DETERMINATION OF PAIRINGS ON A
CURVE USING AGGREGATED INVERSIONS
BACKGROUND
[0001] Computers have become increasingly interconnected via networks (such as the
Internet), and security and authentication concerns have become increasingly important.
Cryptographic techniques that involve a key-based cipher, for example, can take
sequences of intelligible data (e.g., typically referred to as plaintext) that form a message
and mathematically transform them into seemingly unintelligible data (e.g., typically
referred to as ciphertext), through an enciphering process. In this example, the
enciphering can be reversed, thereby allowing recipients of the ciphertext with an
appropriate key to transform the ciphertext back to plaintext, while making it very
difficult, if not nearly impossible, for those without the appropriate key from recovering
the plaintext.
[0002] Public -key cryptographic techniques are an embodiment of key -based cipher. In
public -key cryptography, for example, respective communicating parties have a
public/private key pair. The public key of each respective pair is made publicly available
(e.g., or at least available to others who are intended to send encrypted communications),
and the private key is kept secret. In order to communicate a plaintext message using
encryption to a receiving party, for example, an originating party can encrypt the plaintext
message into a ciphertext message using the public key of the receiving party and
communicate the ciphertext message to the receiving party. In this example, upon receipt
of the ciphertext message, the receiving party can decrypt the message using its secret
private key, thereby recovering the original plaintext message.
[0003] An example of public/private key cryptology comprises generating two large
prime numbers and multiplying them together to get a large composite number, which is
made public. In this example, if the primes are properly chosen and large enough, it may
be extremely difficult (e.g., practically impossible due to computational infeasibility) for
someone who does not know the primes to determine them from just knowing the
composite number. However, in order for this method to be secure, the size of the
composite number should be more than 1,000 bits. In some situations, such a large size
makes the method impractical to be used.
[0004] An example of authentication is where a party or a machine attempts to prove
that it is authorized to access or use a product or service. Often, a product ID system is
utilized for a software program(s), where a user enters a product ID sequence stamped on
the outside of the properly licensed software package as proof that the software has been
properly paid for. If the product ID sequence is too long, then it will be cumbersome and
user unfriendly. Other common examples include user authentication, when a user
identifies themselves to a computer system using an authentication code.
[0005] As another example, in cryptography, elliptic curves are often used to generate
cryptographic keys. An elliptic curve is a mathematical object that has a structure and
properties well suited for cryptography. Many protocols for elliptic curves have already
been standardized for use in cryptography. A recent development in cryptography
involves using a pairing, where pairs of elements from one or more groups, such as points
on an elliptic curve, can be combined to generate new elements from another group to
create a cryptographic system.
SUMMARY
[0006] This Summary is provided to introduce a selection of concepts in a simplified
form that are further described below in the Detailed Description. This Summary is not
intended to identify key factors or essential features of the claimed subject matter, nor is it
intended to be used to limit the scope of the claimed subject matter.
[0007] Encryption and decryption are usually performed based on a secret. This secret
can utilize an order of a group of points, or some other characteristic of the group, such as
the generator, or a multiple of the generator. A variety of different groups can be used in
cryptography, such as implementing the points on an elliptic curve for the group's
elements. A group of elements (e.g., points) derived from an elliptic curve can be used in
the encryption/decryption, for example, as the discrete logarithm problem (DLP) for such
a group is considered to be hard. A hard DLP is preferred in cryptography in order to
create a secure encryption/decryption process, for example.
[0008] Currently, when computing pairings on an elliptic curve a lot of operations such
as multiplications or inversions in the finite field that the elliptic curve is defined over can
be required. One can attempt to reduce the number of multiplications to reduce
computation expense, and/or speed the computations up in other ways. One technique for
speeding up the computations is to reduce a number of inversions undertaken when
computing the pairing.
[0009] For example, when working in affine space, both multiplications and inversions
are performed, where inversions are more computationally expensive than multiplications.
In order to reduce the number of inversions current practitioners change the coordinate
system of the curve points from affine space to projective space. This has an effect of
reducing the inversions, while increasing the number of multiplications, which are
computationally cheaper.
[0010] One or more of the techniques and/or systems described herein provide an
alternate to converting the coordinates to projective space, while still reducing a number of
inversions needed to compute the pairing on the elliptic curve. Using these techniques and
systems one may aggregate inversions for coordinates in affine space, for example, and
reuse the aggregated inversions for an additive act used for the pairing computation.
Further, portions of the computations can be parallelized on multi-core systems, for
example, to speed up the overall computation time. In this way, for example, pairings
used in a cryptographic system can be computed using less computational resources and in
a shorter time (e.g., faster) than present implementations.
[0011] In one embodiment, when determining mathematical pairings for a curve for use
in cryptography, a plurality of inversions that are used when determining the mathematical
pairings for the curve are aggregated (e.g., into a single inversion, such as an intermediate
calculation in the pairings computation). The mathematical pairings for the curve are
determined in affine coordinates along a binary representation of a scalar read from right
to left using the aggregated plurality of inversions.
[0012] To the accomplishment of the foregoing and related ends, the following
description and annexed drawings set forth certain illustrative aspects and
implementations. These are indicative of but a few of the various ways in which one or
more aspects may be employed. Other aspects, advantages, and novel features of the
disclosure will become apparent from the following detailed description when considered
in conjunction with the annexed drawings.
DESCRIPTION OF THE DRAWINGS
[0013] Fig. 1 is a block diagram illustrating an exemplary cryptosystem in accordance
with one or more of the methods and/or systems disclosed herein.
[0014] Fig. 2 is an illustration of an exemplary system using a product identifier to
validate software.
[0015] Fig. 3 is a flow-chart diagram of an exemplary method for determining
mathematical pairings for a curve for use in cryptography.
[0016] Fig. 4 is a flow diagram illustrating one embodiment of an implementation of
portions of one or more of the methods described herein.
[0017] Fig. 5 is a flow diagram illustrating an exemplary implementation of one or more
of the techniques and/or systems described herein.
[0018] Fig. 6 is a component block diagram illustrating an exemplary system for
determining mathematical pairings for a curve for use in cryptography.
[0019] Fig. 7 is a component block diagram illustrating an exemplary implementation of
one or more of the systems described herein.
[0020] Fig. 8 is a component block diagram illustrating an exemplary implementation of
one or more of the systems described herein.
[0021] Fig. 9 is an illustration of an exemplary computer-readable medium that may be
devised to implement one or more of the methods and/or systems described herein.
[0022] Fig. 10 is a component block diagram of an exemplary environment that may be
devised to implement one or more of the methods and/or systems described herein.
DETAILED DESCRIPTION
[0023] The claimed subject matter is now described with reference to the drawings,
wherein like reference numerals are used to refer to like elements throughout. In the
following description, for purposes of explanation, numerous specific details are set forth
in order to provide a thorough understanding of the claimed subject matter. It may be
evident, however, that the claimed subject matter may be practiced without these specific
details. In other instances, structures and devices are shown in block diagram form in
order to facilitate describing the claimed subject matter.
[0024] The one or more cryptographic pairings techniques and/or systems described
herein can determine mathematical pairings for an elliptic curve that can be used for
cryptographic applications. For example, they can be used to determine pairings for a
proposed authorization (e.g., an electronic signature) for cryptographic applications.
[0025] Typically, a pairing-based cryptosystem utilizes a group (e.g., of elements and a
binary multiplier derived from an elliptic curve) whose elements are publicly known (e.g.,
by knowing the curve). The scalar that is used to compute the pairing is publicly known.
Unknown secrets are either the input points or implicitly contained in the input points to
the pairing. The basis for the security is the hardness of an associated discrete logarithm
problem. A pairing-based encryption and decryption as illustrated in Fig. 1, as an
example, typically refers to encryption and decryption that uses keys that are generated
based on aspects or characteristics of an algebraic curve. The exemplary cryptosystem of
Figs. 1 and 2 can be based on the curve being publically known but the points generated
being secret, as the points generated from the curve by the scalar are secret (e.g., and
difficult to determine). In one embodiment of the pairing-based cryptography, the curve
may be an elliptic curve, and the elements that comprise the group can be generated from
points on the elliptic curve. As one of ordinary skill in the art may appreciate, in a typical
situation a point P is publicly known and a scalar m is secret. Then a point Q = mP is
made public as well. Because an associated discrete log problem is hard, it is infeasible to
determine m from P and Q. Accordingly, the secrets are usually just the scalars and the
points are public.
[0026] Pairing-based cryptosystems can be used to encrypt a wide variety of
information. For example, a cryptosystem may be used to generate a "short" signature or
product identifier, which is a code that allows validation and/or authentication of a
machine, program or user, for example. The signature can be a "short" signature in that it
uses a relatively small number of characters.
[0027] Fig. 1 is a block diagram illustrating an exemplary cryptosystem 100 in
accordance with certain embodiments of the methods and systems disclosed herein. The
exemplary cryptosystem 100 comprises an encryptor 102 and a decryptor 104. A plaintext
message 106 can be received at an input module 108 of the encryptor 102, which is a
pairing-based encryptor that encrypts message 106 based on a public key—an element of a
publicly known group—generated based on a secret scalar (known merely by decryptor
104). In one embodiment, the group can be a group of points generated from the elliptic
curve used by the encryptor 102, and discussed in more detail below. A plaintext message
106 is typically an unencrypted message, although encryptor 102 can encrypt other types
of messages. Thus, the message 106 may alternatively be encrypted or encoded by some
other component (not shown) or a user.
[0028] An output module 110 of the encryptor 102 outputs an encrypted version of the
plaintext message 106, which can be ciphertext 112. Ciphertext 112, which may comprise
a string of unintelligible text or some other data, can then be communicated to the
decryptor 104, which can be implemented, for example, on a computer system remote
from a computer system on which encryptor 102 is implemented. Given the encrypted
nature of ciphertext 112, the communication link between the encryptor 102 and the
decryptor 104 need not be secure (e.g., it is often presumed that the communication link is
not secure). As an example, the communication link can be one of a wide variety of
public and/or private networks implemented using one or more of a wide variety of
conventional public and/or proprietary protocols, and including both wired and wireless
implementations. Additionally, the communication link may include other non-computer
network components, such as hand-delivery of media including ciphertext or other
components of a product distribution chain.
[0029] The decryptor 104 receives the ciphertext 112 at an input module 114 and,
because the decryptor 104 is aware of the secret key corresponding to the public key used
to encrypt the message 106 (e.g., as well as the necessary generator), can decrypt the
ciphertext 112 to recover the original plaintext message 106, which is output by an output
module 116 as a plaintext message 118. In one embodiment, the decryptor 104 is a
pairing-based decryptor that decrypts the message based on the group of points generated
from the elliptic curve (e.g., a group as was used by encryptor 102), and is discussed in
more detail below.
[0030] In one embodiment, the encryption and decryption are performed in the
exemplary cryptosystem 100 based on a secret, which may be the scalar used to generate
the public key, an element of a group of points from the elliptic curve, thereby allowing
the solution to the problem to be difficult to determine. The secret is known to the
decryptor 104, and a public key can be generated based on the secret known to encryptor
102. In this embodiment, this knowledge may allow the encryptor 102 to encrypt a
plaintext message that can be subsequently decrypted merely by the decryptor 104. Other
components, including the encryptor 102, which do not have knowledge of the secret,
cannot decrypt the ciphertext (e.g., although decryption may be technically possible, it is
not computationally feasible). Similarly, in one embodiment, the decryptor 104 may also
generate a message using the secret key based on a plaintext message; a process referred to
as digitally signing the plaintext message. In this embodiment, the signed message can be
communicated to other components, such as the encryptor 102, which can verify the
digital signature based on the public key.
[0031] Fig. 2 is an illustration of an exemplary system 200 using a product identifier to
validate software in accordance with certain embodiments of the methods and systems
described herein. The exemplary system comprises a software copy generator 202
including a product identifier (ID) generator 204. Software copy generator 202 may
produce software media 210 (e.g., a CD-ROM, DVD (Digital Versatile Disk), etc.) that
can contain files needed to collectively implement a complete copy of one or more
application programs, (e.g., a word processing program, a spreadsheet program, an
operating system, a suite of programs, and so forth). These files can be received from
source files 206, which may be a local source (e.g., a hard drive internal to generator 202),
a remote source (e.g., coupled to generator 202 via a network), or a combination thereof.
Although a single generator 202 is illustrated in Fig. 2, often multiple generators operate
individually and/or cooperatively to increase a rate at which software media 210 can be
generated.
[0032] A product ID generator 204 can generate a product ID 212 that may include
numbers, letters, and/or other symbols. The generator 204 generates a product ID 212
using the pairing-based encryption techniques and/or systems described herein. The
product ID 212 may be printed on a label and affixed to either a carrier containing
software media 210 or a box into which software media 210 is placed. Alternatively, the
product ID 212 may be made available electronically, such as a certificate provided to a
user when receiving a softcopy of the application program via an on-line source (e.g.,
downloading of the software via the Internet). The product ID 212 can serve multiple
functions, such as being cryptographically validated to verify that the product ID is a valid
product ID (e.g., thus allowing the application program to be installed). As a further
example, the product ID 212 may serve to authenticate the particular software media 210
to which it is associated.
[0033] The generated software media 210 and associated product ID 212 can be
provided to a distribution chain 214. The distribution chain 214 can represent one or more
of a variety of conventional distribution systems and methods, including possibly one or
more "middlemen" (e.g., wholesalers, suppliers, distributors, retail stores (either on-line or
brick and mortar), etc.), and/or electronic distribution, such as over the Internet.
Regardless of the manner in which media 210 and the associated product ID 212 are
distributed, the media 210 and product ID 212 are typically purchased by (e.g., licensed)
or distributed to, the user of a client computer 218, for example.
[0034] The client computer 218 can include a media reader 220 that is capable of
reading the software media 210 and installing an application program onto client computer
218 (e.g., installing the application program on to a hard disk drive or memory (not
shown) of client computer 218). In one embodiment, part of the installation process can
involve entering the product ID 212 (e.g., to validate a licensed copy). This entry may be
a manual entry (e.g., the user typing in the product ID via a keyboard), or alternatively an
automatic entry (e.g., computer 218 automatically accessing a particular field of a license
associated with the application program and extracting the product ID therefrom). The
client computer 218 can also include a product ID validator 222 which validates, during
installation of the application program, the product ID 212. In one embodiment, the
validation can be performed using the pairing-based decryption techniques and/or systems
described herein. If the validator 222 determines that the product ID is valid, an
appropriate course of action can be taken (e.g., an installation program on software media
210 allows the application to be installed on computer 2 18). However, if the validator 222
determines that the product ID is invalid, a different course of action can be taken (e.g.,
the installation program terminates the installation process preventing the application
program from being installed).
[0035] In one embodiment, the product ID validator 222 can also optionally authenticate
the software media (e.g., application program) based on the product ID 212. This
authentication verifies that the product ID 212 entered at computer 218 corresponds to the
particular copy of the application be accessed, for example. As an example, the
authentication may be performed at different times, such as during installation, or when
requesting product support or an upgrade. Alternatively, in this embodiment, the
authentication may be performed at a remote location (e.g., at a call center when the user
of client computer 218 calls for technical support, the user may be required to provide the
product ID 2 12 before receiving assistance).
[0036] In one embodiment, if an application program manufacturer desires to utilize the
authentication capabilities of the product ID, the product ID generated by generator 204
for each copy of an application program can be unique. As an example, unique product
IDs can be created by assigning a different initial number or value to each copy of the
application program (e.g., this initial value is then used as a basis for generating the
product ID). The unique value associated with the copy of the application program can be
optionally maintained by the manufacturer as an authentication record 208 (e.g., a
database or list) along with an indication of the particular copy of the application program.
The indication of the copy can be, for example, a serial number embedded in the
application program or on software media 210, and may be hidden in any of a wide variety
of conventional manners. Alternatively, for example, the individual number itself may be
a serial number that is associated with the particular copy, thereby allowing the
manufacturer to verify the authenticity of an application program by extracting the initial
value from the product ID and verifying that it is the same as the serial number embedded
in the application program or software media 210.
[0037] A method can be devised that allows a mathematical pairing to be determined for
a curve, where a first set of elements submitted as a cryptographic key (e.g., points on an
elliptic curve) can be compared with known points on the curve, and used for
cryptographic purposes. Effective cryptosystems are typically based on groups where the
Discrete Logarithm Problem (DLP) for the group is hard (e.g., difficult to calculate), such
as a group of points from the elliptic curve. The DLP can be formulated in a group, which
is a collection of elements together with a binary operation, such as a group multiplication.
As an illustrative example, the DLP may be: given an element g in a finite group G and
another element h that is an element of G, find an integer x such that gx = h. Generating
pairings for use in cryptography typically requires a lot of underlying multiplications in a
finite field over which the elliptic curve is defined.
[0038] Fig. 3 is a flow-chart diagram of an exemplary method 300 for determining
mathematical pairings for a curve for use in cryptography. The exemplary method 300
begins at 302 and involves aggregating a plurality of inversions used in determining the
mathematical pairings for the curve, at 304. Because pairing operations on an elliptic
curve utilize a lot of underlying multiplications in the finite field, in order to make pairing
operations more efficient, for example, a number of multiplications can be mitigated,
and/or efficiencies can be created in other pairing operations.
[0039] In one embodiment, in the finite field for the curve, both multiplications and
inversions (e.g., identifying multiplicative inverses or reciprocals) are performed for the
pairing operation. Inversions are typically more computationally expensive to perform
than the multiplications. For example, an inversion to multiplication ratio for
computations can often be eighty to one, as a coordinate system used for the curve points
is commonly changed from affine to projective in order to reduce a number of inversions.
As an illustrative example of an inversion determination, to approximate a multiplicative
inverse of a nonzero real number x, a number y can be repeatedly replaced with 2y-xy 2.
In this example, when changes to y remain within a threshold, y is an approximation of the
multiplicative inverse of x. It will be appreciated that this example is merely for
illustration purposes, and that there are other techniques for determining inversions,
particularly for other types of numbers, such as complex numbers.
[0040] In the exemplary method 300, for example, while working in the affine
coordinate system, a number of inversions can be greatly mitigated by combining the
inversions, and determining them at a same time. In one embodiment, when using affine
coordinates for the pairing computation, respective doubling (e.g., multiplication) and
addition acts use a finite field inversion to compute a slope value for a line that is
evaluated in a subsequent act. In this embodiment, the inversions can be aggregated, for
example, using "Montgomery's trick" to replace 1finite field inversions by a single
inversion and 3(1-1) multiplications.
[0041] As an illustrative example of Montgomery's trick, in order to determine
inversions for elements x and y, instead of determining two inversions the product xy can
be determined and its inverse computed. In this example, the inverse of x and y can then
be determined by the multiplications: x 1 = (xy)_1y a d y 1 = (xy) 1• In hi way, in this
example, the inversions of the two elements x and y can be determined by one inversion
and three multiplications. Where inversions are to be determined for n elements, merely
one inversion and 3(n-l) multiplications can be performed. Therefore, in one
embodiment, where the pairing computation comprises a plurality of inversions (n), the n
inversions can be aggregated into one inversion.
[0042] In one embodiment, let [al . . .,as] be a sequence of elements of which reciprocals
[a 1, . . .,as
_1] are to be computed. The reciprocals can be computed by first computing a
product ai . . .as, its reciprocal (ai . . .as)_1 , the products ai a;_i a + 1 as for 1 < i < s, and the
reciprocals of single elements by
a 1 = (ai a _i ai+i as) (a as)_1 .
The acts can be performed in 1 inversion and 3(s-l) multiplications. That is, s inversions
can be replaced by 1 inversion and 3(s-l) multiplications.
[0043] In one embodiment, the product ai a2. . .as can be computed in a binary tree with
s-1 multiplications, for example, where the s-1 products can be stored for use in the
inversion aggregation. Further, in this embodiment, the reciprocals (ai &2. . .as)_1 are
computed, and subsequent reciprocals are computed along a same tree with 2(s-l)
multiplications.
[0044] Returning to the exemplary method 300 of Fig. 3, at 306, the mathematical
pairings are determined for the curve in affine coordinates along a binary representation of
a scalar read from right to left using the aggregated inversions. When computing a Tate
pairing for a curve a typical Miller loop algorithm goes through the scalar from left to
right (or from top down). As an illustrative example, assume that k>l so that
denominators in a Miller algorithm are eliminated.
[0045] For the following exemplary embodiment and illustrative examples utilize the
following notations: Let p>3 be a prime and Fq be a finite field of characteristic p. Let E
be an elliptic curve defined over Fq having a Weierstrass equation E:y 2=x 3+ax+b. For a
prime r with r|#E(Fq), let k be an embedding degree of E with respect to r, i. e. k is a
smallest positive integer with r|q k-l. A set of r-torsion points on E can be denoted by
E[r] and a set of F_(q i )-rational r-torsion points by E(F_(q i ))[r] for i>0. Let _q be a
q-power Frobenius endomorphism on E.
[0046] Further, define
G_l=E[r]nker ( q-[l])=E(F _q)[r],
Let k>l . A reduced Tate pairing is a map:
e_r=G_l xG_2G_3,
(P,Q) f_(r,P) (Q)-((q-k- l)/r),
where f_(r,P) F_q (E) is a function with divisor r(P)-r(0). Denote the function in F_q (E)
given by a line through two points R and S on E by 1_(R,S). If R=S, then the line is given
by the tangent to the curve passing through R.
[0047] The following illustrates one embodiment of a typical Miller loop for computing
a Tate pairing (using the above notations):
Input: P G G , Q G G2 , r = rm _ , ... , r 0) 2
Output: e P , Q) = ?)
R - P , f - 1
for ( - m —2; > 0;
f f 2 - R,R Q
R - [2]R
if (r = l)then
R R + P
end if
end for
10: / - f
11: return /
[0048] In this illustrative example, Lines 3 and 4 in the above algorithm together are
commonly called a doubling act, and Lines 6 and 7 are commonly called an addition act.
[0049] In one embodiment of the act 306 of the exemplary method 300, the Miller loop
algorithm can be modified, where the binary representations are read from right to left (or
bottom up). The following is an illustrative example of the right to left (or bottom up)
approach:
Input: P GG , Q = (r,
Output: ', ?
1 : - , / - 1
2 : - 0 , - 1
3 : for(i - 0 ; < —1; i + +)do
4 : if = 1) then
6 - +
7 end if
8
9 R - [2]P
10: end for
[0050] In this illustrative example, the doubling act is in Lines 8 and 9, and the addition
act in Lines 5 and 6. The above algorithm does m doubling acts and h addition acts. In
this example, although the loop can be done m times, merely m-1 doubling acts are used, a
last one may not influence the computation.
[0051] Further, in this embodiment, when using the "bottom-up" approach, the addition
acts can be postponed in line 5 and 6 (in the bottom-up algorithm, above). Here, for
example, pairs of relevant function values and corresponding points f R , R) can be stored
in a list L (e.g., in a database), and a computation of a final function value can be
computed later. As an illustrative example, the following algorithm provides a bottom-up
approach with postponement of the addition acts, by storing the values and points (see
Line 5 of the following algorithm), and carries out the computation of the final function
value later (see Line 10):
Input: P G G , Q GG2 , r = (r,
Output: e P, Q) = ?)
1 : - , / - 1
2 : - [ ]
3 : For ( - 0 ; i m - 1; i + + ) do
4 : if = 1) then
5: Append f R , R to L .
6 : end if
8 : R - [ 2 ]R
9 : End for
10: Compute f r >p Q from the pairs in L .
qk - l
i i : / - / —
12: return /
In this example, the approach can be more efficient than the present top-down algorithm,
as postponing the computation allows to save costs equivalent to h-1) multiplications in
F_(q k).
[0052] In one embodiment, using the aggregated inversions, line 10 of the above
algorithm, "Compute f r >p Q from the pairs in ," can also be carried out along a binary
tree. In this embodiment, in each layer of the binary tree, the aggregated inversion
technique can be applied. In this way, for example,as described above, (h- 1) inversions
can be substituted for [log(h)] inversions and 3(h-l -[log(h)]) multiplications when
computing mathematical pairings for the curve. Therefore, the number of inversions is
dramatically reduced while a small number of multiplications are added, which are
computationally cheaper.
[0053] Having computed the mathematical pairings for the curve, the exemplary method
300 of Fig. 3 ends at 308.
[0054] Fig. 4 is a flow diagram illustrating one embodiment 400 of an implementation
of portions of one or more of the methods described herein. Determining the
mathematical pairings for the curve in affine coordinates along a binary representation of a
scalar read from right to left using the aggregated plurality of inversions can comprise
reading a binary representation of the scalar from right to left, at 402. As described above,
the binary representation of the scalar can be read from right to left, for example, when the
curve point coordinates are in the affine space.
[0055] At 404, a multiple of a curve point is determined by computing a scalar multiple
of a curve point, where the scalar multiple, for example, is a m-fold sum of the curve point
in affine space. That is, for example, the multiplication acts of the bottom up approach
algorithm can be performed, a plurality of curve point multiples can be determined. This
act is often called the doubling act, for example, where for (i - 0; i < m — 1 ; i + + )
do. . .f R - f R • lR R (Q), and R - [2] (using the above described notations). Notably, in
this embodiment, the multiplication is performed on the coordinates in affine space, unlike
current commonly used techniques that switch the coordinates to projective space in order
to mitigate a number of inversions, for example.
[0056] At 406, the inversions of the additions of the curve point can be determined for
the finite field. That is, for example, while reading the scalar from right to left, curve
points are added depending on the binary representation of the scalar. In this example, for
the curve point additions, inversions are determined in the finite field. As an example, an
inversion is often referred to as a multiplicative inverse, or a reciprocal. As described
above, the inversions are aggregated, for example, into a single inversion for respective
acts in the additive process. In this way, a plurality of inversions are combined at
respective levels in the binary tree representation of the pairing portion of the algorithm,
for example, merely using one inversion for respective levels.
[0057] At 408, an output of the aggregated inversion is determined, for example, as a
slope value for a line function, and the line function is updated with the outputted slope
value. For example, when computing pairings for elements (e.g., curve points) for a curve
over a finite field, line functions are evaluated to compute the pairings, such as to a new
element in a different group. As such, in this example, the aggregated inversion output is
used as the slope value for the line function, for example, at respective levels of the binary
tree representation, and the line is evaluated with the slope value to determine the pairings.
[0058] Fig. 5 is a flow diagram illustrating one embodiment 500 of an implementation
of one or more of the techniques and/or systems described herein. In this embodiment
500, two elements are received from an electronic signature 550, such as submitted by a
user to authenticate their identity, as input for a pairing computation, at 502. Further, two
elements from a known group 552 (e.g., a shared secret cryptographic key for security) are
submitted as input for pairing computation. In this embodiment, pairings will be
computed for the elements from the signature 550 and pairings will be computed for the
elements from the group 552.
[0059] At 504, multiples of the curve points submitted as elements of the group are
determined, as described above. In this embodiment 500, binary representations of the
scalar read from right to left 554 are used to determine the multiples of the curve point.
The inversions are aggregated from the multiples at 506, into a single inversion, and are
stored 556, such as in a remote or local database. In one embodiment, the scalar, which is
read from right to left in a binary representation 554, may be comprised in a cryptographic
key, such as a public key.
[0060] In one embodiment, the determining of pairings for the curve in affine
coordinates along a binary representation of a scalar read from right to left can be
parallelized on a plurality of processors, at 508. For example, computers commonly have
multi-core processors, which may allow the computations to be parallelized on more than
one core in order to speed up the computation and free resources. In one embodiment, the
parallelization may comprise two or more instances for the determination of the multiples
of the curve point on two or more processors, for example, at a same time.
[0061] At 510, an output for the aggregated inversions is retrieved, for example, as a
slope value. As described above, an aggregated inversion may be used at respective levels
in a binary tree representation of the additive act in the pairings computations. Further, in
one embodiment, the stored aggregated inversions 556 may be reused in a subsequent
pairing computation of a set of coordinates in affine space. As an example, the aggregated
inversions can be retrieved from the remote or local database and reused. In this way, a
number of computations can be mitigated by reducing the inversion aggregation act.
[0062] In one embodiment, the aggregated inversion may be reused when a first curve
point that was used in determining the aggregated inversions is a same element as a second
curve point for which the aggregated inversions are reused. That is, the aggregated
inversions can be determined using a curve point submitted as an element in the pairings
computation. In this embodiment, if a second set of elements submitted after the first set
is computed, and the second set comprises an element that is the same as the element from
the first set, the aggregated inversions may be reused in computing the pairings for the
second set of elements, for example.
[0063] At 512, the line function is updated using the aggregated inversions output as a
slope value. In one embodiment, the output of the aggregated inversions may be used to
update a coefficient in the function of the line used for the pairing computation. At 514,
the pairings can be determined for the elements, for example, by evaluating the updated
line function, resulting in a mathematical pairing for the encrypted signature authorization
element 558, and a mathematical pairing for the secret elements 560.
[0064] At 516, the respective pairings 558, 560, can be compared to determine whether
they are equal, for example, to determine authenticity of the submitted electronic
signature. In this embodiment, 500, if the elements are found not to be equal, at 518, (or
they are not from the same group) the submitted signature is not authenticated, at 520. If
the elements are found to be equal, at 518, (and from the same group) the submitted
signature is authenticated, at 522. In this way, for example, the computation of the
pairings for elements can be used for cryptographic purposes, and the one or more
techniques described herein may be used to facilitate a more efficient and faster pairing
computation.
[0065] One or more systems may be devised for determining mathematical pairings for a
curve, in order to compare submitted elements for cryptographic purposes, for example.
Because the computation of pairings for use in cryptography can require a lot of
underlying multiplications and inversions in a finite field over which the elliptic curve is
defined , the one or more systems described herein can be devised to mitigate the time and
resources used to compute these pairings. Fig. 6 is a component block diagram illustrating
an exemplary system 600 for determining mathematical pairings for a curve for use in
cryptography.
[0066] The exemplary system 600 comprises an inversion aggregation component 602,
which aggregates inversions that are used to determine the mathematical pairings for the
curve. The inversion aggregation component 602 is operably coupled with one or more
programmed processors 650, which reside in one or more computing devices, and with a
data storage component 654 that can store one or more of the aggregated inversions 656.
Further, in the exemplary system 600, a mathematical pairings determination component
604 is operably coupled with the data storage component 654, and can determine the
mathematical pairings for the curve in affine coordinates along a binary representation of a
scalar read from right to left using the aggregated inversions 656 stored thereon.
[0067] In one embodiment, the inversion aggregation component 602, pairings
determination component 604, and data storage component 654 may be comprised in a
same computing device, such as the computing device 652 that comprises the one or more
processors 650. Alternatively, the components of the system may be disposed on different
devices, or in some combination thereof.
[0068] In one embodiment, the inversion aggregation component 602 can be configured
to aggregate the plurality of inversions into a single inversion for use in the mathematical
pairing determination. For example, respective inversions for a level of a binary
representation of multiples of the curve point can be combined by the inversion
aggregation component 602 into a single inversion for that level. In this example, the
combined (aggregated inversion, e.g., 656) can be stored in the data storage component
654, and used by the pairings determination component 604 for computing pairings.
[0069] Fig. 7 is a component block diagram illustrating one embodiment 700 of an
exemplary implementation of the one or more systems described herein. A cryptographic
system 702, such as illustrated in Figs. 1 and 2 (e.g., 104, 222), may comprise a pairing on
an elliptic curve-based determiner 750, for example, which utilizes one or more
implementations of the systems described herein. Further, the cryptographic system 702
can comprise a group that is publically known (e.g., by knowing the curve) that utilizes the
cryptographic system (e.g., for authentication, security, encryption, etc.).
[0070] In this exemplary embodiment 700, an input component 704, such as a
component that can read an incoming document that uses cryptographic authentication,
receives a document 754 that comprises cryptographic elements 708 and a public key 706.
As an example, the document 754 may be an encrypted document that is being submitted
to a decryptor (e.g., in order to be read), the cryptographic elements 708 are points on a
curve (e.g., group elements comprised from the group if the document is authentic), and a
scalar used in the computation of the pairings. The public key is usually a point on the
curve, while the secret key is a scalar.
[0071] In this embodiment 700, the pairing on an elliptic curve-based determiner 750
can determine a pairing for the submitted cryptographic elements 708, and for elements
from the private key 710, to determine whether the submitted document is authentic, for
example. That is, if the document is authentic, for example, the cryptographic elements
708 will match to the same element as those from the private key 710, when pairings are
computed for each using the scalar from the public key 706. In this way, the
cryptographic system can output an authentication 752 for the document 754, for example,
in order for the document 754 to be decrypted for viewing.
[0072] Fig. 8 is a component block diagram illustrating another exemplary embodiment
800 of an implementation of the one or more systems described herein. In this
embodiment 800, the mathematical pairings determination component 504 is operably
coupled with a plurality of processors 802a-802n that can parallelize a determination of
addition relationships for multiples of the curve point 850 in affine coordinates along a
binary representation of a scalar read from right to left. That is, for example, various parts
of the pairings determination (e.g., the addition acts) can be run on several processors at a
same time in order to reduce an overall time for computing the pairings 852.
[0073] Further, in the exemplary embodiment 800, the mathematical pairings
determination component 504 can be configured to reuse the aggregated plurality of
inversions 656, such as where a first curve point that was used in determining the
aggregated plurality of inversions is a same element as a second curve point for which the
aggregated inversions are reused. In this embodiment, an inversion reuse determination
component 804 can determine whether the second curve point is the same element as the
first curve point, such as by comparing a stored version of the first curve point to a second
one received 850. Additionally, the inversion reuse determination component 804 can
retrieve the aggregated inversions 656 from the data storage component 654 that
correspond to the first curve point. In this way, the retrieved inversions 656 can be reused
by the pairing determination component 504, for example, instead of computing new
aggregated inversions.
[0074] Still another embodiment involves a computer-readable medium comprising
processor-executable instructions configured to implement one or more of the techniques
presented herein. An exemplary computer-readable medium that may be devised in these
ways is illustrated in Fig. 9, wherein the implementation 900 comprises a computerreadable
medium 908 (e.g., a CD-R, DVD-R, or a platter of a hard disk drive), on which is
encoded computer-readable data 906. This computer-readable data 906 in turn comprises
a set of computer instructions 904 configured to operate according to one or more of the
principles set forth herein. In one such embodiment 902, the processor-executable
instructions 904 may be configured to perform a method, such as the exemplary method
300 of Fig. 3, for example. In another such embodiment, the processor-executable
instructions 904 may be configured to implement a system, such as the exemplary system
600 of Fig. 6, for example. Many such computer-readable media may be devised by those
of ordinary skill in the art that are configured to operate in accordance with the techniques
presented herein.
[0075] Although the subject matter has been described in language specific to structural
features and/or methodological acts, it is to be understood that the subject matter defined
in the appended claims is not necessarily limited to the specific features or acts described
above. Rather, the specific features and acts described above are disclosed as example
forms of implementing the claims.
[0076] As used in this application, the terms "component," "module," "system,"
"interface," and the like are generally intended to refer to a computer-related entity, either
hardware, a combination of hardware and software, software, or software in execution.
For example, a component may be, but is not limited to being, a process running on a
processor, a processor, an object, an executable, a thread of execution, a program, and/or a
computer. By way of illustration, both an application running on a controller and the
controller can be a component. One or more components may reside within a process
and/or thread of execution and a component may be localized on one computer and/or
distributed between two or more computers.
[0077] Furthermore, the claimed subject matter may be implemented as a method,
apparatus, or article of manufacture using standard programming and/or engineering
techniques to produce software, firmware, hardware, or any combination thereof to control
a computer to implement the disclosed subject matter. The term "article of manufacture"
as used herein is intended to encompass a computer program accessible from any
computer-readable device, carrier, or media. Of course, those skilled in the art will
recognize many modifications may be made to this configuration without departing from
the scope or spirit of the claimed subject matter.
[0078] Fig. 10 and the following discussion provide a brief, general description of a
suitable computing environment to implement embodiments of one or more of the
provisions set forth herein. The operating environment of Fig. 10 is only one example of a
suitable operating environment and is not intended to suggest any limitation as to the
scope of use or functionality of the operating environment. Example computing devices
include, but are not limited to, personal computers, server computers, hand-held or laptop
devices, mobile devices (such as mobile phones, Personal Digital Assistants (PDAs),
media players, and the like), multiprocessor systems, consumer electronics, mini
computers, mainframe computers, distributed computing environments that include any of
the above systems or devices, and the like.
[0079] Although not required, embodiments are described in the general context of
"computer readable instructions" being executed by one or more computing devices.
Computer readable instructions may be distributed via computer readable media
(discussed below). Computer readable instructions may be implemented as program
modules, such as functions, objects, Application Programming Interfaces (APIs), data
structures, and the like, that perform particular tasks or implement particular abstract data
types. Typically, the functionality of the computer readable instructions may be combined
or distributed as desired in various environments.
[0080] Fig. 10 illustrates an example of a system 1000 comprising a computing device
1012 configured to implement one or more embodiments provided herein. In one
configuration, computing device 1012 includes at least one processing unit 1016 and
memory 1018. Depending on the exact configuration and type of computing device,
memory 1018 may be volatile (such as RAM, for example), non-volatile (such as ROM,
flash memory, etc., for example) or some combination of the two. This configuration is
illustrated in Fig. 10 by dashed line 1014.
[0081] In other embodiments, device 1012 may include additional features and/or
functionality. For example, device 1012 may also include additional storage (e.g.,
removable and/or non-removable) including, but not limited to, magnetic storage, optical
storage, and the like. Such additional storage is illustrated in Fig. 10 by storage 1020. In
one embodiment, computer readable instructions to implement one or more embodiments
provided herein may be in storage 1020. Storage 1020 may also store other computer
readable instructions to implement an operating system, an application program, and the
like. Computer readable instructions may be loaded in memory 1018 for execution by
processing unit 1016, for example.
[0082] The term "computer readable media" as used herein includes computer storage
media. Computer storage media includes volatile and nonvolatile, removable and non
removable media implemented in any method or technology for storage of information
such as computer readable instructions or other data. Memory 1018 and storage 1020 are
examples of computer storage media. Computer storage media includes, but is not limited
to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, Digital
Versatile Disks (DVDs) or other optical storage, magnetic cassettes, magnetic tape,
magnetic disk storage or other magnetic storage devices, or any other medium which can
be used to store the desired information and which can be accessed by device 1012. Any
such computer storage media may be part of device 1012.
[0083] Device 1012 may also include communication connection(s) 1026 that allows
device 1012 to communicate with other devices. Communication connection(s) 1026 may
include, but is not limited to, a modem, a Network Interface Card (NIC), an integrated
network interface, a radio frequency transmitter/receiver, an infrared port, a USB
connection, or other interfaces for connecting computing device 1012 to other computing
devices. Communication connection(s) 1026 may include a wired connection or a
wireless connection. Communication connection(s) 1026 may transmit and/or receive
communication media.
[0084] The term "computer readable media" may include communication media.
Communication media typically embodies computer readable instructions or other data in
a "modulated data signal" such as a carrier wave or other transport mechanism and
includes any information delivery media. The term "modulated data signal" may include a
signal that has one or more of its characteristics set or changed in such a manner as to
encode information in the signal.
[0085] Device 1012 may include input device(s) 1024 such as keyboard, mouse, pen,
voice input device, touch input device, infrared cameras, video input devices, and/or any
other input device. Output device(s) 1022 such as one or more displays, speakers,
printers, and/or any other output device may also be included in device 1012. Input
device(s) 1024 and output device(s) 1022 may be connected to device 1012 via a wired
connection, wireless connection, or any combination thereof. In one embodiment, an
input device or an output device from another computing device may be used as input
device(s) 1024 or output device(s) 1022 for computing device 1012.
[0086] Components of computing device 1012 may be connected by various
interconnects, such as a bus. Such interconnects may include a Peripheral Component
Interconnect (PCI), such as PCI Express, a Universal Serial Bus (USB), firewire (IEEE
1394), an optical bus structure, and the like. In another embodiment, components of
computing device 1012 may be interconnected by a network. For example, memory 1018
may be comprised of multiple physical memory units located in different physical
locations interconnected by a network.
[0087] Those skilled in the art will realize that storage devices utilized to store computer
readable instructions may be distributed across a network. For example, a computing
device 1030 accessible via network 1028 may store computer readable instructions to
implement one or more embodiments provided herein. Computing device 1012 may
access computing device 1030 and download a part or all of the computer readable
instructions for execution. Alternatively, computing device 1012 may download pieces of
the computer readable instructions, as needed, or some instructions may be executed at
computing device 1012 and some at computing device 1030.
[0088] Various operations of embodiments are provided herein. In one embodiment,
one or more of the operations described may constitute computer readable instructions
stored on one or more computer readable media, which if executed by a computing device,
will cause the computing device to perform the operations described. The order in which
some or all of the operations are described should not be construed as to imply that these
operations are necessarily order dependent. Alternative ordering will be appreciated by
one skilled in the art having the benefit of this description. Further, it will be understood
that not all operations are necessarily present in each embodiment provided herein.
[0089] Moreover, the word "exemplary" is used herein to mean serving as an example,
instance, or illustration. Any aspect or design described herein as "exemplary" is not
necessarily to be construed as advantageous over other aspects or designs. Rather, use of
the word "exemplary" is intended to present concepts in a concrete fashion. As used in
this application, the term "or" is intended to mean an inclusive "or" rather than an
exclusive "or." That is, unless specified otherwise, or clear from context, "X employs A
or B" is intended to mean any of the natural inclusive permutations. That is, if X employs
A; X employs B; or X employs both A and B, then "X employs A or B" is satisfied under
any of the foregoing instances. In addition, the articles "a" and "an" as used in this
application and the appended claims may generally be construed to mean "one or more"
unless specified otherwise or clear from context to be directed to a singular form.
[0090] Also, although the disclosure has been shown and described with respect to one
or more implementations, equivalent alterations and modifications will occur to others
skilled in the art based upon a reading and understanding of this specification and the
annexed drawings. The disclosure includes all such modifications and alterations and is
limited only by the scope of the following claims. In particular regard to the various
functions performed by the above described components (e.g., elements, resources, etc.),
the terms used to describe such components are intended to correspond, unless otherwise
indicated, to any component which performs the specified function of the described
component (e.g., that is functionally equivalent), even though not structurally equivalent to
the disclosed structure which performs the function in the herein illustrated exemplary
implementations of the disclosure. In addition, while a particular feature of the disclosure
may have been disclosed with respect to only one of several implementations, such feature
may be combined with one or more other features of the other implementations as may be
desired and advantageous for any given or particular application. Furthermore, to the
extent that the terms "includes," "having," "has," "with," or variants thereof are used in
either the detailed description or the claims, such terms are intended to be inclusive in a
manner similar to the term "comprising."
CLAIMS
1. A computer-based method for determining mathematical pairings for a curve for
use in cryptography, comprising:
aggregating a plurality of inversions used in determining the mathematical pairings
for the curve using one or more micro-processors; and
determining the mathematical pairings for the curve in affine coordinates along a
binary representation of a scalar read from right to left using the aggregated plurality of
inversions.
2. The method of claim , aggregating a plurality of inversions comprising
aggregating the plurality of inversions into a single inversion for mathematical pairing
determination.
3. The method of claim , comprising reusing the aggregated plurality of inversions
for one or more pairing calculations when determining the mathematical pairing for the
curve.
4. The method of claim 1, comprising parallelizing the determination of two or more
mathematical pairings for the curve in affine coordinates along a binary representation of a
scalar read from right to left on a plurality of processors.
5. The method of claim 1, the output of the aggregated inversions comprising a slope
value for the updating of a line function in the pairing determination.
6. The method of claim 1, comprising using the aggregated plurality inversions as one
or more slope values for updating a line function in the pairing determination.
7. The method of claim 1, comprising using an output of the aggregated inversions to
update a coefficient of a line used to determine the mathematical pairing.
8. The method of claim 1, comprising reading a binary representation of the scalar for
the curve from right to left.
9. The method of claim 1, the curve point comprising an element of a group used for
a cryptographic application.
10. The method of claim 1, determining the mathematical pairings for the curve in
affine coordinates along a binary representation of a scalar read from right to left using the
aggregated plurality of inversions comprising:
determining a multiple of a curve point by a scalar using a binary representation of
the scalar read from right to left;
determining an inversion of an aggregation of additions; and
using an output of the aggregated inversions to update a function of a line used to
determine the mathematical pairing.
11. A system for determining mathematical pairings for a curve for use in
cryptography, comprising:
an inversion aggregation component, operably coupled with one or more
programmed processors disposed in one or more computing devices, and configured to
aggregate a plurality of inversions used in determining the mathematical pairings for the
curve, and operably coupled with a data storage component configured to store one or
more of the aggregated inversions; and
a mathematical pairings determination component, operably coupled with the data
storage component, and configured to determine the mathematical pairings for the curve in
affine coordinates along a binary representation of a scalar read from right to left using the
aggregated plurality of inversions.
12. The system of claim 11, comprised in a cryptographic system comprising:
an input receiving component configured to receive at least two elements;
a cryptographic key comprising the scalar; and
configured to compare results of a first mathematical pairing on an elliptic curve
for the received elements with a second mathematical pairing for at least two points on the
elliptic curve, using the scalar.
13. The system of claim 11, the inversion aggregation component configured to
aggregate the plurality of inversions for the mathematical pairing determination into a
single inversion for use in the mathematical pairing determination for two points on the
curve.
14. The system of claim 11, the mathematical pairings determination component
operably coupled with a plurality of processors configured to parallelize a determination
of addition relationships for multiples of the curve point in affine coordinates along a
binary representation of a scalar read from right to left.
15. The system of claim 11, the mathematical pairings determination component
configured to reuse the aggregated plurality of inversions if a first curve point that was
used in determining the aggregated plurality of inversions is a same element as a second
curve point for which the aggregated inversions are reused.