Secure probabilistic analytics using homomorphic encryption

Abstract

An example method for performing a secure probabilistic analytic includes acquiring, by a client, an analytic, at least one analytic parameter associated with the analytic, and an encryption scheme. The encryption scheme can include a public key for encryption and a private key for decryption. The method further includes generating, using the encryption scheme, at least one analytical vector based on the analytic and analytic parameter, and sending the analytical vector and the encryption scheme to at least one server. The method includes generating, by the server based on the encryption scheme, a set of terms from a data set, evaluating the analytical vector over the set of terms to obtain an encrypted result; estimating, by the server, a probabilistic error of the encrypted result; and sending, by the server, the encrypted result and the probabilistic error to the client where the encrypted result is decrypted.

Description

TECHNICAL FIELD

This disclosure relates to the technical field of encryption and decryption of data. More specifically, this disclosure relates to systems and methods for performing secure probabilistic analytics using a homomorphic encryption.

BACKGROUND

With development of computer technologies, many sensitive data, such as financial information and medical records can be kept on remote servers or cloud-based computing resources. Authorized users can access the sensitive data using applications running, for example, on their personal computing devices. Typically, personal computing devices are connected, via data networks, to servers or cloud-based computing resources. Therefore, the sensitive data can be subject to unauthorized access.

Encryption techniques, such as a homomorphic encryption, can be applied to the sensitive data to prevent unauthorized access. The encryption techniques can be used to protect “data in use”, “data in rest”, and “data in transit”. A homomorphic encryption is a form of encryption in which a specific algebraic operation (generally referred to as addition or multiplication) performed on plaintext, is equivalent to another operation performed on ciphertext. For example, in Partially Homomorphic Encryption (PHE) schemes, multiplication in ciphertext is equal to addition of the same values in plaintext.

SUMMARY

This summary is provided to introduce a selection of concepts in a simplified form that are further described in the Detailed Description below. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

Generally, the present disclosure is directed to the technology for secure data processing. Some embodiments of the present disclosure may facilitate a secure transmission of analytics from a client device to remote computing resource(s) for performing analytics over a data source and a secure transmission of results of the analytics from the computing resources back to the client device.

According to one example embodiment of the present disclosure, a method for performing a secure probabilistic analysis using homomorphic encryption is provided. The method may include receiving, from a client, by at least one server, at least one analytic vector, a term generation function, and a keyed hash function. The at least one analytic vector can be encrypted using a homomorphic encryption scheme. The homomorphic encryption scheme can include a public key for encryption and a private key for decryption. The method may further include extracting, by the at least one server, a set of term components from a data set using the term generation function and the keyed hashed function. The method may further include evaluating, by the at least one server, the at least one analytic vector over the set of term components to obtain at least one encrypted result. The method may further include estimating, by the at least one server, a probabilistic error of the at least one encrypted result. The method may allow sending, by the at least one server, the at least one encrypted result and the probabilistic error to the client, wherein the client is configured to decrypt the at least one encrypted result using the homomorphic encryption scheme.

In some embodiments, the homomorphic encryption scheme includes a partially homomorphic encryption scheme. The partially homomorphic encryption scheme may include at least one of the following: a Rivest, Shamir and Adleman cryptosystem, Elgamal cryptosystem, Benaloh cryptosystem, Goldwasser-Micali cryptosystem, and Pallier cryptosystem. In certain embodiments, the homomorphic encryption scheme may include a fully homomorphic encryption scheme.

In some embodiments, the at least one analytic vector is generated based on an analytic and at least one parameter associated with the analytic. The generation of the analytic vector may include extracting, using the term generation function, a set of term elements from the analytic and the at least one analytic parameter. The generation of the analytic vector may further include generating, using the keyed hash function, the set of hashes from the set of term elements. The generation may further include determining whether an index of at least one element of the analytical vector is present in the set of hashes. If the index is present in the set of hashes, the at least one element is assigned a non-zero value. The non-zero value can include an encrypted value of a non-zero bitmask of a term element selected from the set of term elements, wherein the hash of the term element is equal to the index. The encrypted value can be obtained using the homomorphic encryption scheme. If the index is not present in the set of the hashes, the at least one element is assigned a zero value.

In certain embodiments, a dimension of the at least one analytic vector is greater than the number of elements in the set of the term elements. In some embodiments, estimating the probabilistic error is performed based on a hash collision rate of the hash function over the data set and a dimension of the at least one analytic vector. In various embodiments, the data set can be in a plaintext form, deterministically encrypted, and/or semantically encrypted.

According to one example embodiment of the present disclosure, a system for performing a secure probabilistic analysis using homomorphic encryption is provided. The system may include at least one processor and a memory storing processor-executable codes, wherein the at least one processor can be configured to implement the operations of the above-mentioned method for performing a secure probabilistic analysis using homomorphic encryption.

According to yet another example embodiment of the present disclosure, the operations of the above-mentioned method for performing secure probabilistic analytics using a homomorphic encryption are stored on a machine-readable medium comprising instructions, which when implemented by one or more processors perform the recited operations.

Other example embodiments of the disclosure and aspects will become apparent from the following description taken in conjunction with the following drawings.

BRIEF DESCRIPTION OF DRAWINGS

Exemplary embodiments are illustrated by way of example and not limitation in the figures of the accompanying drawings, in which like references indicate similar elements.

FIG. 1 is a block diagram of an example environment suitable for practicing methods for secure probabilistic analytics using a homomorphic encryption as described herein.

FIG. 2 is a block diagram showing details of a homomorphic encryption scheme, according to an example embodiment.

FIG. 3 is a flow chart of an example method for performing secure probabilistic analytics using a homomorphic encryption.

FIG. 4 is a computer system that can be used to implement some embodiments of the present disclosure.

DETAILED DESCRIPTION OF EXAMPLARY EMBODIMENTS

The technology disclosed herein is concerned with methods and systems for performing secure probabilistic analytics over data source using a homomorphic encryption. Embodiments of the present disclosure may facilitate a secure transmission of analytics from a client device to computing resource(s) providing a target data source and secure transmission of results of analytics from the computing resource(s) back to the client device.

Some embodiments of the present disclosure may be used to encrypt an analytic on a client device using homomorphic encryption techniques. The encrypted analytic can be sent to computing resource(s) providing desired data source(s). The encrypted analytics can be performed over desired data source(s) to produce encrypted results. The encrypted results can be returned to the client device and decrypted using the homomorphic encryption techniques. Embodiments of the present disclosure may allow performing of an analytic over desired data sources in a secure and private manner because neither content of the analytic nor results of the analytic are revealed to a data owner, observer, or attacker.

According to one example embodiment of the present disclosure, a method for performing secure probabilistic analytics using a homomorphic encryption may commence with acquiring, by a client, an analytic, at least one analytic parameter associated with the analytic, and an encryption scheme. The encryption scheme may include a public key for encryption and a private key for decryption. The method may further include generating, by the client and using the encryption scheme, at least one analytical vector based on the analytic and the at least one analytic parameter. The method may further include sending, by the client, the at least one analytical vector and the encryption scheme, to at least one server.

The method may also include acquiring, by the at least one server, a data set for performing the analytic. The method may allow extracting, by the at least one server and based on the encryption scheme, a set of terms from the data set. The method may further include, evaluating, by the at least one server, the at least one analytical vector over the set of terms to obtain at least one encrypted result. The method may further allow estimating, by the at least one server, a probabilistic error of the at least one encrypted result. The method may also include sending, by the at least one server, the at least one encrypted result and the probabilistic error to the client. The method may also include decrypting, by the client and based on the encryption scheme, the at least one encrypted result to generate at least one result of the analytic.

Referring now to the drawings, various embodiments are described in which like reference numerals represent like parts and assemblies throughout the several views. It should be noted that the reference to various embodiments does not limit the scope of the claims attached hereto. Additionally, any examples outlined in this specification are not intended to be limiting and merely set forth some of the many possible embodiments for the appended claims.

FIG. 1 shows a block diagram of an example environment 100 suitable for practicing the methods described herein. It should be noted, however, that the environment 100 is just one example and is a simplified embodiment provided for illustrative purposes, and reasonable deviations of this embodiment are possible as will be evident for those skilled in the art.

As shown in FIG. 1, the environment 100 may include at least one client device 105 (also referred to as a client 105) and at least one server 110. The client(s) 105 can include any appropriate computing device having network functionalities allowing the device to communicate to server(s) 110. In some embodiments, the client(s) 105 can be connected to the server(s) 110 via one or more wired or wireless communications networks. In various embodiments, the client 105 includes, but is not limited to, a computer (e.g., laptop computer, tablet computer, desktop computer), a server, cellular phone, smart phone, gaming console, multimedia system, smart television device, set-top box, infotainment system, in-vehicle computing device, informational kiosk, smart home computer, software application, computer operating system, modem, router, and so forth. In some embodiments, the client(s) 105 can be used by users for Internet browsing purposes.

In some embodiments, the server(s) 110 may be configured to store or provide access to at least one data source 115. In certain embodiments, the server(s) 110 may include a standalone computing device. In various embodiments, the data source(s) 115 may be located on a single server 110 or distributed over multiple server(s) 110. The data source(s) 115 may include plaintext data, deterministically encrypted data, semantically encrypted data, or a combination of thereof.

In some embodiments, the server(s) 110 may be implemented as cloud-based computing resource shared by multiple users. The cloud-based computing resource(s) can include hardware and software available at a remote location and accessible over a network (for example, the Internet). The cloud-based computing resource(s) can be dynamically re-allocated based on demand. The cloud-based computing resources may include one or more server farms/clusters including a collection of computer servers which can be co-located with network switches and/or routers.

In various embodiments, the client(s) 105 can make certain client inquires within the environment 100. For example, the client(s) 105 may be configured to send analytics to the server(s) 110 to be performed over the data source(s) 115. The server(s) 110 can be configured to perform the analytics over the data source(s) 115 and return the results of analytics to the client(s) 105.

To protect the content of the analytics, the client(s) 105 can be configured to encrypt the analytics using a homomorphic encryption scheme. The homomorphic encryption scheme can include a partially homomorphic encryption scheme and a fully homomorphic encryption scheme. The partially homomorphic encryption scheme can include one of a Rivest, Shamir and Adleman cryptosystem, Elgamal cryptosystem, Benaloh cryptosystem, Goldwasser-Micali cryptosystem, and Pallier cryptosystem. The analytics can be encrypted with a public (encryption) key of the homomorphic encryption scheme. The encrypted analytics and the public key can be sent to the server 110. The encrypted analytics can be only decrypted with a private (decryption) key of the homomorphic encryption scheme. The decryption key can be kept on the client 105 and never provided to the server(s) 110.

To protect the content of the results of the analytic, the server(s) 110 can be configured to perform the encrypted analytics on the data source using the same homographic encryption scheme and the public key received from the client 105 and, thereby, obtain encrypted results of the analytics. The encrypted results can be sent to the client(s) 105. The client(s) 105 can decrypt the encrypted results using the private key. Because the private key is always kept on the client(s) 105, neither encrypted analytic nor encrypted results of the analytics can be decrypted on the server 110 or when intercepted while in transition between the client(s) 105 and the server(s) 110.

FIG. 2 is a block diagram showing details of homomorphic encryption scheme 200, according to some example embodiments. The modules of the homomorphic encryption scheme 200 can be implemented as software instructions stored in memory of the client 105 and executed by at least one processor of the client 105. The client 105 may be configured to acquire a desired analytic A to be executed over data source 115. The analytic A can be associated with analytic parameter set {A_P}. The analytic A and analytic parameter set {A_P} can be further encrypted into a sequence of homomorphic analytical vectors {A_V} using a homomorphic encryption scheme E.

The homomorphic encryption scheme 200 may include a term generation (TG) function 210. The term generation function 210 can be used to extract a set of term elements {T} of analytic A that correspond to an analytic parameter A_P For, example, if the analytic parameter A_P is a frequency distribution for database elements in <row:column> pairs where row=Y, then the set {T} reflects the frequency distribution of these elements from the database.

The homomorphic encryption scheme 200 may further include a keyed hash function H(T) 220. The hash function H(T) can be used to obtain a set H(T)={H(T): T in {T}}. The set H(T) is the range of the hash function H(T) over the set of term elements {T}. The keyed hash function H(T) can be associated with a public key used for the encryption. The number of distinct elements in the set H(T) is equal to the number of distinct elements in the set of term elements {T}.

The homomorphic encryption scheme 200 may further include an analytic vector construction module 230. The analytic vector construction module 230 can be used to construct an analytical vector A_V for the analytic parameter A_P. The desired size s of the analytical vector A_V can be selected to be greater than the number of distinct elements in the set of term elements {T}. For index j=0, . . . , (s−1): if H(T)=j for a term element T in the set {T}, then vector component A_V[j]=E(B_j) where B_j is a nonzero bit mask corresponding to the term element T, wherein E is the homographic encryption scheme. If there is no T in {T} such that H(T)=j, then A_V[j]=E(0). In this manner, the analytical vector A_V includes encryptions of nonzero bitmasks for only the term elements present in the set {T}. The analytic A cannot be recovered from the analytical vectors {A_V} without a private key associated with the homomorphic encryption scheme E.

The client 105 can be further configured to send the analytical vectors {A_V}, the term generation function TG, and the hash function H(T) with the public key to the server(s) 110.

In some embodiments, the server(s) 110 can be configured to extract a set of term elements {T} from the data source 115 using the term generation function TG and the keyed hash function H(T). The server(s) 110 can be further configured to evaluate the encrypted analytical vectors {A_V} over the set of term elements {T} to produce encrypted results E(R). The server(s) 110 can be further configured to estimate a probabilistic error b of the encrypted results E(R) based on a hash collision rate of the hash function H(T) over data source 115 and dimension of analytical vectors {A_V}. The server(s) 110 can be further configured to send the encrypted results E(R) and the probabilistic error b to the client 105.

The client 105 can be configured to decrypt the encrypted results E(R) in order to obtain the results R using the private key of the homomorphic encryption scheme E. Because the analytical vector {A_V} includes nonzero entries for terms in set {T}, the homomorphic properties of E ensure that only results corresponding to the nonzero elements of the analytical vector {A_V} are present in results R.

FIG. 3 is a flow chart of an example method 300 for performing secure probabilistic analytics using a homomorphic encryption, according to some example embodiments. The method 300 may be performed within environment 100 illustrated in FIG. 1. Notably, the steps recited below may be implemented in an order different than described and shown in the FIG. 3. Moreover, the method 300 may have additional steps not shown herein, but which can be evident to those skilled in the art from the present disclosure. The method 300 may also have fewer steps than outlined below and shown in FIG. 3.

The method 300 may commence in block 305 with receiving, by at least one server, from a client, at least one analytic vector, a term generation function, and a keyed hash function. The at least one analytic vector can be encrypted using the homomorphic encryption scheme. The homomorphic encryption scheme may include a public key for encryption and a private key for decryption.

In block 310, the method 300 may proceed with extracting, by the at least one server, a set of term components from a data set using the term generation function and the keyed hashed function.

In block 315, the method 300 may evaluate, by the at least one server, the at least one analytic vector over the set of term components to obtain at least one encrypted result.

In block 320, the method 300 may estimate, by the at least one server, a probabilistic error of the at least one encrypted result. The estimate can be based on a hash collision of the keyed hash function over the data set and the length of the analytic vector.

In block 325, the method may proceed with sending, by the at least one server, the at least one encrypted result and the probabilistic error to the client. The client can be configured to decrypt the at least one encrypted result using the homomorphic encryption scheme.

FIG. 4 illustrates an exemplary computer system 400 that may be used to implement some embodiments of the present disclosure. The computer system 400 of FIG. 4 may be implemented in the contexts of the likes of the client 105, the server(s) 110, and the data source 115. The computer system 400 of FIG. 4 includes one or more processor units 410 and main memory 420. Main memory 420 stores, in part, instructions and data for execution by processor units 410. Main memory 420 stores the executable code when in operation, in this example. The computer system 400 of FIG. 4 further includes a mass data storage 430, portable storage device 440, output devices 450, user input devices 460, a graphics display system 470, and peripheral devices 480.

The components shown in FIG. 4 are depicted as being connected via a single bus 490. The components may be connected through one or more data transport means. Processor unit 410 and main memory 420 is connected via a local microprocessor bus, and the mass data storage 430, peripheral device(s) 480, portable storage device 440, and graphics display system 470 are connected via one or more input/output (I/O) buses.

Mass data storage 430, which can be implemented with a magnetic disk drive, solid state drive, or an optical disk drive, is a non-volatile storage device for storing data and instructions for use by processor unit 410. Mass data storage 430 stores the system software for implementing embodiments of the present disclosure for purposes of loading that software into main memory 420.

Portable storage device 440 operates in conjunction with a portable non-volatile storage medium, such as a flash drive, floppy disk, compact disk, digital video disc, or Universal Serial Bus (USB) storage device, to input and output data and code to and from the computer system 400 of FIG. 4. The system software for implementing embodiments of the present disclosure is stored on such a portable medium and input to the computer system 400 via the portable storage device 440.

User input devices 460 can provide a portion of a user interface. User input devices 460 may include one or more microphones, an alphanumeric keypad, such as a keyboard, for inputting alphanumeric and other information, or a pointing device, such as a mouse, a trackball, stylus, or cursor direction keys. User input devices 460 can also include a touchscreen. Additionally, the computer system 400 as shown in FIG. 4 includes output devices 450. Suitable output devices 450 include speakers, printers, network interfaces, and monitors.

Graphics display system 470 include a liquid crystal display (LCD) or other suitable display device. Graphics display system 470 is configurable to receive textual and graphical information and processes the information for output to the display device.

Peripheral devices 480 may include any type of computer support device to add additional functionality to the computer system.

The components provided in the computer system 400 of FIG. 4 are those typically found in computer systems that may be suitable for use with embodiments of the present disclosure and are intended to represent a broad category of such computer components that are well known in the art. Thus, the computer system 400 of FIG. 4 can be a personal computer (PC), hand held computer system, telephone, mobile computer system, workstation, tablet, phablet, mobile phone, server, minicomputer, mainframe computer, wearable, or any other computer system. The computer may also include different bus configurations, networked platforms, multi-processor platforms, and the like. Various operating systems may be used including UNIX, LINUX, WINDOWS, MAC OS, PALM OS, QNX ANDROID, IOS, CHROME, TIZEN, and other suitable operating systems.

The processing for various embodiments may be implemented in software that is cloud-based. In some embodiments, the computer system 400 is implemented as a cloud-based computing environment, such as a virtual machine operating within a computing cloud. In other embodiments, the computer system 400 may itself include a cloud-based computing environment, where the functionalities of the computer system 400 are executed in a distributed fashion. Thus, the computer system 400, when configured as a computing cloud, may include pluralities of computing devices in various forms, as will be described in greater detail below.

In general, a cloud-based computing environment is a resource that typically combines the computational power of a large grouping of processors (such as within web servers) and/or that combines the storage capacity of a large grouping of computer memories or storage devices. Systems that provide cloud-based resources may be utilized exclusively by their owners or such systems may be accessible to outside users who deploy applications within the computing infrastructure to obtain the benefit of large computational or storage resources.

The cloud may be formed, for example, by a network of web servers that comprise a plurality of computing devices, such as the computer system 400, with each server (or at least a plurality thereof) providing processor and/or storage resources. These servers may manage workloads provided by multiple users (e.g., cloud resource customers or other users). Typically, each user places workload demands upon the cloud that vary in real-time, sometimes dramatically. The nature and extent of these variations typically depends on the type of business associated with the user.

The present technology is described above with reference to example embodiments. Therefore, other variations upon the example embodiments are intended to be covered by the present disclosure.

Claims

1. A method for performing a secure probabilistic analysis using homomorphic encryption, the method comprising: receiving, from a client, by at least one server, at least one analytic vector, the at least one analytic vector being encrypted using a homomorphic encryption scheme, a term generation function, and a keyed hash function, wherein the homomorphic encryption scheme includes a public key for encryption and a private key for decryption;extracting, by the at least one server, a set of term components from a data set using the term generation function and the keyed hash function;evaluating, by the at least one server, the at least one analytic vector over the set of term components to obtain at least one encrypted result;estimating, by the at least one server, a probabilistic error of the at least one encrypted result; andsending, by the at least one server, the at least one encrypted result and the probabilistic error to the client, wherein the client is configured to decrypt the at least one encrypted result using the homomorphic encryption scheme, wherein the at least one analytic vector is generated based on an analytic and at least one parameter associated with the analytic, the generation of the at least one analytic vector including: extracting, using the term generation function, a set of term elements from the analytic and the at least one analytic parameter;generating, using the keyed hash function, a set of hashes from the set of term elements;determining whether an index of at least one element of the analytic vector is present in the set of hashes;if the index is present in the set of hashes, assigning the at least one element a non-zero value; andif the index is not present in the set of hashes, assigning the at least one element a zero value.

2. The method of claim 1, wherein the homomorphic encryption scheme includes a partially homomorphic encryption scheme.

3. The method of claim 2, wherein the partially homomorphic encryption scheme includes at least one of the following: a Rivest, a Shamir and Adleman cryptosystem, an Elgamal cryptosystem, a Benaloh cryptosystem, a Goldwasser—Micali cryptosystem, and a Pallier cryptosystem.

4. The method of claim 1, wherein the homomorphic encryption scheme includes a fully homomorphic encryption scheme.

5. The method of claim 1, wherein a dimension of the at least one analytic vector is greater than the number of elements in the set of term elements.

6. The method of claim 1, wherein the non-zero value is an encrypted value of a non-zero bitmask of a term element of the set of term elements, the hash of the term element being equal to the index, the encrypted value being obtained using the homomorphic encryption scheme.

7. The method of claim 1, wherein estimating the probabilistic error is based on a hash collision rate of the keyed hash function over the data set and dimension of the at least one analytic vector.

8. The method of claim 1, wherein the data set is in a plaintext form.

9. The method of claim 1, wherein the data set is deterministically encrypted or semantically encrypted.

10. A system for performing a secure probabilistic analysis using homomorphic encryption, the system comprising: at least one processor; anda memory communicatively coupled with the at least one processor, the memory storing instructions, which when executed by the at least processor perform a method comprising: receiving, from a client, at least one analytic vector, the at least one analytic vector being encrypted using a homomorphic encryption scheme, a term generation function, and a keyed hash function, wherein the homomorphic encryption scheme includes a public key for encryption and a private key for decryption;extracting, by at least one server, a set of term components from a data set using the term generation function and the keyed hash function;evaluating the at least one analytic vector over the set of term components to obtain at least one encrypted result;estimating a probabilistic error of the at least one encrypted result; andsending the at least one encrypted result and the probabilistic error to the client, wherein the client is configured to decrypt the at least one encrypted result using the homomorphic encryption scheme, wherein the at least one analytic vector is generated based on an analytic and at least one parameter associated with the analytic and the generation includes: extracting, using the term generation function, a set of term elements from the analytic and the at least one analytic parameter;generating, using the keyed hash function, a set of hashes from the set of term elements;determining whether an index of at least one element of the analytic vector is present in the set of hashes;if the index is present in the set of hashes, assigning the at least one element a non-zero value; andif the index is not present in the set of hashes, assigning the at least one element a zero value.

11. The system of claim 10, wherein the homomorphic encryption scheme includes a partially homomorphic encryption scheme.

12. The system of claim 10, wherein the homomorphic encryption scheme includes at least one of a Rivest, a Shamir and Adleman cryptosystem, an Elgamal cryptosystem, a Benaloh cryptosystem, a Goldwasser—Micali cryptosystem, and a Pallier cryptosystem.

13. The system of claim 10, wherein the homomorphic encryption scheme includes a fully homomorphic encryption scheme.

14. The system of claim 10, wherein a dimension of the at least one analytic vector is greater than the number of elements in the set of term elements.

15. The system of claim 10, wherein the non-zero value is an encrypted value of a non-zero bitmask of a term element of the set of term elements, the hash of the term element being equal to the index, the encrypted value being obtained using the homomorphic encryption scheme.

16. The system of claim 10, wherein estimating the probabilistic error is based on a hash collision rate of the keyed hash function over the data set and dimension of the at least one analytic vector.

17. The system of claim 10, wherein the data set is in one of the following forms: a plaintext, deterministically encrypted, and semantically encrypted.

18. A non-transitory computer-readable storage medium having embodied thereon instructions, which when executed by at least one processor, perform steps of a method, the method comprising: receiving, by at least one server from a client, at least one analytic vector, the at least one analytic vector being encrypted using a homomorphic encryption scheme, a term generation function, and a keyed hash function, wherein the homomorphic encryption scheme includes a public key for encryption and a private key for decryption;extracting, by the at least one server, a set of term components from a data set using the term generation function and the keyed hash function;evaluating, by the at least one server, the at least one analytic vector over the set of term components to obtain at least one encrypted result;estimating, by the at least one server, a probabilistic error of the at least one encrypted result; andsending, by the at least one server, the at least one encrypted result and the probabilistic error to the client, wherein the client is configured to decrypt the at least one encrypted result using the homomorphic encryption scheme, wherein the at least one analytic vector is generated based on an analytic and at least one parameter associated with the analytic, the generation of the at least one analytic vector including: extracting, using the term generation function, a set of term elements from the analytic and the at least one analytic parameter;generating, using the keyed hash function, a set of hashes from the set of term elements;determining whether an index of at least one element of the analytic vector is present in the set of hashes;if the index is present in the set of hashes, assigning the at least one element a non-zero value; andif the index is not present in the set of hashes, assigning the at least one element a zero value.