U.S. patent application number 15/468621 was filed with the patent office on 2017-10-12 for system and methods for validating and performing operations on homomorphically encrypted data.
This patent application is currently assigned to THE GOVERNING COUNCIL OF THE UNIVERSITY OF TORONTO. The applicant listed for this patent is THE GOVERNING COUNCIL OF THE UNIVERSITY OF TORONTO. Invention is credited to Glenn GULAK, Alhassan KHEDR.
Application Number | 20170293913 15/468621 |
Document ID | / |
Family ID | 59998250 |
Filed Date | 2017-10-12 |
United States Patent
Application |
20170293913 |
Kind Code |
A1 |
GULAK; Glenn ; et
al. |
October 12, 2017 |
SYSTEM AND METHODS FOR VALIDATING AND PERFORMING OPERATIONS ON
HOMOMORPHICALLY ENCRYPTED DATA
Abstract
A system and method of validating and performing operations on
homomorphically encrypted data are described herein. The methods
include processing a secure financial transaction by receiving a
transaction request to complete a financial transaction, with at
least a portion of the request encrypted according to a homomorphic
encryption scheme, and the transaction request comprising
confidential cardholder data including an account number,
non-confidential cardholder data, and transaction data, and
retrieving one or more sets of encrypted comparison cardholder data
encrypted according to a homomorphic encryption scheme. The
confidential cardholder data is then compared to each set of the
comparison cardholder data using one or more homomorphic operations
to determine which set of comparison cardholder data matches the
confidential cardholder data and validating the confidential
cardholder data. An encrypted indicator is generated indicating
authorization or rejection of the request and forwarded to a party
seeking authorization to complete the financial transaction.
Inventors: |
GULAK; Glenn; (Toronto,
CA) ; KHEDR; Alhassan; (Toronto, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
THE GOVERNING COUNCIL OF THE UNIVERSITY OF TORONTO |
Toronto |
|
CA |
|
|
Assignee: |
THE GOVERNING COUNCIL OF THE
UNIVERSITY OF TORONTO
Toronto
CA
|
Family ID: |
59998250 |
Appl. No.: |
15/468621 |
Filed: |
March 24, 2017 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
62321411 |
Apr 12, 2016 |
|
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|
62417490 |
Nov 4, 2016 |
|
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G06Q 20/08 20130101;
G16H 10/60 20180101; H04L 63/0414 20130101; G06Q 2220/00 20130101;
H04L 9/3093 20130101; A61B 5/72 20130101; G06Q 20/382 20130101;
H04L 9/008 20130101; G06N 20/00 20190101; G16H 10/40 20180101; H04L
63/0428 20130101; G06Q 20/38215 20130101; G06Q 20/3829
20130101 |
International
Class: |
G06Q 20/38 20060101
G06Q020/38 |
Claims
1. A method of processing a secure financial transaction,
comprising: receiving a transaction request to complete a financial
transaction, with at least a portion of the request encrypted
according to a homomorphic encryption scheme, and the transaction
request comprising confidential cardholder data including an
account number, non-confidential cardholder data, and transaction
data; retrieving one or more sets of encrypted comparison
cardholder data from a cardholder data database, each set of
encrypted comparison cardholder data encrypted according to a
homomorphic encryption scheme; comparing the encrypted confidential
cardholder data to each set of the encrypted comparison cardholder
data using one or more homomorphic operations to determine which
set of comparison cardholder data matches the confidential
cardholder data and validating the confidential cardholder data;
generating an encrypted indicator indicating authorization or
rejection of the request to complete the financial transaction
based upon at least the validation of the confidential cardholder
data; and forwarding the encrypted indicator authorization or
rejection of the request to complete the financial transaction to a
party seeking authorization to complete the financial transaction,
wherein the confidential cardholder data is never decrypted during
the method.
2. The method of claim 1, wherein the homomorphic encryption scheme
is a fully homomorphic encryption scheme or a somewhat homomorphic
encryption scheme.
3. The method of claim 1, wherein the confidential cardholder data
includes one or more of: a portion of the account number, the
account number, an expiry date and a Card Verification Value (CVV)
number.
4. The method of claim 1, wherein the non-confidential cardholder
data includes one or more of: a bank name, a cardholder name, a
Bank Identification Number (BIN) and the last four digits of the
account number.
5. The method of claim 1, wherein the transaction data includes one
or more of: a transaction amount, a transaction date and a merchant
identifier.
6. The method of claim 1, further including executing a step of
retrieving the transaction data and executing the step of comparing
the transaction data using one or more homomorphic operations to
determine whether the transaction amount can be authorized.
7. The method of claim 1, wherein a portion of the request is
compared to the set of comparison data to reduce the size of the
set of comparison data prior to comparing the encrypted
confidential cardholder data.
8. The method of claim 1, wherein the one or more homomorphic
operations combine to form an XNOR operation.
9. The method of claim 1, wherein the financial transaction is a
credit card transaction or a debit card transaction or a
stored-value card transaction.
10. The method of claim 1, wherein the request to complete the
financial transaction is received by one or more of: a bank, a
credit card company, a card issuer or a payment processor.
11. A non-transient computer-readable medium containing
computer-readable instructions which when executed by a computer
processor perform the method of: receiving a request to complete a
financial transaction with at least a portion of the request
encrypted according to a homomorphic encryption scheme, and the
transaction request comprising confidential cardholder data
including an account number, non-confidential cardholder data and
transaction data; retrieving one or more sets of encrypted
comparison cardholder data from a cardholder data database, each
set of encrypted comparison cardholder data encrypted according to
a homomorphic encryption scheme; comparing the confidential
cardholder data to each set of the comparison cardholder data using
one or more homomorphic operations to determine which set of
comparison cardholder data matches the confidential cardholder data
and validating the confidential cardholder data; generating an
encrypted indicator indicating authorization or rejection of the
request to complete the financial transaction based upon at least
the validation of the confidential cardholder data; and forwarding
the encrypted indicator indicating authorization or rejection of
the request to complete the financial transaction to a party
seeking authorization to complete the financial transaction,
wherein the confidential cardholder data is never decrypted during
the method.
12. The method of claim 1, wherein the homomorphic encryption
scheme is a fully homomorphic encryption scheme or a somewhat
homomorphic encryption scheme.
13. The computer-readable medium of claim 11, wherein the
confidential cardholder data includes one or more of: a portion of
the account number, an account number, an expiry date and a Card
Verification Value (CVV) number.
14. The computer-readable medium of claim 11, wherein the
non-confidential cardholder data includes one or more of: a bank
name, a cardholder name, a Bank Identification Number (BIN) and the
last four digits of the account number.
15. The computer-readable medium of claim 11, wherein the
transaction data includes one or more of: a transaction amount, a
transaction date and a merchant identifier.
16. The computer-readable medium of claim 11, further including
executing a step of retrieving the transaction data and executing
the step of comparing the transaction data using one or more
homomorphic operations to determine whether the transaction amount
can be authorized.
17. The computer-readable medium of claim 11, wherein a portion of
the request is compared to the set of comparison data to reduce the
size of the set of comparison data prior to comparing the encrypted
confidential cardholder data.
18. The computer-readable medium of claim 11, wherein the one or
more homomorphic operations combine to form an XNOR operation.
19. The computer-readable medium of claim 11, wherein the financial
transaction is a credit card transaction or a debit card
transaction or a stored-value card transaction.
20. The computer-readable medium of claim 11, wherein the request
to complete the financial transaction is received by one or more
of: a bank, a credit card company, a card issuer or a payment
processor.
Description
FIELD OF THE INVENTION
[0001] This disclosure relates to homomorphically encrypted data
systems and methods, and more specifically, to validation of, and
performing operations on, homomorphically encrypted confidential
data without decryption of the confidential data, such as financial
data.
BACKGROUND OF THE INVENTION
[0002] Privacy of sensitive personal information is an increasingly
important topic as more personal data is transmitted and shared,
particular via the use of wireless transmissions and cloud data
services. Privacy issues arise due to the fear of having a security
breach on cloud servers or due to the fear that the service
providers themselves misuse this sensitive information. Standard
encryption schemes try to address these concerns by devising
encryption schemes that are harder to break, yet they do not solve
the possible misuse of this sensitive data by the service providers
themselves.
[0003] While privacy of confidential and personal data, such as
financial data, is a paramount concern, access to this data for
legitimate purposes, such as to execute a financial transaction, is
also needed.
[0004] Accordingly, there is a need for improvement in the art.
SUMMARY OF THE INVENTION
[0005] According to an embodiment of the invention, there is
provided a method of processing a secure financial transaction,
comprising: receiving a transaction request to complete a financial
transaction, with at least a portion of the request encrypted
according to a homomorphic encryption scheme, and the transaction
request comprising confidential cardholder data including an
account number, non-confidential cardholder data and transaction
data; retrieving one or more sets of encrypted comparison
cardholder data encrypted according to a homomorphic encryption
scheme; comparing the encrypted confidential cardholder data to
each set of the encrypted comparison cardholder data using one or
more homomorphic operations to determine which set of comparison
cardholder data matches the confidential cardholder data and
validating the confidential cardholder data; generating an
encrypted indicator indicating authorization or rejection of the
request to complete the financial transaction based upon at least
the validation of the confidential cardholder data; and forwarding
the encrypted indicator indicating authorization or rejection of
the request to complete the financial transaction to a party
seeking authorization to complete the financial transaction,
wherein the confidential cardholder data is never decrypted during
the method.
[0006] According to a further embodiment of the invention, there is
provided a non-transient computer-readable medium containing
computer-readable instructions which when executed by a computer
processor perform the method of: receiving a transaction request to
complete a financial transaction, with at least a portion of the
request encrypted according to a homomorphic encryption scheme, and
the transaction request comprising confidential cardholder data
including an account number, non-confidential cardholder data and
transaction data; retrieving one or more sets of encrypted
comparison cardholder data encrypted according to a homomorphic
encryption scheme; comparing the confidential cardholder data to
each set of the comparison cardholder data using one or more
homomorphic operations to determine which set of comparison
cardholder data matches the confidential cardholder data and
validating the confidential cardholder data; generating an
encrypted indicator indicating authorization or rejection of the
request to complete the financial transaction based upon at least
the validation of the confidential cardholder data; and forwarding
the encrypted indicator indicating authorization or rejection of
the request to complete the financial transaction to a party
seeking authorization to complete the financial transaction,
wherein the confidential cardholder data is never decrypted during
the method.
[0007] Other aspects and features according to the present
application will become apparent to those ordinarily skilled in the
art upon review of the following description of embodiments of the
invention in conjunction with the accompanying figures.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] The drawings illustrate, by way of example only, embodiments
of the present disclosure.
[0009] FIG. 1 is a diagram of a secure computing environment
according to an embodiment of the present invention;
[0010] FIG. 2 is a diagram of a remote device according to an
embodiment of the present invention;
[0011] FIG. 3 is a diagram of a data system according to an
embodiment of the present invention;
[0012] FIG. 4 is a diagram of a key authority system according to
an embodiment of the present invention;
[0013] FIG. 5 is a diagram of another data system according to an
embodiment of the present invention;
[0014] FIGS. 6A-6J show equations for performing computations
according to an embodiment of the present invention;
[0015] FIG. 7 is a table of data used in an example for an
embodiment of the present invention;
[0016] FIGS. 8 and 9 show testing parameters and results of tests
conducted;
[0017] FIG. 10 is a diagram of a four-party credit card transaction
system;
[0018] FIG. 11 is a diagram of a payment processing chain according
to an embodiment of the present invention;
[0019] FIG. 12 is a diagram of a secure payment processing system
according to an embodiment of the present invention;
[0020] FIG. 13 is a chart showing secret key size reduction
according to an embodiment of the present invention;
[0021] FIG. 14 is a chart showing ciphertext size reduction and
obviating a flatten function;
[0022] FIG. 15 shows pseudocode for a ciphertext multiplication
operation;
[0023] FIG. 16 is a table of example parameter selection according
to an embodiment of the present invention; and
[0024] FIGS. 17A-17G show expressions/equations according to an
embodiment of the present invention.
[0025] Like reference numerals indicate like or corresponding
elements in the drawings.
DETAILED DESCRIPTION OF THE EMBODIMENTS
[0026] Although financial applications form the basis of the
inventive examples discussed herein, the inventive techniques
discussed have application to other forms of confidential data. In
this context, the different devices and parties involved in data
acquisition, storage, and analysis may be reconsidered as
appropriate for the type of confidential data being processed.
[0027] Homomorphic Encryption/Decryption
[0028] The homomorphic encryption/decryption operations executed on
confidential data may be performed as shown in FIGS. 1-5 and FIGS.
6A-6J. Key generation may be performed at a secure system, such as
the key authority system 36. Public keys 68 may be pre-loaded onto
the remote devices 20 containing the confidential information (such
as a credit card, remote sensor, etc.) prior to deployment of
remote devices 20. Alternatively, or additionally, public keys 68
may be made available to the remote devices 20 via a network
resource (such as a public cloud), which may be configured to
require credential verification or authorization (e.g., user name
and password) to access public keys 68. The encryption engines 62
on board the remote devices 20, and in the data system 120, may be
configured to implement the homomorphic encryption techniques
discussed below. The decryption engine 104 at the key authority
system 36 may be configured to implement the homomorphic decryption
techniques discussed below. The calculation engine 88 at the data
systems 22, 120 may be configured to implement the computational
techniques discussed below, including the storage and processing of
equations, calculation parameters, upper/lower bounds, and
intermediate/accumulated results.
[0029] Regarding notation for the below discussion, for an odd
prime number q, the ring Z/qZ (or Z.sub.q) with the interval (-q/2,
q/2).andgate.Z is identified. The notation [x].sub.q denotes
reducing x modulo q. The examples discussed herein use polynomial
rings defined by the quotient polynomials R=Z[X]/.PHI..sub.m(X),
where .PHI..sub.m(X)=x.sub.n+1 is the irreducible m.sup.th
cyclotomic polynomial, in which n is a power of 2 and m=2n. Let
R.sub.q=R/qR. Any type of multiplication including matrix and
polynomial multiplication is denoted herein by the multiplication
operator ` `. Rounding up to the nearest integer is denoted by |a|.
Matrices of rings are defined as A.sub.M.times.N, where A.sub.ij
.epsilon.R.sub.q and M, N are the matrix dimensions.
I.sub.l.times.l represents the identity matrix of rings. Row
vectors are represented as [a b], where a and b are the vector
elements. Column vectors on the other hand are represented as [a;
b].
[0030] The parameters of the cryptographic scheme are n, the degree
of the number field; q, the modulus; .sigma..sub.k and
.sigma..sub.c, the standard deviation of the discrete Gaussian
error distribution in the keyspace and ciphertext space,
respectively; l.left brkt-top.log q.right brkt-bot. that governs
the number of ring elements in a ciphertext. The setting of these
parameters depends on the security level .lamda. (e.g., .lamda.=80
or 128 bits) as well as the complexity of functions contemplated to
evaluate on ciphertexts.
[0031] A bit decompose function BD (integer) takes an l-bit input
integer, then outputs a row vector with size l containing the bit
decomposition of this integer. (Note that the letter "l" referred
to herein, including the figures, is a lowercase script letter
"L".) Similarly, BD (polynomial) takes an input polynomial of size
n, where each coefficient is an i-bit integer, then outputs an
l-sized row vector of polynomials (each of size n) containing the
bit decomposition of each coefficient of the input polynomial,
yielding a matrix of size l.times.n. Finally, BD (Matrix of
polynomials) takes an input matrix of polynomials of size x.times.y
(each polynomial is of size n with integer coefficients), then
outputs a matrix of polynomials expanded by a factor l in the
column dimension, yielding a matrix of size x.times.yl, where each
consecutive l elements along the row contain the bit representation
of each coefficient of each of the input polynomials. For example,
the bit decompose of the input polynomial matrix
B.sub.x.times.y.times.n is BD
(B.sub.x.times.y.times.n)=.beta..sub.x.times.yl.times.n. It is
noted that despite the fact that the polynomial coefficients of
matrix .beta..sub.x.times.yl.times.n are single bit values, the
storage requirement of matrix .beta. in CPU or GPU memory is not
equal to x.times.yl.times.n bits. This is due to the fact that the
smallest addressable unit of memory is a byte (i.e., Byte
Addressable). Hence, .beta. requires x.times.yl.times.n bytes of
storage. This results in the further observation that the storage
requirement of .beta..sub.x.times.yl.times.n is at least 8 times
the storage requirement of B.sub.x.times.y.times.n.
[0032] A bit decompose Inverse function BDI( ) is the inverse of
the bit decompose function BD( ). The BDI( ) function groups
consecutive l coefficients along a row (the coefficients don't need
to be binary), and outputs the integer corresponding to those l
bits. Mathematically, the BDI( ) function can be defined as
multiplying the expanded matrix of polynomials
.beta..sub.x.times.yl from the right by the matrix
.alpha..sub.yl.times.y defined in the equation of FIG. 6A. (The
polynomial dimension n is omitted from this point forward for
clarity). Hence B.sub.x.times.y=BDI
(.beta..sub.x.times.yl)=.beta..sub.x.times.yl.alpha..sub.yl.times.y.
[0033] Ring Learning with Errors (RLWE) Based Encryption Scheme
[0034] The method and system may employ a RLWE (Ring Learning With
Errors) cryptographic scheme, and the general principles of such
scheme will now be described. However, this scheme is not
particularly limiting and other suitable polynomial-based fully
homomorphic cryptographic schemes may be used. Moreover, any gaps
in the below would be well understood by those skilled in
cryptography in view of the known art.
[0035] The system and method is configured to generate keys,
encrypt information, and decrypt information.
[0036] The key generator is configured to implement a Keygen
(1.sup..lamda.) function as follows. A polynomial
t.rarw.D.sub.R.sub.q.sub.,.sigma..sub.k is chosen. The secret key
becomes sk=s.sub.2.times.1.rarw.[1; -t] .epsilon.R.sub.q.sup.2. The
public key is pk=A.sub.1.times.2=[b a], based on a uniform sample
a.rarw.R.sub.q, e.rarw.D.sub.R.sub.q.sub.,.sigma..sub.k, set
b=at+e. It is noted that the expression in FIG. 17A holds.
[0037] As shown in FIG. 13, this is advantageous over a known
secret key sk=v=PO2(s) based on a powers-of-two expansion such as
PO2(x) defined as [x, 2x, . . . , 2.sup.l-1x]. Hence, the key
generator generates smaller secret keys by a theoretical factor of
l times.
[0038] The encryption engine 62 is configured to implement an
Enc(pk, .mu.) function as follows. The message space is R.sub.q. A
uniform vector r.sub.N.times.1 is sampled where each coefficient in
the polynomials in r sampled from {0,1},
E.sub.N.times.2.rarw.D.sub.R.sub.q.sub.,.sigma..sub.c.sup.N.times.2.
The plaintext polynomial .mu..epsilon.R.sub.q is encrypted by
calculating the expression in FIG. 17B. As shown in FIG. 14, this
is advantageous over prior techniques that use C.sub.N.times.N, as
the encryption engine 62 results in smaller ciphertext by a
theoretical factor of l times.
[0039] The decryption engine 104 is configured to implement a
Dec(sk, C) function as follows. Given the ciphertext C, the
plaintext .mu..epsilon.R.sub.q is restored by multiplying C by the
secret-key s according to the expression in FIG. 17C.
[0040] This is advantageous over prior techniques that implement
Dec(sk, C)=C.sub.N.times.Nv.sub.N.times.1, as the decryption engine
104 requires the performance of fewer operations a theoretical
factor of l times.
[0041] It is noted that the first l coefficients in the first term
of the expression in FIG. 17C are in the form .mu., 2.mu., . . . ,
2.sup.l-1.mu.. This means that the element at location
i.epsilon.[0, l-1] is in the form .mu.2.sup.i+error. That is, the
most significant bit of each entry carries a single bit from the
number .mu. assuming that error <q/2 and there is theoretically
no wrap-around mod q as may be found in prior techniques.
[0042] It is now possible to perform operations on ciphertext
without first decrypting the ciphertext. For input ciphertexts
C.sub.N.times.2 and D.sub.N.times.2.epsilon.R.sub.q.sup.N.times.2
encrypting .mu..sub.1 and .mu..sub.2 respectively, homomorphic
operations are implemented as follows.
[0043] The addition operator implements an ADD(C, D) function to
add two ciphertexts C.sub.N.times.2 and D.sub.N.times.2 by
performing the entry-wise addition
C.sub.N.times.2+D.sub.N.times.2.
[0044] The multiplication operator and bitwise decomposition
function implement a MULT(C, D) function to multiply two
ciphertexts C.sub.N.times.2 and D.sub.N.times.2 by performing the
bitwise decomposition function (or BD) on one ciphertext and then
executing the multiplication, as
BD(C.sub.N.times.2)D.sub.N.times.2.
[0045] As shown in FIG. 14, this is advantageous over prior
techniques that define MULT(C,
D)=FLATTEN(C.sub.N.times.ND.sub.N.times.N), where FLATTEN(A) is
defined as BD(BDI(A)). The present technique requires fewer
operations by a theoretical factor of at least l times.
[0046] Correctness of the above homomorphic addition should readily
apparent to those skilled in the art. The multiplication is
asymmetric in the input ciphertexts C and D. That is, the
components of D are treated as a whole, whereas the components of C
are broken up into their bit-wise decompositions. The
multiplication is correct, as discussed below, and gives a slow
noise-growth rate.
[0047] The correctness of the multiplication operation should
readily apparent to those skilled in the art in view of the
expression in FIG. 18D, in which matrix dimensions are removed for
clarity. In the last line of the manipulation of expression in FIG.
18D, it is apparent that the encryption of
.mu.=.mu..sub.2.mu..sub.1.
[0048] Correct decryption depends on the ciphertext noise being
bounded. Thus, it is important to understand how homomorphic
operations increase ciphertext noise. Taking C as a fresh
ciphertext, it is apparent that homomorphic addition of v
ciphertexts increases the noise by a factor of v in the worst case.
In various contemplated implementations, since the coefficients of
the error polynomials are contemplated to follow a Gaussian
distribution, the factor is closer to O( {square root over
(.nu.)}).
[0049] It is further apparent that homomorphic multiplication of
two ciphertexts C=Enc(.mu..sub.1) and D=Enc(.mu..sub.2) with error
magnitudes B.sub.1 and B.sub.2, respectively, increases the error
to O(B.sub.1.parallel..mu..sub.2.parallel..sub.2+B.sub.2nlog q) in
the worst case, and
O(B.sub.1.parallel..mu..sub.2.parallel..sub.1+B.sub.2 {square root
over (nlog(q))}) in various contemplated implementations. Here,
.parallel..mu..parallel..sub.1 denotes the l.sub.1 norm of the
message polynomial .mu.. It is advantageous that error dependence
on the two ciphertexts is asymmetric, as evident from the
above.
[0050] To multiply .nu. ciphertexts the order of multiplication is
contemplated to play a role in error. In techniques described
herein, input .mu. will typically be 0 or 1, meaning that the
growth is simply additive with respect to B.sub.1. Thus, it is
advantageous to multiply .nu. ciphertexts with (the same) error
level B through an accumulator-like function as shown in FIG. 15,
rather than using a binary tree of multiplications, which tends to
grow error at superpolynomial rates. The resulting error growth is
O(Bvn log (q)) in the worst case, and O(B {square root over
(vnlog(q))}) in various contemplated implementations. Hence, the
control logic may be configured to implement accumulative
multiplications, as shown in FIG. 15 and as may be required by
various contemplated implementations.
[0051] For example, reference is now made to the expression in FIG.
17E, in which x.sub.1, . . . , x.sub..nu. are v-tuples of input
encrypted bits, y.sub.1, . . . , y.sub..nu. are v-tuples of bits in
some set S, and operation (xl.sym.yl) represents binary XNOR
between bits x.sub.i and y.sub.i. Since the form of the expression
in FIG. 17E stipulates that exactly one of the terms may survive
(F=1 when x.sub.1, . . . , x.sub..nu. .epsilon.S, otherwise F=0),
the small total error growth can result, even though the component
computing based on the expression in FIG. 17E may not be able to
determine precisely which term will survive.
[0052] It is apparent that noise grows to O(Bvn log q|S|) in the
worst case, or O(B {square root over (vn log (q)|S|)}) in various
contemplated implementations. This is in contrast to O(B {square
root over ((n log (q)).sup.log(.nu.)|S|)}) when using the known
Brakerski-Gentry-Vaikuntanathan encryption scheme, implemented in
IBM HElib. Indeed, such expressions, as in the expression in FIG.
17E, are far from atypical, and they occur quite naturally in
evaluating decision trees and PIR-like functions as will be
discussed further below.
[0053] Another source of improvement afforded by the presently
disclosed techniques is evident from the error term
B.sub.1.parallel..mu..sub.2.parallel..sub.1+B.sub.2n log q. When
multiplication is performed using an accumulator, as shown in FIG.
15, B.sub.2 represents the smaller error in the fresh ciphertexts
C.sub.i, and B.sub.1 represents the larger error in the accumulated
ciphertext C.sub.accum. If C.sub.i encrypts .mu..sub.2=0, then the
larger error term B.sub.1 disappears from the error expression.
[0054] This error reduction is also apparent from the expression in
FIG. 17E. When evaluating each of the products in the expression in
FIG. 17E, the error can be seen to grow proportional not to .nu.,
the total number of multiplications, but rather with k, the longest
continuous chain of 1's starting from the end. It is contemplated
that this is because the last time a zero is encountered in the
multiplication chain, the error is reduced, by the observation
above. Assuming that S is an expected set, the expected length of a
continuous chain of trailing 1's is
.SIGMA..sub.i=1.sup..nu.i2.sup.i<2. In other words, the
multiplicative factor of .nu. disappears from the error expression
as well, and error growth becomes close to O(B {square root over (n
log(q)|S|)}). This is substantially the same effect as if |S|
ciphertexts were added.
[0055] Further, when f is taken as a function to be evaluated, for
example, the expression in FIG. 17E, the error.sub.j(B,n,q) denotes
how much the error grows when evaluating the function f on
ciphertexts in R.sub.q with an initial error of magnitude B. For
correct decryption, it is expected that the expression in FIG. 17F
holds. Since errors tend to grow slower using the present
techniques, q can be set to be correspondingly smaller to meet a
security level equivalent to that of prior techniques. Following
the analysis of Lindner and Peikert, for a security level of
.lamda. bits, it is expected that the following expression
holds:
n>log q(.lamda.+110)/7.2 (7)
[0056] Because log q in the present techniques is smaller, n can be
set to be smaller, for the same security level .lamda.. In turn, a
smaller n can result in a error.sub.j(B,n,q) that is smaller,
leading to an even smaller q, and so on. Suitable parameters are
obtained by solving both the above inequalities in FIGS. 17F and
17G together. FIG. 16 summarizes an example of such a parameter
selection.
[0057] NTRU Based FHE Scheme
[0058] As an alternative to the RLWE-based FHE scheme, in some
examples of the invention, an NTRU variant of the encryption scheme
is used to reduce computational complexity and to speedup
operations, as will be detailed below. The encryption system works
as follows.
[0059] A key generation function, Keygen (1.sup..lamda.), requires
the choosing of two polynomials f.sub.1.times.1,
g.sub.(1.times.1).rarw.D.sub.R.sub.q.sub.,.sigma..sub.k such that
(a) f is invertible in the ring R.sub.q; and (b) f.ident.1 (mod 2).
This can be done by sampling the polynomial f from the distribution
D.sub.R.sub.q.sub.,.sigma..sub.k until it satisfies conditions (a)
and (b).
[0060] The public key pk and the private (secret) key sk can be
computed from the equation shown in FIG. 6B.
[0061] For the encryption function, Enc(pk, m), the message space
is R.sub.q. A plain text polynomial .mu..epsilon.R.sub.q is
encrypted by evaluating the equation of FIG. 6C, where
S.sub.l.times.1,
E.sub.l.times.1.rarw.D.sub.R.sub.q.sub.,.sigma..sub.c.sup.l.times.1
are sampled from a discrete Gaussian distribution with standard
deviation .sigma..sub.c. This encryption function can be
implemented at the encryption engine 62.
[0062] Concerning the decryption function, Dec(sk, C), for a given
the ciphertext C, the plaintext .mu..epsilon.R.sub.q is restored by
multiplying C by the secret-key f using the equation of FIG. 6D.
This decryption function can be implemented at the decryption
engine 104.
[0063] With reference to FIG. 6D, the l polynomials in the first
term of the last line are in the form .mu.f, 2 .mu.f, . . . , 2 l-1
.mu.f. This means that the element at location i.epsilon.[0, l-1]
is in the form .mu.f2.sup.i+error. That is, the most significant
bit of each entry carries a single bit from each coefficient in
.mu.f assuming that error <q/2 and there is no wrap-around mod q
as is found in the prior art. As such, ".mu.f" can be fully
recovered from C which can then be multiplied by f.sup.-1 to
recover .mu..
[0064] In addition to the ability to recover the message polynomial
.mu., the l ring elements in the ciphertext facilitate and manage
noise growth in homomorphic operations (in particular, homomorphic
multiplication) as described below.
[0065] Regarding homomorphic operations, for input ciphertexts
C.sub.l.times.1 and D.sub.l.times.1.epsilon.R.sub.q.sup.l.times.1
encrypting .mu..sub.1 and .mu..sub.2 respectively, homomorphic
operations are defined as follows. For an addition operation,
ADD(C, D), to add two ciphertexts C.sub.l.times.1 and
D.sub.l.times.1, the output is C.sub.l.times.1+D.sub.l.times.1,
which is an entry-wise addition. For a multiplication operation,
MULT(C, D), to multiply two ciphertexts C.sub.l.times.1 and
D.sub.l.times.1, output is BD(C.sub.l.times.1)D.sub.l.times.1. The
addition and multiplication operations can be implemented at the
calculation engine 88.
[0066] The correctness of the above homomorphic addition and
homomorphic multiplication should be apparent to those skilled in
the art in view of this disclosure. It is clear that the
multiplication operation is asymmetric in the input ciphertexts C
and D. That is, the components of D are treated as a whole, whereas
the components of C are broken up into their "bit-wise
decompositions". It is shown below that this multiplication method
is correct and gives a slow noise-growth rate.
[0067] The correctness of the multiplication operation is evident
from the decryption operation shown in FIG. 6E, in which matrix
dimensions are removed for clarity. Note that the last line shown
in FIG. 6E is the encryption of .mu.=.mu..sub.2.mu..sub.if. Note
also that BD
(C.sub.l.times.1)BDI(I.sub.l.times.l)=I.sub.l.times.lC.sub.l.times.1=C.su-
b.l.times.1.
[0068] Concerning noise analysis, taking C as a fresh ciphertext,
the following observations can be made in view of the operation
shown in FIG. 6E. Homomorphic addition of v ciphertexts increases
the noise by a factor of v, in the worst case. In some
implementations, since the coefficients of the error polynomials
follow a Gaussian distribution, the factor is closer to O( {square
root over (.nu.)}). In addition, homomorphic multiplication of two
ciphertexts C=Enc(.mu..sub.1) and D=Enc(.mu..sub.2) with error
magnitudes B1 and B2, respectively, increases the error to
O(B.sub.1.parallel..mu..sub.2.parallel..sub.1+B.sub.2n log(q)) in
the worst case, and
O(B.sub.1.parallel..mu..sub.2.parallel..sub.1+B.sub.2 {square root
over (n log(q))}) in various contemplated applications. Here,
.parallel..mu..sub.2.parallel..sub.1 denotes the l.sub.1 norm of
the message polynomial .mu.. As can be seen, error dependence on
the two ciphertexts is asymmetric.
[0069] Regarding the setting of parameters, taking f to be a
function that is being evaluated and that computes the
multiplication of v ciphertexts, error.sub.f(B, n, q) denotes how
much the error grows when evaluating a function f on ciphertexts in
Rq with an initial error of magnitude B. For correct decryption,
the equation of FIG. 6F should be satisfied.
[0070] Since error in accordance with the present invention may
grow slower than in some known schemes and since there may be no
need for a chain of moduli to control the noise growth, q can be
set to be correspondingly smaller for the same security level
afforded by such known schemes. For a security level of .lamda.
bits, the equation of FIG. 6G should be satisfied.
[0071] Because log q according to the present invention may be
smaller relative to known schemes, n may be set smaller for the
same security level .lamda.. In turn, with a smaller n, the new
error.sub.f(B, n, q) is smaller, leading to an even smaller q, and
so on. Optimal parameters can be obtained by solving the above
inequalities of FIGS. 6F and 6G together. The table of FIG. 7
summarizes an example parameter selection.
[0072] In addition, the encryption scheme according to the present
embodiments does not use the flatten operation introduced by the
GSW scheme. A flattened ciphertext (Ctxt) takes up a large memory
space. It also needs considerable computation time to combine
entries and decompose them back into bits (or even decompose them
into groups of "m" bits for a higher radix Bit Decompose operator).
Further, since the encryption scheme according to the present
invention does not use a flatten operation, it does not have single
bits representing ciphertexts. Rather, ciphertexts are represented
as packed numbers mod q. To multiply ciphertexts, it is
advantageous to use the fast NTT algorithm to speed up the
ciphertext multiplication operation, as opposed to using regular
polynomial circular convolution. The encryption scheme according to
the present embodiment decrypts the most significant bit from all 1
polynomials in the ciphertext, as opposed to decrypting only a
single bit from a single polynomial in the ciphertext. In this way
1 bits, one from each polynomial, can be decrypted. These bits are
combined back into the encrypted polynomial using the formula shown
in FIG. 6H.
[0073] Types of encrypted confidential data that can be collected
and homomorphic calculations that can be performed thereon are now
discussed. Specific data/calculations discussed below include
analysis of financial transaction data, particularly credit card
data, and evaluation of relational operations. Although some of the
examples discussed pertain to specific data, it is noted that some
computations can be used for other types of data (e.g., relational
operations). Further, it is noted that the examples discussed below
are not limiting and other examples within the scope of the present
invention are contemplated.
[0074] Financial Transactions
[0075] One area of application for an embodiment of the homomorphic
encryption system is financial transactions, particularly credit
card transactions. Attacks on credit card information have
escalated tremendously in recent years, with major breaches
resulting in millions of client record being exposed. Once exposed,
the client information may be resold (e.g. on the dark web), used
for fraudulent transactions, particularly card not present (CNP)
transactions, or used for direct attacks on Point-of-Sale (POS)
systems.
[0076] These attacks happen because credit card information, either
in the databases or for the client, are present in a plain text
format at some point in the credit card authorization process.
According to an embodiment of the present invention, the system may
encrypt the credit card information from the point at which it is
acquired and never decrypt the ciphertext even at the
authentication step.
[0077] Payment Processing
[0078] Major credit card companies, such as VISA.TM. and
MasterCard.TM., structure bank card transactions in what is called
the four-party model. The parties in this model are the cardholder
1500, the merchant 1510 (the service provider), the payment
processor 1520 (the acquirer), and the card issuer 1540. In
addition, there is potentially a fifth party which is the credit
card company (payment brand) 1530 all as shown in FIG. 10.
[0079] The transaction starts when the cardholder 1500, who wishes
to purchase something, uses his credit card information on an
online merchant or presents his credit card 1505 to the merchant
1510 at the store. The merchant 1510, online or using the point of
sale system, acquires and encrypts the credit card information and
sends it 1515 to the payment processor 1520 over the network. The
encryption at the merchant may be done at the POS terminal or may
be done using an application installed on the merchant's systems
using the public encryption key. The payment processor 1520 then
decrypts the credit card information and forwards it 1525 to the
card issuer 1530 for authorization. The merchant 1510 charges the
credit card and provides the service or product to the cardholder
1500 once the credit card authorization is received. The payment
processor 1520 reimburses the merchant for the service, after that,
the card issuer 1530 pays back the payment processor 1520 within 24
or 48 hours. A tokenized system may be utilized between the payment
processor and the merchant to eliminate the need for the merchant
to store the cardholder information. The merchant stores only a
token corresponding to either the cardholder account or to the
individual transactions.
[0080] Information, such as cardholder and transaction data, as
well as internal fees, such as network and interchange fees, may be
passed 1545 between the payment processor 1520 and credit card
company 1540, or between 1555 the card issuer 1530 and credit card
company 1540, or both, either as required during the transaction,
or after the transaction.
[0081] There are multiple points of vulnerability in this credit
card system that may allow attackers to steal credit card
information.
[0082] Point of Sale System (POS): When the cardholder presents his
card at the merchant store, his/her card is swiped in a credit card
machine or a card reader that transfers this information to a
computer. There may be some points in the system where this card
information is not encrypted for some time before sending it to the
payment processor. At this point, malware installed in the system
can gather this information and send it back to the attacker. To
address this vulnerability, credit card information must be
encrypted the moment it is read using the card reader. This way,
malware will not be able to gather card information using this
method. Currently, point to point encryption (P2PE) is used to
securely transmit credit card information from the POS system to
the payment processor, which then decrypts it to send to the card
issuer for verification. P2PE uses 3DES or AES for encryption which
may be considered safe for the time being but may not be secure
against quantum computers in the future.
[0083] Payment Processor: When the payment processor receives the
encrypted credit card information from the merchant, it decrypts it
and sends it in plaintext to the card issuer for authentication.
This is a clear point of weakness. Any malware installed in the
payment processor system, or any attacker who broke the secure
channel between the payment processor and the card issuer, can
gather the credit card information while being in plaintext form.
To address this vulnerability, credit card information should never
be decrypted at any point. Additionally, in the case of a tokenized
system, the presence of the secure vault which translates the
transaction into a token and vice versa. This vault stores all the
cardholder information and their corresponding tokens. If this
vault is hacked, valuable information will be at risk.
[0084] The card issuer has two points of vulnerability: 1) it
receives the client credit card information in plaintext. This
exposes it to the possibility of attack; and 2) the card issuer
database of credit card information for all its clients sits in a
secure server in plaintext to compare it to the incoming card
information. This is the largest security threat on credit card
information. This is because an attack on this secure database will
result in the loss of all credit card information.
[0085] These vulnerabilities may be solved if all the credit card
information in the database and also the incoming card information
in plaintext were encrypted and were compared while in
ciphertext.
[0086] According to an embodiment as shown in FIG. 11, there is
presented an embodiment of the present invention which is a change
in the credit card authorization system that may provide a much
higher level of security than current systems. The embodied system
may be based on encrypting the credit card information at the POS
system using quantum secure Fully Homomorphic Encryption (FHE)
which is based on lattice based cryptography. The need for FHE is
based on the need for both homomorphic addition and multiplication
operations on encrypted data. If only one type of operation is
required, somewhat homomorphic encryption may be used. Once the
card information is encrypted at the online website or at the POS,
there may not be a need to decrypt at any point afterwards. The
encrypted credit card information may be compared against credit
card database data (e.g. sets of cardholder data for various
cardholders held in a database) at the card issuer which may also
be encrypted using the same FHE encryption.
[0087] Thus, the system may eliminate any possible attacks on
credit card information, either in the network, or in the database
servers because they may all be encrypted using a quantum secure
FHE encryption scheme.
[0088] The cardholder data from the consumer 1610 (the primary
account number (PAN)) is captured by the online transaction data
entry (or at the POS), the data is encrypted using a public-key
Fully Homomorphic Encryption (FHE) scheme. The credit card number
(except for the first digit and the last 4 digits), month, year,
and CVV numbers may be encrypted using the FHE public encryption
key published by the credit card company. The name of the
cardholder and the first digit and the last 4 digits of the credit
card number may be encrypted using the regular encryption schemes.
The merchant 1620 then sends the encrypted PAN to the
acquirer/payment processor 1630.
[0089] The payment processor 1630 forwards the encrypted PAN to the
appropriate payment brand 1640 using the credit card first digit
(e.g., Visa.TM., MasterCard.TM., American Express.TM., etc.). The
payment brand 1640 then forwards the encrypted PAN to the card
issuer 1650 (issuing bank) using the next five digits (BIN: Bank
Identification Number).
[0090] The card issuer 1650 may use the cardholder name and last 4
credit card digits to narrow down the credit card entries in the
database that match this information. Each entry in the narrowed
down list may be enumerated, compared against the encrypted PAN
using FHE algorithms, and the encrypted matching indicator will be
multiplied by the number corresponding to each entry in the list.
The encrypted result from each entry may all be added together to
get a final result. The final result may encrypt the number of the
matching account, the numbers corresponding to all other accounts
will be multiplied by `0` (due to mismatch), which will nullify the
result. The encrypted result is sent back to the payment brand
1640, which decrypts it and sends the final result, which is a
single digit corresponding to the matching account, back to the
card issuer 1650 to verify that the card is not reported lost or
stolen, and that the account has the appropriate amount of
credit/funds available to pay for the transaction. The card entries
may be ordered by most frequently used and the rest of the entries
may be ignored once a hit is found to reduce the verification
time.
[0091] If approved, the issuer 1650 generates an authorization
number and routes this number back along with a card-specific FHE
encrypted PAN to the payment brand 1640. The payment brand 1640 the
forwards the authorization code and the encrypted PAN back to the
acquirer/processor 1630.
[0092] The acquirer/processor 1630 accesses a secure vault 1660 to
retrieve/generate a token corresponding to the encrypted PAN. Note
that the secure vault 1660 may also be FHE encrypted except for the
tokens.
[0093] The acquirer/processor 1630 returns the token back to the
merchant 1620. The merchant may retain the token long term in a
merchant database 1670 for the processing of returns, retrieval
requests or chargebacks, as well as for business intelligence
reasons such as analysis of consumer buying behavior and creation
of marketing programs.
[0094] This embodiment is an example of a centralized system where
only one entity, the credit card company 1640, owns the secret
decryption key and distributes public encryption keys. Another
embodiment is a decentralized system where each card issuer 1650
holds its own secret decryption key and that the POS systems
recognize which card issuer the credit card belongs to and encrypts
the card's information using the appropriate public encryption key
corresponding to this card issuer. This decentralized system may
need another entity which is solely responsible for decrypting the
encrypted comparison result. This may be required to decouple the
card issuer servers holding the encrypted credit card information
from the servers which hold the valuable secret decryption key.
[0095] Another embodiment is the use of FHE multiparty computation.
Different secret keys will be generated for each user and the
public key will be stored in the credit card itself. The cardholder
data will be encrypted by the user's public key and may be sent to
the system for matching against other accounts. Multiple secret
decryption keys will be required to decrypt the final result.
Additionally, with multiple keys, each party may be limited to
decryption only of the information associated with their key, and
not be capable of decrypting any other information, improving
overall security of the data.
[0096] The system may be applied on multiple different levels: the
first version is a full system where information is encrypted from
the origin and never decrypted at any point in the system.
[0097] The second version is a partial system where, in order to
accommodate the current payment systems, and to reduce the risk of
data attacks, at the payment processor, after card information
decryption, card information can be re-encrypted using FHE before
sending to the card issuer. This may reduce the possibility of
attacks but there is a brief moment in time where data is in the
clear between information decryption and re-encryption where data
can reside in the device memory vulnerable to attacks. However, the
credit card database at the card issuer may always remain encrypted
using FHE.
[0098] The system also supports online transactions using online
payment systems using credit cards or PayPal or other online
transaction services where payment information may be encrypted on
the client PC prior sending over the Internet.
[0099] According to an embodiment, credit cards consist of the
following key information: Bank name (Front); Credit card number of
13-19 digits (Front); Credit card expiry date (Front); Cardholder
name (Front); and Card Verification Value (CVV) (Back). The credit
card number consists of a leading 6-digit Bank Identifier Number
(BIN), also known as an Issuer Identifier Number (IIN), and a 6- to
12-digit account number, and a single digit checksum number. The
terminating 3 digits of the account number and the checksum number
are encrypted with the same FHE scheme but using a different key
which may be given to specific parties for decryption. Other cards
based on a BIN or IIN, such as debit cards, reward cards, and
merchant-specific cards may also be used and have an equivalent
numbering system.
[0100] In an embodiment, the following fields may be encrypted
using quantum secure lattice-based Fully Homomorphic Encryption
(FHE): the middle 3-9 digits of the credit card number (e.g. the
secure portion of the account number), the expiry date and the CVV
number. These fields may never be decrypted at any point in the
verification process. They may always be in the ciphertext (Ctxt)
form.
[0101] The remaining items in the credit card may be encrypted
using the same encryption scheme but with one or more different
public/secret key pairs. A proper key management system is crucial
for the overall security of the system. The other keys may be used
by intermediate parties to decrypt the needed fields. Though the
remaining fields may be encrypted using an FHE scheme, no
homomorphic operations may be applied to them. Every character of
the remaining fields may be encoded in adjacent coefficients in the
same polynomial. The remaining fields are: bank name, cardholder
name, and the first 6 digits (BIN number) and last 4 digits of the
credit card number.
[0102] These remaining fields may be in plaintext (Ptxt) form at
some points in the verification process. They help identify the
credit card company (first digit) and well as narrow down the
credit card entries in the card issuer credit card database (last
four digits+Cardholder name). This choice narrows down the probable
matching cards to just a few. Additionally, by partitioning the
fields, the system may accommodate future changes to the credit
card number system, such as the 8-digit and alphanumerical BIN
proposals developed by the International Organization for
Standardization (ISO).
[0103] Additionally, further fields may be included in the
transaction request and handled without disruption as, since the
data is not decrypted, the actual source and content of the data is
not significant, only the ability to validate and authenticate it.
Thus, developments such as biometric (e.g. fingerprint)
identification, rotating or variable CVV numbers, and one-time card
numbers may all be validated and authentication via the present
method and system, and, in some cases, without any substantive
changes.
[0104] As mentioned earlier, some fields in the credit card
information may never be decrypted at any time in the verification
process. Performing the credit card authentication using these
encrypted items uses the properties of the FHE encryption
scheme.
[0105] Fully homomorphic encryption permits addition and
multiplication operations on encrypted data to generate an
encrypted result, which, when decrypted, gives the correct result
if the same set of operations were applied on unencrypted data.
[0106] To perform blind matching between two single bits x and y,
it is possible to apply the XNOR binary operation on these two bits
z=x.sym.y. To implement the XNOR operator homomorphically, it needs
to be broken down into addition and multiplication operations,
specifically, z=x.sym.y=xy+xy.
[0107] To match multiple-bit inputs, x=x.sub.ix.sub.2 . . . x.sub.n
and y=y.sub.1y.sub.2 . . . y.sub.n, this can simply be done by
matching the corresponding, same-order, individual bits, and
multiplying the result of the individual bit matching together
z=z.sub.i.times.z.sub.2.times. . . . .times.z.sub.n. Where the
final result z is a single bit indicating if x is equal to y or
not.
[0108] Implementing the XNOR operation using x.sym.y=xy+xy is used
for other homomorphic encryption schemes which support multiplying
multiple accumulator Ctxts (Ctxts that encrypts the result of
previous Ctxt multiplications). This multiple accumulator
multiplication is not suitable for the embodied encryption scheme
since the parameter values are set to be as small as possible,
while keeping the security level constant, that it supports
multiplying an accumulator only with either fresh Ctxts, or Ctxts
that are results from Ctxt addition/subtraction operations. This
may reduce the memory and network usage of the Ctxts, and may also
speedup Ctxt operations.
[0109] To address this, the XNOR operation can be re-written to
make it suitable for the embodied encryption scheme. The XNOR is
reformulated to be z=x.sym.y=(1-x-y)(1-x-y) and applying this
formula on x and y will result in z=1 only when x=y. This addresses
the system requirements, since for multi-bit inputs x and y, the
matching bit z equals
z=(1-x.sub.1-y.sub.1)(1-x.sub.1-y.sub.1)(1-x.sub.2-y.sub.2)(1-x.sub.2-y.s-
ub.2) . . . (1-x.sub.n-y.sub.n)(1-x.sub.n-y.sub.n). Now each
individual bracket consists only of Ctxt addition/subtraction
operation which is implementable using the embodied encryption
scheme.
[0110] Secure relational operation where z=1 if x>y, and z=0
otherwise is also possible using the FHE encryption scheme using
the secure relational operations as discussed above in FIGS. 6I and
6J. This can be very useful to also be able to check if the amount
needed for the credit card transaction "y" is exceeds the credit
limit for this account "x" (or exceeds the account limit in case of
a debit card).
[0111] According to an embodiment, in performing the secure credit
card authentication, the middle digits (for this example, 6 digits
for a 16-digit card number are assumed) of the credit card number
are first converted into bits where each bit is then encrypted
using FHE encryption. The same process is performed for the expiry
date and the CVV number. These encrypted bits are then sent on the
network along with the cardholder name, bank name, BIN number, and
last 4 digits of the credit card to the payment processor which
then forwards them to the appropriate card issuer. The card issuer
will then receive this information, and using the cardholder name
and last 4 digits of the card number, it will narrow down the
search to just a few possible matching accounts (due to the match
in the name and the last 4 digits) and the corresponding encrypted
information will be fetched. Secure authentication is then applied
between the incoming encrypted information and the possible
matching accounts. The authentication operation done on the
non-matching accounts will result in an encrypted indicator which
is z=0. The matching account, if all the bits of the credit card
number, expiry date, and CVV numbers match (and possibly BIN number
if used), and if the requested transaction amount is less than the
account balance or limit, an encrypted indicator of z=1 will be
generated. The encrypted indicator, either `0` or `1` can then be
multiplied by a number corresponding to the order of this entry
against the reduced list. To decrypt this final result and to get a
confirmation for this transaction, this encrypted indicator, which
encrypts the order of the matching account in the reduced list,
needs to be decrypted using the secret decryption key available at
the credit card company (or on a separate secure server owned by
the card issuer in the case of decentralized scheme). After
decryption, only a single digit is returned which corresponds to
the order of the matching account in the reduced list.
[0112] In another embodiment, the sensitive bits for the input card
and for each of the matching accounts are partitioned into "k"
parts, where "k" corresponds to the number of processors (e.g.
GPUs) available in each machine and each part is sent to a
different GPU. Each GPU may be enumerated by a number ranging from
0 to k-1. The result from each GPU may be rotated by a number of
polynomial slots corresponding to its GPU number by multiplying by
a polynomial with a 1 shifted to the corresponding location. When
the results are available from each GPU, all the results are added.
This will produce a result where k slots of the encrypted
polynomial each has a bit corresponding to a match or non-match for
the corresponding bits. After decryption, if all the bits are
non-zero, this means that this account is a match. When a matching
account is found, it may be put at the top of the list in the
database to make matching time faster in the next times, also the
remaining accounts may be ignored.
[0113] The quantum-secure, lattice-based FHE scheme used for this
system may be computationally intensive. To speed up the
homomorphic operations, graphical processing units (GPUs) are used
alongside the CPUs in the system. Each GPU is capable of supporting
a number of transactions per second. To achieve high transactions
per second required by large transaction companies like VISA.TM.
and MasterCard.TM. (each may require up to 4000 transactions per
second), many GPU cards are required. To realize such a performance
throughput, a multi-server environment may be implemented that can
be scaled to serve any level of demand.
[0114] Furthermore, while the above embodiment presents a simple
matching of the credit card in the transaction request against a
list of account names and numbers to determine the presence of a
valid account, other lists and types of validation may also take
place. For example, there may be a separate list of stolen/lost
cards and cardholder data against which the transaction request may
be compared as well. In that case, the card number may be valid,
but the authentication denied, if the card also appears on the
stolen/lost list. A similar process may also be used for other
validation and authentication lists, such as blacklisted countries
for transaction, while remaining compliant with laws regarding
disclosure of such information.
[0115] This may also be directed to the benefit of the user and/or
merchant as well, such as by matching data in the request to a
merchant rewards program, either by extracting the data from the
request, or by matching it in a list. This reduces the need for
multiple transactions and separate processing, potentially leading
to greater transaction efficiency. Further extensions and
variations may only be limited by data bandwidth and processing
time requirements.
[0116] Effectively, any encrypted data may be validated and
authenticated against any set of encrypted data without the need
for decryption. Changes in the data formats, data structure, or
even the data itself may be more easily accommodated, since
decryption information need not be shared.
[0117] Homomorphic encryption does not natively provide data
integrity. Hackers may be able tamper with the encrypted data by
apply their mathematical operations on the data while at rest or in
motion. This may affect the result of the manipulated data and may
change the result to something that may aid the hacker in an
attack. For example, applying a homomorphic OR operation on the
final encrypted flag with an encrypted "1" which will make the
result always a "1" or "Transaction Granted". The protection of
data integrity from the modification of unauthorized parties may be
provided by applying extra mathematical machinery to provide a
tamper resistant credit card authentication system. As an example,
without limitation, a Hash-based Message Authentication Code (HMAC)
can be appended to the encrypted message to provide the required
authentication of the message integrity.
[0118] In addition, as an extra security protection layer to the
credit card transaction, the system may support encrypting a credit
card CVV number that is periodically changing over time. The system
may homomorphically encrypt the CVV number generator key and store
it in the bank database. When a new customer credit card is to be
verified and purchase authorized, mathematical operations are
applied homomorphically on the CVV generator key to generate the
new valid CVV number to be matched against the customer credit card
to be verified.
[0119] For example, other forms of static or quasi-static private
identification data which required both confidential/encrypted
treatment and validation and authentication for a request, may be
used within this system and method. Not only bank debit cards
(similar data structure to credit cards), but other private
identification which is typically found in a card or similar format
may be used. As with credit cards, the data deemed confidential and
the data which may be shared as cleartext depends on the decisions
of the governing body responsible for the identification data, e.g.
the government in the case of a passport.
[0120] The full system according to an embodiment is shown in FIG.
12. The authentication process may take place when the payment
processor 1710 sends the credit card information to the card
issuer. The card issuer may have a front-end server 1720 that
accepts incoming requests. This front-end server 1720 may be
connected to a network of GPU machine farms 1730 that send frequent
updates to the front-end server 1720 about its occupancy (if it is
free or not). When the front-end server 1720 receives a request, it
may check its server table for a free machine 1740, and fetch its
corresponding IP address. This IP address is then sent back to the
payment processor 1710. The payment processor 1710 then connects to
this free computing node 1740 and sends the cardholder card
information to it. When the computing node 1740 gets that
information, it may fetch from the database the data of the credit
cards which match the cardholder name and the last four digits. The
authentication process may then take place on that GPU on this
specific machine 1740. The result of the authentication operation
may then be sent to the credit card company 1750 for decryption.
The decrypted authentication digit may then be sent back to the
card issuer 1710 to indicate the final result. This system may be
implemented to ease the network connection requirements for the
front-end server 1720 and to distribute it on all the computing
nodes 1730.
[0121] According to a more general embodiment, FIG. 1 shows a
networked computer system embodying various aspects of the present
invention for use with confidential data. A plurality of remote
devices 20 are connected to a data system 22 via a wide-area
network 24. The remote devices 20 may include devices such as
wearable devices 26, terminals 28, terminals 30, and the like.
Wearable devices 26 may include devices such as special-purpose
devices, smart-watches, wearable computers, smart clothing, and
similar. Wearable devices 26 may include, but are not limited to,
sensors such as global positioning system (GPS) receivers, and
similar. Terminals 28 and terminals 30 may include devices such as
smartphones, tablet computers, desktop/laptop computers, and
similar. Features that mutually distinguish the remote devices 20
such as wearable devices 26, terminals 28, and terminals 30 are not
contemplated to limit the present invention. Functionality of these
devices 26-30 may overlap. Wearable devices 26 and terminals 28
collect data directly from users and may do so without the need for
manual input by users. Wearable devices 26 and terminals 28 may
allow for manual input by a user 32 (e.g., typing or selecting) of
data. Terminals 30 may require some form of manual input by
personnel 34 to enter data related to users 32.
[0122] The remote devices 20 may be configured to collect data.
[0123] The wide-area network 24 may include one or a combination of
data networks, such as a local-area network, a wireless network, a
cellular network, an intranet, a virtual private network (VPN), and
the Internet.
[0124] The data system 22 may include one or more computers, which
may be known as servers, configured with program code that is
stored in memory and is executed by one or more processors to
perform homomorphic calculations on data collected from the remote
devices 20, as discussed in detail below.
[0125] The system may further include a key authority system 36 and
analyst terminals 38.
[0126] The key authority system 36 stores one or more cryptographic
keys, such as one or more private (secret) keys for a fully
homomorphic asymmetric cryptographic scheme, such as a
ring-learning with errors (RWLE) or NTRU homomorphic encryption
scheme. The key authority system 36 includes one or more servers
configured to store such private keys and restrict the use of the
private keys to authorized users.
[0127] Each private key controlled by the key authority system 36
may correspond to a public key that is distributed to the remote
devices 20. Data encrypted by the remote devices 20 could be
decrypted by users with access to the respective private key,
although this is not central to the present embodiment. According
to the present embodiment, the results of homomorphic calculations
performed on encrypted data may be decrypted by users with access
to the respective private key. It is contemplated that a large
number of remote devices 20 may share the same public key and thus
form a large and continuous source of data for homomorphic
calculations that do not require the decryption of the data.
Rather, the private key may only be needed for the decryption of
calculation results.
[0128] Any number of public-private key pairs may be used. It may
be advantageous to segment devices 20 into different sets,
according to device type or other factor, by providing such devices
with different public keys. Different sets or types of devices 20
may be given different public keys based on other factors, such as
the group/organization, device manufacturer, etc. Further, for a
device 20 that collects multiple types of data, each type of data
may be assigned to a different public key for encryption by that
public key. Again, this may reduce exposure of user data. For sake
of clarity, the examples discussed herein reference a single
public-private key pair, but it should be noted that the present
invention contemplates various public-private key pairs. Further,
it is important to note that collaboration and computation using a
set of data is limited to a set of encrypted data that is able to
be decrypted by the same private key. Hence, to facilitate
wide-ranging collaboration and computation, limiting the system to
a single private key may be advantageous.
[0129] Analyst terminals 38 may include devices such as
smartphones, tablet computers, desktop/laptop computers, and
similar operable by analysts 40 such as administrators,
researchers, and the like. Analyst terminals 38 may initiate
homomorphic calculations performed at the data system 22 and may
have the encrypted results of such calculations decrypted by the
private key held by the key authority system 36. Plaintext results
of the calculations may then be outputted at the analyst terminals
38 for further calculation and study.
[0130] The private key can be provided to decrypt calculation
results in various ways, depending on specific requirements of
various implementations according to the present invention.
Encrypted results of calculations may be transmitted to the key
authority system 36 for decryption at the key authority system 36,
with the decrypted plaintext results (DR) 42 of the calculations
being transmitted to one or more analyst terminals 38 for output
via a secure channel 44, such as a secure subnetwork, a VPN
operating through the wide-area network 24, or similar network that
offers increased security. Additionally, or alternatively, such a
secure channel 44 may be used to transmit the private key from the
key authority system 36 to the analyst terminals 38 to decrypt
calculation results at one or more analyst terminals 38. The secure
channel 44 need not be limited to network communications. For
example, analyst terminals 38 may be situated at physically secure
locations, thereby offering a physical aspect to the secure channel
44, if the private key or encrypted data is to be transmitted over
a network. Alternatively, the secure channel 44 may be mainly or
exclusively physical and the private key or encrypted data can be
copied onto physical key cards, memory sticks, or similar devices
that can be used to manually convey the private key or encrypted
data to the analyst terminals 38.
[0131] In operation, data may be continually collected by the
remote devices 20, encrypted at the remote devices 20, and
transmitted to the data system 22 as encrypted data (ED) 50. In
general, any device transmitting data to the data system 22 may be
configured to encrypt its data prior to transmission. The data
system 22 may store the encrypted data 50 for as long as desired.
At any time, an analyst terminal 38 may be used to select a set of
data for analysis and to configure a calculation to be performed on
the selected set of data. This information may be sent by the
analyst terminal 38 to the data system 22 as a calculation command
(CC) 52 that triggers the data system 22 to perform the calculation
on the selected encrypted data, according to a homomorphic
technique, without decrypting the data to obtain encrypted results
54. The encrypted results 54 of the calculation may then be
transmitted to the analyst terminal 38. The analyst terminal 38 may
then obtain decrypted results 42 using the secure channel 44 to
communicate with the key authority system 36.
[0132] Advantageously, data may not be decrypted during the
performance of calculations. User privacy may be improved and it is
contemplated that more users may volunteer their data to be used in
studies knowing that their data is better protected. In addition,
opportunities for man-in-the-middle and other types of attacks may
be mitigated due to the data and calculation results being
transmitted in encrypted form and due to tight control of the
private key.
[0133] In other applications, the remote devices 20 such as
wearable devices 26, terminals 28, terminals 30, data system 22,
terminals 38, remote devices 130 and other components may be
specifically general purposes devices or devices made specific to a
chosen application.
[0134] FIG. 2 shows an example remote device 20. The remote device
20 may include a sensor 60 and/or input device, an encryption
engine 62, and a network interface 64. The remote device 20 may
further include memory (e.g., RAM, hard-drive, solid-state drive,
etc.) for storing captured data 66, a public key 68, and encrypted
data 50. The remote device 20 may also include a processor
configured to execute the encryption engine 62 and control
operations of the remote device 20.
[0135] The sensor and/or input device 60 may be configured to
capture data of an individual. Example input devices include a
keyboard or touchscreen, for manual entry of data. An input device
may be used together with a sensor, such that the collected data
includes manually inputted data as well as directly measured data.
The type and nature of the data captured is not particularly
limited.
[0136] The encryption engine 62 may be configured to apply the
public key 68 to encrypt the captured data 66 to generate encrypted
data 50. The encryption engine 62 may be configured to perform
fully homomorphic encryption as discussed above.
[0137] The network interface 64 may be configured to communicate
encrypted data 50 to the network 24 and specifically to the data
system 22 (FIG. 1). After the encrypted data 50 has been
transmitted to the data system 22, it may be deleted to save
memory.
[0138] FIG. 3 shows an example of the data system 22. The data
system 22 may include a network interface 80, a data accumulator
82, a query constructor 84, a user interface 86, and a calculation
engine 88. The data system 22 may further include memory (e.g.,
RAM, hard-drive, solid-state drive, etc.) for storing encrypted
data 50, encrypted calculation results (ER) 54, data 90, and
authorizations 92. The data system 22 may also include a processor
configured to execute the data accumulator 82, the query
constructor 84, the user interface 86, and the calculation engine
88 and to control operations of the data system 22. It is noted
that no encryption key need be stored at the data system 22, as the
data system 22 performs its calculations on encrypted data.
[0139] The network interface 80 may be configured to receive data
and commands from the network 24. The network interface 80 may be
configured to receive encrypted data 50 and commands from the
remote devices 20. The network interface 80 may be configured to
receive calculation commands from analyst terminals 38.
[0140] The data accumulator 82 may be configured to control the
capture of encrypted data 50 from the plurality of remote devices
20. The data accumulator 82 may be configured to periodically
interrogate each remote device 20 for new encrypted data, receive
such encrypted data in response, and store such encrypted data 50
in memory at the data system 22. The data accumulator 82 may be
configured to reference the authorizations 92 as a condition for
collecting data.
[0141] The query constructor 84 may be configured to receive
calculation commands from analyst terminals 38. A calculation
command triggers commencement of a calculation by the calculation
engine 88. A calculation command may include parameters specifying
a set of the encrypted data 50 on which to conduct a calculation as
well as parameters specifying the nature of the calculation. The
query constructor 84 may be configured to provide to the analyst
terminals 38 a summary of the encrypted data 50 available for
calculation. The query constructor 84 may be configured to
reference the authorizations 92 as a condition for using elements
of encrypted data 50 in calculations.
[0142] The user interface 86 may be configured to receive commands
and data from remote devices 20, analyst terminals 38, or other
devices and to output information about the encrypted data 50.
Specifically, the user interface 86 may be configured to receive
data 90 that is not necessarily encrypted. Data 90 may also include
associations to the encrypted data 50, such that elements of
encrypted data 50 may be linked to the data 90 that is considered
useful for designing studies. For example, an association of data
90 to encrypted data 50 may include a unique identifier, such as a
hash or serial number, of the user in both the encrypted data 50
and the data 90. It may be beneficial to include the unique
identifier in the encrypted data 50 in plaintext form, such as via
an unencrypted metadata field attached to the encrypted data 50, a
file name, or an unencrypted database field in association with
encrypted data 50 stored in a database.
[0143] The user interface 86 may be configured to receive commands
to control authorizations 92 that are granted by users or other
individuals to, for example, analysts at terminals 38.
Authorizations 92 may include data indicative of the consent to
collect and store encrypted data 50 and consent to make encrypted
data 50 available to the calculation engine 88. In applications,
authorizations 92 may include one or more many-to-many mappings
that map users to data and further to individuals or organizations,
such that each (or his/her legal representative) can give consent
to provide any type of data to any individual or organization.
Authorizations 92 may also include time windows for consent, such
that consent is automatically withdrawn after expiry of a selected
time.
[0144] The user interface 86 may include an authentication system,
such as a username and password log-in authentication system, for
verifying that users who modify data 90 and authorizations 92 are
authorized to make such changes.
[0145] The calculation engine 88 may be configured to perform
homomorphic calculations on encrypted data 50 according to received
parameters defining the set of encrypted data 50 and the
calculations to perform. The calculation engine 88 outputs
encrypted results 54 that are transmitted via the network interface
to the key authority system 36 or the analyst terminal 38. The
calculation engine 88 can be configured to perform any suitable
calculation in the encrypted domain.
[0146] Such calculations are contemplated to include addition,
multiplication, discrete calculations, continuous calculations,
comparisons using relational operations, combinations of such, and
similar. Further, such calculations may be modeled as polynomial
series. To achieve this, the calculation engine 88 may be
configured as described below.
[0147] FIG. 4 shows an example key authority system 36. The key
authority system 36 may include a network interface 100, an
authorization processor 102, and a decryption engine 104. The key
authority system 36 may further include memory (e.g., RAM,
hard-drive, solid-state drive, etc.) for storing encrypted
calculation results 54, a private key 106, and decrypted results
42. The private key 106 corresponds to the public key 68 (FIG. 2)
stored in the remote devices 20. The key authority system 36 may
also include a processor configured to execute the authorization
processor 102 and the decryption engine 104, as well as to control
operations of the key authority system 36.
[0148] The network interface 100 may be configured to receive
encrypted results 54 from analyst terminals 38 via the secure
channel 44 (FIG. 1). The network interface 100 may be further
configured to transmit the decrypted results 42 to analyst
terminals 38. Alternatively, or additionally, the network interface
100 may be configured to transmit the private key 106 to the
analyst terminals 38 via the secure channel 44.
[0149] The authorization processor 102 may be configured to
restrict access to the key authority system 36 to authorized users.
The authorization processor 102 may include an authentication
system, such as a username and password log-in authentication
system or an electronic credential verification system, for
verifying users who attempt to access the decrypted results 42 or
the private key 106 or both.
[0150] The decryption engine 104 may be configured to apply the
private key 106 to decrypt the encrypted calculation results 54 to
generate the decrypted results 42. The decryption engine 104 may be
configured to perform homomorphic decryption as discussed
herein.
[0151] FIG. 5 shows a data system 120 according to an embodiment.
The data system 120 may include components discussed herein, such
as data system 22. The description for such components can be
referenced with like reference numerals indicating like components.
For sake of clarity, only differences will be discussed in detail.
According to an embodiment, the data system 120 may replace the
data system 22 in the system of FIG. 1.
[0152] The data system 120 may be configured to generate alerts
based on calculations performed on encrypted data 50 received from
remote devices 20. The data system 120 may transmit these alerts
via the network 24 to remote devices 130 (FIG. 1) operated by
professionals who may act on such alerts. It is contemplated that
generated alerts remain in the encrypted domain. As such, alerts
may be routed through the key authority system 36 for decryption
prior to being transmitted to the relevant remote devices 130 in
plaintext form. In such case, privacy may be enhanced by having the
plaintext form of the alert be a general indication of the nature
of the alert (e.g., a text alert) rather than specific numerical
values, as it is contemplated that recipients of alerts may have
background knowledge of the specific condition and perhaps
knowledge of specific alert trigger values. Alternatively, remote
devices 130 may be provided with the private key to decrypt
received alerts.
[0153] The data system 120 may include alert triggers 122
executable by a processor and configurable by authorized users. The
alert triggers 122 and the calculation engine 88 may perform
comparisons between encrypted results 54 and encrypted alert
conditions 124 stored in memory and configurable by authorized
users. Encrypted alert conditions 124 may be initially received in
unencrypted form via the network interface 80 before being
encrypted by an encryption engine 62 using the same public key 68
that encrypts the data at the remote devices 20. Encrypted alert
conditions 124 may be applied to data represented by the encrypted
results 54 and alert triggers 122 issue alerts for data that meet
the conditions. Such alerts may be configured to be transmitted via
the network interface 80 and the network 24 to remote devices 130
of selected authorized users, such as professionals. Alerts may be
communicated via the key authority system 36 for centralized
decryption prior to being forwarded to the selected authorized
users in plaintext form. Alternatively, the remote devices 130 may
store the private key 106 and may be configured with a decryption
engine 104 to decrypt the alerts. For example, a professional may
set an alert condition 124 for a user, identified by data 90, using
a equation homomorphically evaluated in the encrypted domain by the
calculation engine 88 using data continuously collected by a
wearable device 26. The encrypted alert condition 124 may be a
particular maximum, minimum, or interval, if met or exceeded,
causes the alert trigger 122 to send an electronic alert message to
remote devices 130 operated by a professional who may be otherwise
be unaware of the specific alert conditions. Advantageously, the
specific evaluation and the values evaluated may remain in the
encrypted domain, so as to improve privacy.
[0154] The alert triggers 122 may be evaluated on a periodic basis
or upon detecting new encrypted data 50. The alert triggers 122 may
store information concerning delivery of the alert, such as network
addresses (e.g., email addresses, phone numbers, etc.) of the
destination remote devices 130 that are to receive the alerts.
[0155] Various components of the data systems 22, 120, and
specifically the calculation engine 88, may be implemented as one
or more hardware devices. Such hardware devices may be configured
to implement the computational techniques discussed herein using
only hardware or by using hardware that executes program code. A
suitable hardware device may be configured to implement the
computational techniques discussed herein using, for example,
Chinese Remainder Theorem (CRT), Number Theoretic Transform (NTT),
one or more memory blocks, one or more memory interfaces, matrix
multiplications, matrix additions, or a combination of such. One
such suitable hardware device to achieve this is a Graphics
Processing Unit (GPU). Other examples of suitable hardware device
include a field-programmable gate array (FPGA) and an
application-specific integrated circuit (ASIC).
[0156] Tests were conducted on a system constructed according to
the present invention. FIG. 8 details the test system used. FIG. 9
summarizes performance results of the homomorphic operations for
the present invention compared to Khedr, A., Gulak, G., and
Vaikuntanathan, V. SHIELD: Scalable Homomorphic Implementation of
Encrypted Data-Classifiers, Cryptology ePrint Archive, Report
2014/838, 2014 (http://eprint.iacr.org/2014/838.pdf) shown in FIG.
9 at "Khedr 2014"; Lauter, K., Lopez-Alt, A., and Naehrig, M.
Private Computation on Encrypted Genomic Data. Tech. Rep,
MSR-TR-2014-93, June 2014 shown in FIG. 9 at "Lauter 2014"; and
LBos, J. W., Lauter, K., and Naehrig, M. Private predictive
analysis on encrypted medical data, In Journal of biomedical
informatics, Elsevier Inc., 2014, pp. 234-243 shown in FIG. 9 at
"LBos 2014". The results reported for "Lauter 2014" are for larger
parameters with larger circuit depth, as the circuit depth used to
test the present invention is higher than in "Lauter 2014" with the
chosen parameters. As can be seen, a CPU implementation of the
present invention may achieve a 58 times speedup for a
multiplication operation as compared to "LBos 2014". By
additionally exploring the parallelizable properties of the present
invention, a 104 times speedup may be achieved by distributing
computations on GPU cores. This resulted in an overall 6085 times
speedup for the multiplication operation compared to "LBos 2014"
and a 286 times speedup compared to "Lauter 2014".
[0157] The present invention may be advantageously scalable across
multiple GPU cards. Experiments were made using four GPU cards
connected to the same computer to measure loss in performance due
to cross-GPU communication. By partitioning large problems into
small ones, computations can be scheduled among all four GPUs to
obtain a speedup of 3.946 times, which indicates that communication
overhead was reduced.
[0158] Additionally, according to an embodiment, the encryption
scheme described herein may be used in privacy-preserving machine
learning applications, in privacy-preserving data mining including
privacy-preserving data clustering applications, and in general
secure computations on financial or other confidential data. For
example, the calculation engine 88 may be configured to perform
homomorphic calculations related to privacy-preserving machine
learning applications, in privacy-preserving data mining including
privacy-preserving data clustering applications on encrypted
data.
[0159] Numerous advantages may be apparent from the above
description of the present invention. Concerning wearable and
portable devices and the "Internet of Things" (IoT), the present
invention can be used to encrypt all data measured by wearable and
portable devices prior to uploading such data to the cloud. This
can be very useful to help researchers conduct research on
confidential data, in a manner that preserves privacy. As
discussed, these devices can store public encryption keys produced
by a centralized entity, which is also responsible for the control
and distribution of private/secret keys to facilities where
computation results/alerts are to be decrypted. Wearable/portable
devices need only encrypt the captured data, and the modest
processing power that is known in these kinds of devices is not a
significant hindrance to implementation. Since performance of the
encryption function is not necessarily time critical, embedded
processors can encrypt measured data within seconds instead of
milliseconds and still have acceptable performance.
[0160] The present invention may be embodied in other specific
forms without departing from the spirit or essential
characteristics thereof. Certain adaptations and modifications of
the invention will be obvious to those skilled in the art.
Therefore, the presently discussed embodiments are considered to be
illustrative and not restrictive, the scope of the invention being
indicated by the appended claims rather than the foregoing
description and all changes which come within the meaning and range
of equivalency of the claims are therefore intended to be embraced
therein.
* * * * *
References