1 Introduction
Matrix factorization CanRec09 ; KesMonOh08 is a popular method to build a recommender system. As exemplified by the Netflix Prize competition NetflixPrize , it has become a dominant technology within collaborativefiltering recommenders. Matrix factorization provides a better predictive accuracy compared to classical neighborhood methods while at the same time is scalable and offers much flexibility for modeling a variety of reallife situations KorBelVol09 .
The coldstart problem
A major problem facing collaborativefiltering recommender systems is how to provide recommendations when rating data is too sparse for a subset of users or items. As a special case, the socalled coldstart problem SPUP02 is how to make recommendations to new users who have not yet rated any item or to deal with new items that have not yet been rated by users.
The coldstart problem is usually addressed by incorporating additional input sources to compensate for the lack of rating data. In addition to ratings, the analyst may for example collect certain user attributes, such as gender, age, or other demographic information AdoTuz05 ; Koren08 .
Another approach for dealing with the coldstart problem is to ask users to rate a minimum number of (well chosen) items RACLMKR02 .
Inference attacks
Relying on additional input sources to address the coldstart problem may be difficult to deploy in practice as privacyconscious users may be reluctant to supply some of their attributes. The second approach does not require to collect extra information beyond the ratings and is very efficient. Unfortunately, the additional received ratings may reveal a lot of information about a user to the analyst. Recent research has indeed demonstrated that this can be used by the analyst to infer private user attributes such as political affiliation KosStiGra13 ; SAPBDKOT13 , sexual orientation KosStiGra13 , age WBIT12 , gender SAPBDKOT13 ; WBIT12 , and even drug use KosStiGra13 . Further privacy threats are reported in BWIT14 ; CKNFS11 ; SP:NarShm08 .
A natural question therefore raised in IMWBFT14 is how a privacyconscious user can benefit from recommender systems while preventing the inference of her private information.
Contributions
In this paper, we show how a privacyconscious user can learn her profile without revealing any information to the analyst. The protocol is practical and proven secure against semihonest adversaries. The communication complexity of the protocol only grows with the squareroot of the number of items.
Once the privacyconscious user learns her profile , she can run a straightforward protocol to learn the predicted rating of any item in the database. This indeed only requires to compute the inner product between the user profile (known to the user) and the item profile (known to the analyst) (CCS:NIWJTB13, , Sect. 4.1).
Related work
In IMWBFT14 , Ioannidis et al. propose a learning protocol which enables the user to prevent the analyst from learning some (previously defined) private user attributes. This protocol perfectly hides these chosen attributes to the analyst, in an informationtheoretic way. The authors also prove that no such protocol can be more accurate, when the analyst ends up knowing the resulting profile, nor can disclose less information for the same accuracy.
Unfortunately, this protocol has also several drawbacks, most of them inherent to the fact it is informationtheoretically secure and does not rely on computational assumptions. First, this protocol still needs to disclose some information about the analyst database to everybody. Second, this protocol is not as accurate as a nonprivacypreserving protocol would be. This is inherent to the fact that Ioannidis et al. restricted themselves to protocols where the analyst learns an approximate profile of the user at the end, so that the resulting user profile shall not contain any information about the private attribute. Third, it can only hide a small fixed set of attributes: all attributes which are not explicitly hidden may be recovered by the analyst. And it may be hard for a user to decide which attributes are really important to her, due to the wide range of possible attributes. Finally, the analyst needs to ask users^{1}^{1}1In the simplest scenario, we have to restrict to nonprivacyconscious users. But it would also be possible to compute item profiles using privacypreserving matrix factorization CCS:NIWJTB13 . to reveal which attributes they deem private. This may not only bother a lot these users, but also brings up the question of the reliability of these data. No user will be likely admitting she is a drug addict, for example, even if she is ensured that this data will not be disclosed.^{2}^{2}2Notice in particular that, in the privacypreserving matrix factorization protocol in CCS:NIWJTB13 , in case of collusion between the CSP (Crypto Service Provider) and the analyst, it is possible to recover all data sent by the user. This means that governmental agencies may force the recommendation systems to disclose these private user attributes.
2 Preliminaries
is the field of real numbers. For any integer , is the ring of integers modulo , while is its multiplicative group. Vectors are always column vectors and are denoted as or . Matrices are denoted with capital letters.
2.1 Cryptographic tools
Publickey encryption
A publickey encryption scheme is defined by three algorithms: , , and . generates a matching pair of public key and secret key , given a security parameter (unary notation). The public key is used to encrypt a message into a ciphertext : . The secret key is used to decrypt a ciphertext : . We assume that the encryption scheme is perfectly correct and semantically secure (i.e., INDCPA) GolMic84 .
Homomorphic encryption
An additively homomorphic encryption scheme is such that the message set is an additive group, and there exists a randomized operation such that is distributed identically to a fresh ciphertext of . This operation can be extended to a scalar multiplication by an integer : is a fresh ciphertext of ; that is, ( times).
To simplify the notation, we will sometimes use for and omit when clear from the context. We so have and .
Example 1 (Paillier encryption scheme).
We recall the Paillier encryption scheme EC:Paillier99 , which is an homomorphic encryption scheme that is semantically secure under the Decisional Composite Residuosity (DCR) assumption. generates two large equallength primes and , computes , and sets and . The public key is while the secret key is . picks a uniformly random integer and returns . returns where . The scheme is additively homomorphic: given and , with .
Oblivious transfer
A outof oblivious transfer (OT) protocol is a cryptographic protocol between two parties: a sender and a receiver. The receiver has an index as input. The sender knows a database . At the end of the protocol, the receiver learns , while the sender learns nothing.
As in our protocol is the number of items in the database, we need to use practical OT protocols with communication complexity sublinear in . We propose to use as outof OT the basic PIR (Private Information Retrieval) protocol in (PKC:OstSke07, , Sect. 2.2) using the Paillier homomorphic encryption scheme, together with a classical 1outof OT SODA:NaoPin01 which is used to mask the PIR database. The resulting OT has two rounds (one message from the receiver to the sender followed by one message from the sender to the receiver) and its communication complexity is proportional to .
2.2 Matrix factorization
The goal of matrix factorization is to predict unobserved ratings for some user and some item , given access to a set of user/item pairs for which a rating has been generated. Matrix factorization provides dimensional vectors such that
(1) 
This allows the analyst to predict missing ratings (i.e., those with ). Vector is referred to as the profile of user while vector as the profile of item .
2.3 Learning the profile of a user
Specifically, when a new user wishes to use the service, she submits a batch of ratings for a subset of
items. Upon receiving these ratings, the analyst can estimate her profile
through the following leastsquares estimation,^{3}^{3}3To ease the presentation, linear regression is considered but the proposed techniques readily apply to the more general setting of ridge regression.
(2) 
and subsequently predict ratings for items , using Eq. (1).
Defining matrix and column vector , the profile of a user can be computed as follows:
(3) 
3 Our learning protocol
We design a tworound learning protocol between a privacyconscious user and an analyst, allowing the user to learn her profile from her (private) ratings , where . At the end of the protocol, the analyst will learn nothing (except the size of ), while the user will only learn her profile and nothing else about the analyst database (except the dimension , the database of items and its size , and bounds and on entries of ratings and of profiles of items , respectively).
We insist that our protocol hides the set of actual items that the user is rating as they might already leak significant information about her. If an upper bound on is known, the exact size of can trivially be masked by adding fake items (with profile and fake rating ) so that the protocol always uses a set of size .
3.1 Protocol
Consider the ring . We assume that is either a prime or is hard to factor, so that for all intents and purposes behaves as a field (since a nonzero noninvertible element of would yield a factor of ). Up to using fixed point arithmetic (e.g., by multiplying values by some integer ), we suppose that the entries of and are integers, and so can be considered as elements of .
Round 1
The user generates a key pair for the homomorphic encryption scheme and encrypts her ratings : for . She also initiates independent OT protocols as a receiver with respective selection indexes .
Round 2
The analyst generates and computes the following matrices (over and over the ciphertext space respectively):^{4}^{4}4We slightly abuse notation here. For vectors, the bracket notation and and operators are applied componentwise.
where and are uniformly random matrices and vectors summing up to zero in and respectively, and is a uniform matrix in (the group of invertible matrices in ).
The analyst then answers the th OT message from the user, as an OT sender with database .
Final step
The user receives through the OT protocols. We write and . We then remark that, in :
So if is large enough, the user can compute back using rational reconstruction WanGuyDav82 ; FC:FouSteWac02 (we recall that satisfies Eq. 3 over the rationals, and that is not necessarily an integer).
Bounds for correctness
The scheme is correct when the above rational reconstruction succeeds. From WanGuyDav82 ; FC:FouSteWac02 and Hadamard’s inequality, we can show correctness when , where and are upper bounds on the absolute values of the coefficients of the item profiles and of the ratings , respectively. For example, if , , , , this is already satisfied for an integer of bits.
Security
Security against semihonest adversaries follows from the security of the OT protocol, the INDCPA property of the homomorphic encryption scheme, and from the following fact: since is invertible and is a group, and only reveal .
3.2 Instantiation using Paillier homomorphic encryption scheme
The scheme can be instantiated using the Paillier encryption scheme and the OT described in Section 2. We can use the internal construction of the OT, to avoid sending ciphertexts of . Concretely, in the OT construction, the user encrypts a vector used to “select” the correct value to be received. If we use two OT protocols for each , one for and one for (instead of a single one for the pair ), then for the second OT, the user just encrypts instead of , she will receive times the value to be received.
The resulting protocol for items, dimension , and ratings from the user (modulus of size bits for Paillier encryption scheme and an elliptic curve over a 256bit prime field for the base OT SODA:NaoPin01 ), has the following performance on a nonoptimized singlethread implementation (on a laptop, CPU Intel® i77567U, GHz, turbo GHz): less than s to generate the first round by the user, less than s to generate the second round by the analyst, less than s to finalize the protocol by the user. The user requires less than s of computation (excluding communication). The analyst time is mostly spent in the exponentiations required in the OT protocol (modulo ): there are of them. These exponentiations can be trivially parallelized. The communication complexity is less than MB and essentially grows linearly with .
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