Learning with errors
Learning with errors (LWE) is the computational problem of inferring a linear -ary function  over a finite ring from given samples  some of which may be erroneous.
The LWE problem is conjectured to be hard to solve, and thus be useful in cryptography.

More precisely, the LWE problem is defined as follows. Let  denote the ring of integers modulo  and let
 denote the set of -vectors over . There exists a certain unknown linear function , and the input to the LWE problem is a sample of pairs , where  and , so that with high probability . Furthermore, the deviation from the equality is according to some known noise model. The problem calls for finding the function , or some close approximation thereof, with high probability.

The LWE problem was introduced by Oded Regev in 2005 (who won the 2018 Gödel Prize for this work), it is a generalization of the parity learning problem. Regev showed that the LWE problem is as hard to solve as several worst-case lattice problems. Subsequently, the LWE problem has been used as a hardness assumption to create public-key cryptosystems, such as the ring learning with errors key exchange by Peikert.

Definition 
Denote by  the additive group on reals modulo one. 
Let  be a fixed vector.
Let  be a fixed probability distribution over .
Denote by  the distribution on  obtained as follows.
 Pick a vector  from the uniform distribution over , 
 Pick a number  from the distribution ,
 Evaluate , where  is the standard inner product in , the division is done in the field of reals (or more formally, this "division by " is notation for the group homomorphism  mapping  to ), and the final addition is in .
 Output the pair .

The learning with errors problem  is to find , given access to polynomially many samples of choice from .

For every , denote by  the one-dimensional Gaussian with zero mean and variance
, that is, the density function is  where , and let  be the distribution on  obtained by considering  modulo one.  The version of LWE considered in most of the results would be

Decision version 

The LWE problem described above is the search version of the problem. In the decision version (DLWE), the goal is to distinguish between noisy inner products and uniformly random samples from  (practically, some discretized version of it). Regev showed that the decision and search versions are equivalent when  is a prime bounded by some polynomial in .

Solving decision assuming search 
Intuitively, if we have a procedure for the search problem, the decision version can be solved easily: just feed the input samples for the decision problem to the solver for the search problem. Denote the given samples by . If the solver returns a candidate , for all , calculate .  If the samples are from an LWE distribution, then the results of this calculation will be distributed according , but if the samples are uniformly random, these quantities will be distributed uniformly as well.

Solving search assuming decision 
For the other direction, given a solver for the decision problem, the search version can be solved as follows: Recover  one coordinate at a time. To obtain the first coordinate, , make a guess , and do the following. Choose a number  uniformly at random. Transform the given samples  as follows. Calculate .  Send the transformed samples to the decision solver.

If the guess  was correct, the transformation takes the distribution  to itself, and otherwise, since  is prime, it takes it to the uniform distribution. So, given a polynomial-time solver for the decision problem that errs with very small probability, since  is bounded by some polynomial in , it only takes polynomial time to guess every possible value for  and use the solver to see which one is correct.

After obtaining , we follow an analogous procedure for each other coordinate .  Namely, we transform our  samples the same way, and transform our  samples by calculating , where the  is in the  coordinate.

Peikert showed that this reduction, with a small modification, works for any  that is a product of distinct, small (polynomial in ) primes.  The main idea is if , for each , guess and check to see if  is congruent to , and then use the Chinese remainder theorem to recover .

Average case hardness 
Regev showed the random self-reducibility of the LWE and DLWE problems for arbitrary  and .  Given samples  from , it is easy to see that  are samples from .

So, suppose there was some set  such that , and for distributions , with , DLWE was easy.

Then there would be some distinguisher , who, given samples , could tell whether they were uniformly random or from .  If we need to distinguish uniformly random samples from , where  is chosen uniformly at random from , we could simply try different values  sampled uniformly at random from , calculate  and feed these samples to .  Since  comprises a large fraction of , with high probability, if we choose a polynomial number of values for , we will find one such that , and  will successfully distinguish the samples.

Thus, no such  can exist, meaning LWE and DLWE are (up to a polynomial factor) as hard in the average case as they are in the worst case.

Hardness results

Regev's result 
For a n-dimensional lattice , let smoothing parameter  denote the smallest  such that  where  is the dual of  and  is extended to sets by summing over function values at each element in the set. Let  denote the discrete Gaussian distribution on  of width  for a lattice  and real . The probability of each  is proportional to .

The discrete Gaussian sampling problem(DGS) is defined as follows: An instance of  is given by an -dimensional lattice  and a number . The goal is to output a sample from . Regev shows that there is a reduction from  to  for any function .

Regev then shows that there exists an efficient quantum algorithm for  given access to an oracle for  for integer  and  such that . This implies the hardness for LWE. Although the proof of this assertion works for any , for creating a cryptosystem, the  has to be polynomial in .

Peikert's result 

Peikert proves that there is a probabilistic polynomial time reduction from the  problem in the worst case to solving  using  samples for parameters , ,  and .

Use in cryptography 

The LWE problem serves as a versatile problem used in construction of several cryptosystems. In 2005, Regev showed that the decision version of LWE is hard assuming quantum hardness of the lattice problems  (for  as above) and  with ). In 2009, Peikert proved a similar result assuming only the classical hardness of the related problem . The disadvantage of Peikert's result is that it bases itself on a non-standard version of an easier (when compared to SIVP) problem GapSVP.

Public-key cryptosystem 
Regev proposed a public-key cryptosystem based on the hardness of the LWE problem. The cryptosystem as well as the proof of security and correctness are completely classical. The system is characterized by  and a probability distribution  on . The setting of the parameters used in proofs of correctness and security is
 , usually a prime number between  and .
  for an arbitrary constant 
  for , where  is a probability distribution obtained by sampling a normal variable with mean  and standard variation  and reducing the result modulo .

The cryptosystem is then defined by:
 Private key: Private key is an  chosen uniformly at random.
 Public key: Choose  vectors  uniformly and independently. Choose error offsets  independently according to . The public key consists of 
 Encryption: The encryption of a bit  is done by choosing a random subset  of  and then defining  as
 
 Decryption: The decryption of  is  if  is closer to  than to , and  otherwise.

The proof of correctness follows from choice of parameters and some probability analysis. The proof of security is by reduction to the decision version of LWE: an algorithm for distinguishing between encryptions (with above parameters) of  and  can be used to distinguish between  and the uniform distribution over

CCA-secure cryptosystem 

Peikert proposed a system that is secure even against any chosen-ciphertext attack.

Key exchange 

The idea of using LWE and Ring LWE for key exchange was proposed and filed at the University of Cincinnati in 2011 by Jintai Ding. The idea comes from the associativity of matrix multiplications, and the errors are used to provide the security. The paper appeared in 2012 after a provisional patent application was filed in 2012.

The security of the protocol is proven based on the hardness of solving the LWE problem. In 2014, Peikert presented a key-transport scheme following the same basic idea of Ding's, where the new idea of sending an additional 1-bit signal for rounding in Ding's construction is also used. The "new hope" implementation selected for Google's post-quantum experiment, uses Peikert's scheme with variation in the error distribution.

See also 
Post-quantum cryptography
Lattice-based cryptography
Ring learning with errors key exchange
Short integer solution (SIS) problem

References

Machine learning
Cryptography
Post-quantum cryptography