solve for \(x\) given the equations, an attacker (who also knows the public key \((A,y)\)) can try itself all pairs of equations and do the same thing.

Our solution for this is simple- just add more equations! If the encryptor adds a random subset of equations then there are \(2^m\) possibilities for that, and an attacker can’t guess them all. That is, if the rows of \(A\) are \(a_1,\ldots,a_m\), then we can pick a vector \(w \in \{0,1\}^m\) at random, and consider the equation \(\langle a ,x \rangle = y\) where \(a = \sum w_i a_i\) and \(y = \sum w_i y_i\). In other words, we can think of this as the equation \(w^\top A x = \langle w,y \rangle\) (note that \(\langle w,y \rangle = w^\top y\) and so we can think of this as the equation that we obtain from \(Ax = y\) by multiplying both sides on the left by the row vector \(w^\top\)).

Thus, at least intuitively, the following encryption scheme would be “secure” in the Gaussian elimination-free world of attackers that haven’t taken freshman linear algebra:

Scheme “LwoE-ENC”: Public key encryption under the hardness of “learning linear equations without errors”.

• Key generation: Pick random \(m\times n\) matrix \(A\) over \(\Z_q\), and \(x\leftarrow_R\Z_q^n\), the secret key is \(x\) and the public key is \((A,y)\) where \(y=Ax\).

• Encryption: To encrypt a message \(b\in\{0,1\}\), pick \(w\in\{0,1\}^m\) and output \(w^\top A,\langle w,y \rangle+\alpha b\) for some fixed nonzero \(\alpha\in\Z_q\).

• Decryption: To decrypt a ciphertext \((a,\sigma)\), output \(0\) iff \(\langle a,x \rangle=\sigma\).

Security in the real world.

Like it or not (and cryptographers typically don’t) Gaussian elimination is possible in the real world and the scheme above is completely insecure. However, the Gaussian elimination algorithm is extremely brittle.
Errors tend to be amplified when you combine equations. This is usually thought of as a bad thing, and numerical analysis is much about dealing with this issue. However, from the cryptographic point of view, these errors can be our saving grace and enable us to salvage the security of the ridiculous scheme above.

To see why Gaussian elimination is brittle, let us recall how it works. Think of \(m=n\) for simplicity. Given equations \(Ax=y\) in the unknown variables \(x\), the goal of Gaussian elimination is to transform them into the equations \(Ix = y'\) where \(I\) is the identity matrix (and hence the solution is simply \(x=y'\)). Recall how we do it: by rearranging and scaling, we can assume that the top left corner of \(A\) is equal to \(1\), and then we add the first equation to the other equations (scaled appropriately) to zero out the first entry in all the other rows of \(A\) (i.e., make the first column of \(A\) equal to \((1,0,\ldots,0)\)) and continue onwards to the second column and so on and so forth.

Now, suppose that the equations were noisy, in the sense that we added to \(y\) a vector \(e\in\Z_q^m\) such that \(|e_i|<\delta q\) for every \(i\).⁴ Even ignoring the effect of the scaling step, simply adding the first equation to the rest of the equations would typically tend to increase the relative error of equations \(2,\ldots,m\) from \(\approx \delta\) to \(\approx 2\delta\). Now, when we repeat the process, we increase the error of equations \(3,\ldots,m\) from \(\approx 2\delta\) to \(\approx 4\delta\), and we see that by the time we’re done dealing with about \(n/2\) variables, the remaining equations have error level roughly \(2^{n/2}\delta\). So, unless \(\delta\) was truly tiny (and \(q\) truly big, in which case the difference between working in \(\Z_q\) and simply working with integers or rationals disappears), the resulting equations have the form \(Ix = y' + e'\) where \(e'\) is so big that we get no information on \(x\).

The Learning With Errors (LWE) conjecture is that this is inherent:

Conjecture (Learning with Errors, Regev 2005): Let \(q=q(n)\) and \(\delta=\delta(n)\) be some functions. The Learning with Error (LWE) conjecture with respect to \(q,\delta\),denoted as \(\ensuremath{\mathit{LWE}}_{q,\delta}\), is the following conjecture: for every polynomial \(m(n)\) and polynomial-time adversary \(R\),

\[\Pr[ R(A,Ax+e) = x ] < negl(n)\]

where for \(q=q(n)\) and \(\delta = \delta(n)\), this probability is taken over \(A\) a random \(m\times n\) matrix over \(\Z_q\), \(x\) a random vector in \(\Z_q^n,\) and \(e\) a random “noise vector” in \(\Z_q^m\) where \(|e_i| < \delta q\) for every \(i \in [m]\).⁵

The LWE conjecture (without any parameters) is that there is some absolute constant \(c\) such that for every polynomial \(p(n)\) there, if \(q(n) > p(n)^c\) then LWE holds with respect to \(q(n)\) and \(\delta(n)=1/p(n)\).⁶

It is important to note the order of quantifiers in the learning with error conjecture. If we want to handle a noise of low enough magnitude (say \(\delta(n) = 1/n^2\)) then we need to choose the modulos \(q\) to be large enough (for example it is believed that \(q > n^4\) will be good enough for this case) and then the adversary can choose \(m(n)\) to be as big a polynomial as they like, and of course run in time which is an arbitrary polynomial in \(n\). Therefore we can think of such an adversary \(R\) as getting access to a “magic box” that they can use \(m=poly(n)\) number of times to get “noisy equations on \(x\)” of the form \((a_i,y_i)\) with \(a_i\in \Z_q^n\), \(y_i \in \Z_q\) where \(y_i = \langle a_i, x \rangle + e_i\).

Search to decision

It turns out that if the LWE is hard, then it is even hard to distinguish between random equations and nearly correct ones:

An LWE based encryption scheme

We can now show the secure variant of our original encryption scheme:

LWE-based encryption LWE-ENC:

• Parameters: Let \(\delta(n)=1/n^4\) and let \(q=poly(n)\) be a prime such that LWE holds w.r.t. \(q,\delta\). We let \(m = n^2\log q\).

• Key generation: Pick \(x\in\Z_q^n\). The private key is \(x\) and the public key is \((A,y)\) with \(y=Ax+e\) with \(e\) a \(\delta\)-noise vector and \(A\) a random \(m\times n\) matrix.

• Encrypt: To encrypt \(b\in\{0,1\}\) given the key \((A,y)\), pick \(w\in\{0,1\}^m\) and output \(w^\top A, \langle w,y \rangle+b\floor{q/2}\) (all computations are done in \(\Z_q\)).

• Decrypt: To decrypt \((a,\sigma)\), output \(0\) iff \(|\langle a,x \rangle-\sigma|<q/10\).

Unlike our typical schemes, here it is not immediately clear that this encryption is valid, in the sense that the decrypting an encryption of \(b\) returns the value \(b\). But this is the case:

We now prove security of the LWE based encryption:

For a public key encryption scheme with messages that are just bits, CPA security means that an encryption of \(0\) is indistinguishable from an encryption of \(1\), even given the public key. Thus Theorem 11.4 will follow from the following lemma:

We now prove Lemma 11.5, which will complete the proof of Theorem 11.4.

But what are lattices?

You can think of a lattice as a discrete version of a subspace. A lattice \(L\) is simply a discrete subset of \(\mathbb{R}^n\) such that if \(u,v\in L\) and \(a,b\) are integers then \(au+bv\in L\).⁹ A lattice is given by a basis which simply a matrix \(B\) such that every vector \(u\in L\) is obtained as \(u=Bx\) for some vector of integers \(x\). It can be shown that we can assume without loss of generality that \(B\) is full dimensional and hence it’s an \(n\) by \(n\) invertible matrix. Note that given a basis \(B\) we can generate vectors in \(L\), as well as test whether a vector \(v\) is in \(L\) by testing if \(B^{-1}v\) is an integer vector. There can be many different bases for the same lattice, and some of them are easier to work with than others (see Figure 11.3).

Some classical computational questions on lattices are:

• Shortest vector problem: Given a basis \(B\) for \(L\), find the nonzero vector \(v\) with smallest norm in \(L\).

• Closest vector problem: Given a basis \(B\) for \(L\) and a vector \(u\) that is not in \(L\), find the closest vector to \(u\) in \(L\).

• Bounded distance decoding: Given a basis \(B\) for \(L\) and a vector \(u\) of the form \(u=v+e\) where \(v\) is in \(L\), and \(e\) is a particularly short “error” vector (so in particular no other vector in the lattice is within distance \(\|e\|\) to \(u\)), recover \(v\). Note that this is a special case of the closest vector problem.

In particular, if \(V\) is a linear subspace of \(\Z_q^n\), we can think of it also as a lattice \(\hat{V}\) of \(\mathbb{R}^n\) where we simply say that that a vector \(\hat{u}\) is in \(\hat{V}\) if all of \(\hat{u}\)’s coordinates are integers and if we let \(u_i = \hat{u}_i \pmod{q}\) then \(u\in V\). The learning with error task of recovering \(x\) from \(Ax+e\) can then be thought of as an instance of the bounded distance decoding problem for \(\hat{V}\).

A natural algorithm to try to solve the closest vector and bounded distance decoding problems is that to take the vector \(u\), express it in the basis \(B\) by computing \(w = B^{-1}u\), and then round all the coordinates of \(w\) to obtain an integer vector \(\tilde{w}\) and let \(v=B\tilde{w}\) be a vector in the lattice. If we have an extremely good basis \(L\) for the lattice then \(v\) may indeed be the closest vector in the lattice, but in other more “skewed” bases it can be extremely far from it.

Ring based lattices

One of the biggest issues with lattice based cryptosystem is the key size. In particular, the scheme above uses an \(m\times n\) matrix where each entry takes \(\log q\) bits to describe. (It also encrypts a single bit using a whole vector, but more efficient “multi-bit” variants are known.) Schemes using ideal lattices are an attempt to get more practical variants. These have very similar structure except that the matrix \(A\) chosen is not completely random but rather can be described by a single vector. One common variant is the following: we fix some polynomial \(p\) over \(\Z_q\) with degree \(n\) and then treat vectors in \(\Z_q^n\) as the coefficients of \(n-1\) degree polynomials and always work modulo this polynomial \(p()\). (By this I mean that for every polynomial \(t\) of degree at least \(n\) we write \(t\) as \(ps+r\) where \(p\) is the polynomial above, \(s\) is some polynomial and \(r\) is the “remainder” polynomial of degree \(<n\); then \(t \pmod{p} = r\).) Now for every fixed polynomial \(t\), the operation \(A_t\) which is defined as \(s \mapsto ts \pmod{p}\) is a linear operation mapping polynomials of degree at most \(n-1\) to polynomials of degree at most \(n-1\), or put another way, it is a linear map over \(\Z_q^n\). However, the map \(A_d\) can be described using the \(n\) coefficients of \(t\) as opposed to the \(n^2\) description of a matrix. It also turns out that by using the Fast Fourier Transform we can evaluate this operation in roughly \(n\) steps as opposed to \(n^2\). The ideal lattice based cryptosystem use matrices of this form to save on key size and computation time. It is still unclear if this structure can be used for attacks; recent papers attacking principal ideal lattices have shown that one needs to be careful about this.

One ideal-lattice based system is the “New Hope” cryptosystem (see also paper) that has been experimented with by Google. People have also made highly optimized general (non ideal) lattice based constructions, see in particular the “Frodo” system (paper here, can you guess what’s behind the name?). Both New Hope and Frodo have been submitted to the NIST competition to select a “post quantum” public key encryption standard.

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Compiled on 11/17/2021 22:36:38

Copyright 2021, Boaz Barak.
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