Acceleration of key agreement protocols

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WO 2010/069063 PCT/CA2009/001846

ACCELERATION OF KEY AGREEMENT PROTOCOLS

TECHNICAL FIELD:

[0001] The following relates to cryptographic systems, and more particularly to a method and apparatus for computing a shared secret key.

DESCRIPTION OF THE PRIOR ART

[0002] Public key cryptography is used to provide security for information transmitted over public networks. Numerous cryptographic protocols are available to provide security, integrity and authentication. Their security is based on the apparent intractability of certain mathematical problems, such as integer factorization and the discrete logarithm problem. These usually have limited computing power and battery power available. Public key schemes sometimes require more computing power than is available. In most cryptographic systems, parameters with a larger number of bits provide greater security but usually incur a penalty of a decrease in speed. In such environments, elliptic curve cryptography (ECC) is particularly appealing since it provides the requisite level of security with parameters having a smaller number of bits relative to other cryptographic systems, such as RSA. Computations are correspondingly faster because of the smaller amount of data that must be manipulated. Security is a requirement in constrained environments, such as mobile computing devices, including cellular phones, pagers, and smart cards. Given the continuing demand for enhanced security, there is a continual need to optimize cryptographic operations to run as quickly as possible and thereby make higher security implementations of existing protocols feasible.

[0003] Digital signatures are a class of cryptographic protocols used to provide authentication. As in all public key systems, a sender has a private key and a corresponding public key that are related to one another mathematically. The public key is made available and authenticated to other users through a certificate or a directory. The sender signs a message using their private key, and a recipient is able to verify the signature by using the authentic public key. Such systems are based on a “hard” mathematical problem, such as the factoring of numbers produced by products of large primes or the difficulty of taking logs in a finite field. The difficulty of the underlying problem provides assurance that only the owner of the private key could generate a signature that will verify using the owner’s public key.

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[0004] The users are computing devices that exchange information over a data communication network and are generally referred to as correspondents. The correspondents are colloquially referred to by personal names Alice, Bob, etc. for ease of description.

[0005] It is often of interest to share a secret key between two users of a public key cryptosystem that may then be used in a symmetric key encryption scheme to exchange encrypted data. Symmetric key protocols are faster at encrypting data than public key protocols, but establishment of a shared secret key between the correspondents is challenging. Public key cryptosystems may be used to establish a shared key so that a pair of correspondents’ may establish a common secret key. This secret key can be used to secure future communications using a symmetric key scheme. One class of key establishment protocols is a key transport protocol, where a single initiator creates the secret key and securely transfers it to the recipients. Another class of key establishment protocol is a key agreement protocol, where all participating entities contribute information which is used to derive the shared secret key.

BRIEF DESCRIPTION OF THE DRAWINGS

[0006] Embodiments will now be described by way of example only with reference to the appended drawings wherein:

[0007] Figure 1 is a schematic representation of a cryptographic system;

[0008] Figure 2 is a flowchart showing a known method of key agreement using the

ECMQV protocol;

[0009] Figure 3 is a flowchart showing details of an embedded key validation step from Figure 2;

[0010] Figure 4 is a flowchart showing an embodiment of an accelerated key agreement according to ECMQV ;

[0011] Figure 5 is a flowchart showing details of the accelerated computation of the shared secret key from Figure 4 performed by one correspondent;

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[0012] Figure 6 is a flowchart showing details of the accelerated computation of the shared secret key from Figure 4 performed by one correspondent;

[0013] Figure 7 is a flowchart showing details of the accelerated computation of the shared secret key from Figure 4 performed by another correspondent;

[0014] Figure 8a is a flowchart showing a known method of simultaneous scalar multiplication using ECMQV protocols;

[0015] Figure 8b is a flowchart showing an accelerated method of simultaneous scalar multiplication using the techniques of Figure 6 and Figure 7;

[0016] Figure 9 is a flowchart showing a known method of key agreement using

MQV protocols over multiplicative groups;

[0017] Figure 10 is a flowchart showing an embodiment of accelerated key agreement using MQV protocols over multiplicative groups;

[0018] Figure 1 la is a flowchart showing a method of simultaneous exponentiation from Figure 9;

[0019] Figure 1 lb is a flowchart showing a method of accelerated simultaneous exponentiation from Figure 10;

[0020] Figure 12 is a flowchart showing an alternative embodiment of accelerated key agreement using the MQV protocol with parallel architecture;

[0021] Figure 13 is a flowchart showing an alternative embodiment of accelerated key agreement using an MQV protocol with single key transfer;

[0022] Figure 14 is a flowchart showing an alternative embodiment of accelerated key agreement using MQV with parallel architecture and single key transfer.

[0023] Figure 15 is a flowchart showing an another alternative embodiment of accelerated key agreement using an MQV protocol with single key transfer;

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PCT/CA2009/001846 (0024] Figure 16 is a flowchart showing an another alternative embodiment of accelerated key agreement using MQV with parallel architecture and single key transfer.

[0025] Figure 17 is a flowchart showing an alternative embodiment of MTI protocol.

DETAILED DESCRIPTION OF THE INVENTION

[0026] The MQV (Menezes, Qu, Vanstone) protocol belongs to the family of key agreement protocols, which provides a method of sharing a key between two users of a public key cryptosystem and that provides authentication of the key. This protocol is described in U.S. Pat. No. 5,761,305, U.S. Pat. No. 5,889,865, U.S. Pat. No. 5,896,455, and U.S. Pat. No. 6,122,736.

[0027] It has a number of variants, some of which are standardized, and therefore is used as a basis for the following description. The MQV key agreement protocol may be implemented in both multiplicative groups defined over a finite field, or using additive groups, such as elliptic curve groups. The general algorithm remains similar in both cases. Notation for MQV is given in the table below.

<td> Term</td><td> Meaning</td>

<td> k_Â</td><td> Alice’s ephemeral private key</td>

<td> k_B</td><td> Bob’s ephemeral private key</td>

<td> Ra</td><td> Alice’s ephemeral public key</td>

<td> Rb</td><td> Bob’s ephemeral public key</td>

<td> d_A</td><td> Alice’s long-term private key</td>

<td> d_B</td><td> Bob’s long-term private key</td>

<td> Qa</td><td> Alice’s long-term public key</td>

<td> Qb</td><td> Bob’s long-term public key</td>

<td> sa</td><td> An intermediate component of the key computed by Alice</td>

<td> sb</td><td> An intermediate component of the key computed by Bob</td>

[0028] A multiplicative group version of the MQV protocol is described below. Specifically, a pair of correspondents Alice and Bob use a two-pass MQV variant to share a key, denoted as K.

1. Alice selects k^ at random from the interval 1 to (#-1), where q is the order of the group.

2. Alice computes R_A - and sends it to Bob, where g is the group generator.

3. Bob selects k_B at random from the interval 1 to (//-1).

4. Bob computes R_B = g^ke and sends it to Alice.

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5. Alice computes s_A = (k_A + d_AR_A)modq and the shared secret key becomes

K = (R_B(Q_B)“°)/

6. Bob computes s_B = (k_B + d_BR_B )modq and the shared secret key becomes

K = (R_a(Q_a77[0029] The computationally intense parts of the key agreement protocol are the exponentiations that are performed in determining K.

[0030] When the MQV protocol was standardized in the ANSI X9.62 and IEEE

Pl363 standards, a truncation operation was introduced to make the protocol more efficient. The MQV protocol as standardized uses a truncation operation to reduce the bit length of an exponent, and is denoted by X .

[0031] The use of the truncation operation speeds up computations since the exponent is shorter and it is believed that this truncation does not affect the security of the protocol.

[0032] Another version of the MQV protocol is the Elliptic Curve MQV (ECMQV) protocol using the three-pass variant with key confirmation, which is described below.

1. Alice selects at random from the interval 1 to (7-I) , where q is the order of the group.

2. Alice computes R_A = k_AP and sends R_A and Q_A to Bob, where P is a point on the curve that generates the elements of the elliptic curve group.

3. Bob performs embedded key validation of R_A to ensure R_A is a point not located at infinity, is an element of the group, and is a point that lies on the elliptic curve.

4. Bob selects at random from the interval 1 to (g-1 ).

5. Bob computes R_B =k_BP and s_B - (k_g + d_B Rn)modq and

K = hs_B (R_a + RaQ_a), where h is the cofactor of the group.

6. K is a point having coordinates (x,y) and Bob performs a Key Derivation Function (KDF) using the x-coordinate of K as its input to derive two ancillary keys, k\ and k^. This is denoted as (k\, U)^KDF(xQ.

7. Bob computes tag t_B using Message Authentication Code (MAC). This is

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PCT/CA2009/001846 denoted as t_B = MAC^ (2,Q_B,Q_A,R_B,R_A).

8. Bob sends t_B,Q_B,R_B to Alice.

9. Alice performs embedded public key validation of R_B to ensure R_B is a point not located at infinite, is an element of the group, and is a point that lies on the elliptic curve.

10. Alice computes s_A -(k_A+ d7? λ ) mod ¢7 and K = hs_A(R_B + RbQ_b), where h is the cofactor of the group.

11. Alice computes (k\, kf) as described at step 6 above.

12. Alice computes t = MAC_k> (2, Q_B,Q_A,R_B,R_A) and verifies t = t_B.

13. Alice computes t_A = MAC_ki (3,Q_B, Q_A, R_B, R_A ) and sends t_A to Bob.

14. Bob computes t = MAC_k (3, Q_B, Q_A, R_B, R_A ) and verifies t = t_A.

15. The agreed session key then becomes fe.

[0033] Similar to the multiplicative group implementation described above, the

ECMQV protocol using additive groups is most computationally intensive when implementing scalar multiplications, also referred to as point-multiplications, to determine K.

[0034] There are several methods used to increase the efficiency when calculating K, in both the exponentiation and scalar multiplication cases. One method for example, increases algorithm efficiency by organizing the correspondents, Alice and Bob, to compute their respective K values in parallel. This architectural change however, does not reduce the computational intensity.

[0035] Referring now to Figure 1, a cryptographic system 10 generally comprises a first correspondent 12 referred to as “Alice”, that communicates with a second correspondent 14 referred to as “Bob”, over a communication channel 16. Each correspondent 12,14 is a computing device, such as a computer, server, cell phone, PDA, ATM or the like, that includes processors, memory, power supplies, input and output devices necessary to perform the designated functions. Each correspondent 12,14 includes a cryptographic module 18, which either has its own memory 20 for storing inputs, outputs and intermediate portions of cryptographic operations, or has access to an external memory 20 that is part of the

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WO 2010/069063 PCT/CA2009/001846 correspondent (12, 14). It can be seen that in the example shown in Figure 1, the first correspondent 12 includes a memory 20 external to the cryptographic module 18 and the second correspondent 14 includes memory 20 that is internal to the cryptographic module 18 to illustrate that the capability to store data can be provided in any suitable arrangement. It will also be appreciated that the memory 20 can be external to and accessible (e.g. via a network connection etc.) to the correspondent 12, 14 if necessary or desired.

[0036] The cryptographic module 18 is configured to perform cryptographic operations such as encryption/decryption, signing and modular arithmetic, etc. Typically, the cryptographic module 18 includes a random number generator 22, secure storage 30 to store private keys and an arithmetic logic unit ALU 28 to perform cryptographic operations.

[0037] It will be appreciated that the memory 30 may comprise all or part of the memory 20 (shown in Figure 1) or may be provided as a separate component in the cryptographic module 18 as shown. The memory 30 may include random access memory (RAM), read only memory (ROM) and/or any other type of suitable memory structure.

[0038] Computational operations of the ALU 28 are controlled via hard-wired or programmed instructions residing in or accessible to a controller 24 and communicated to the ALU 28 via an instruction bus 35. A memory bus 36 is also provided to enable the controller 24 to utilize memory 30 when performing and outputting results of the operations of the ALU 28. The ALU 28 may be used to perform a variety of operations under the direction of controller 24.

[0039] It will be appreciated that any module or component exemplified herein that executes instructions may include or otherwise have access to computer readable media such as storage media, computer storage media, or data storage devices (removable and/or nonremovable) such as, for example, magnetic disks, optical disks, or tape. Computer storage media may include volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information, such as computer readable instructions, data structures, program modules, or other data. Examples of computer storage media include RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium

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WO 2010/069063 PCT/CA2009/001846 which can be used to store the desired information and which can be accessed by an application, module, or both. Any such computer storage media may be part of the controller 24 or accessible or connectable thereto. Any application or module herein described may be implemented using computer readable/executable instructions that may be stored or otherwise held by such computer readable media.

[0040] The cryptographic module also includes a comparator 39 to compare data stored in memory 20.

[0041] The correspondents 12, 14, each share parameters of an elliptic curve cryptosystem with an elliptic curve group E defined over finite field Fp. The parameters, which include a base point P, are stored in the memory 30 and accessible to the controller 24. Similar operations can implement elliptic curves over extension fields or higher genus curves, such as hyperelliptic curves.

[0042] Each correspondent 12,14 has a long-term private key du respectively in the form of computer readable bit strings representing integers, and a corresponding longterm public key Qa, Qb, respectively. The public keys Qa, Qb are each a pair of elements in the underlying field that satisfy the equation of the elliptic curve. The pubic keys represent points on the elliptic curve with the elements representing the (x,y) coordinates of the points. Each correspondent 12,14 has access to an authentic copy of the other correspondent's longterm public key, either as a certificate provided from one correspondent to the other, and signed by a certification authority, (CA), or directly from a CA.

[0043] It is desired to share a secret key between the correspondents using a key agreement protocol that is exemplified as the MQV protocol. It has now been recognized that the exchange of information between the correspondents required by MQV protocol can be organized to provide efficient computations, especially given certain conditions for the keys.

[0044] Specifically, when a first correspondent knows that a short-term public key and long-term public key of the second correspondent are identical in a MQV key agreement exchange, the first correspondent can accelerate the computation of the MQV shared secret key using a single scalar multiplication operation or single exponentiation operation, instead of two operations or a linear combination. The reduced number of intensive computations required to determine the shared secret key K accelerates the MQV protocol.

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[0045] The ephemeral public key and long-term public key of a correspondent may be identical either unintentionally or intentionally. There may be situations where it is suitable to intentionally set the ephemeral public key to be identical with the long-term public key. For example, there may be situations, such as e-mail, where it is not preferable or feasible for the recipient to send an ephemeral public key and therefore, the ephemeral public key of the recipient is made identical to his long-term public key. In yet another situation, an authenticated public key, also referred to as a certificate, may be required and yet unavailable. For example, a financial bank’s on-line server may operate using certificates, while the on-line banking client uses an Internet browser to securely connect to the server without a certificate. The client may not require a certificate to connect with the bank’s server because the client’s ephemeral key and long-term key are intentionally set to be identical.

[0046] A method of computing a shared secret key through the three-pass ECMQV protocol is shown in Figure 2 generally by numeral 200. The steps shown in Figure 2 represent the framework within which an acceleration of the ECMQV protocol may be obtained. Assuming there is no acceleration of ECMQV contemplated, Alice initiates the session and implements a set of instructions (210), in which she first randomly selects an integer as an ephemeral private key k_A between 1 to (&lt;7-1 ) (212), where q is the order of the group. She computes her ephemeral public key Ra as a point on the curve according to R_a = k_AP (214), where the point Ρ is a group generator. Alice then sends her long-term public key Qa and ephemeral public key Ra to Bob (216).

[0047] Bob then performs a set of instructions (220). He performs an embedded public key validation (300) on the received public key R_A, before randomly selecting his own ephemeral private key, kg, between 1 and (q-l) (224). Embedded key validation is shown in detail in Figure 3 by numeral 300. First, the public key, in this case Ra , is verified according to the condition that Ra is not the point at infinity, i.e. FfK (302). If this condition (302) is true, then the next condition (304) is verified, such that the Cartesian coordinates of Ra are properly represented elements within the finite field F_q. Finally, the coordinates of Ra are inserted in to the equation of the elliptic curve to see if Ra lies on the elliptic curve defined by parameters a and b (306). If all conditions are met the algorithm returns “valid” (308). If any of the conditions are not met, then the algorithm returns “invalid” (310).

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[0048] Returning back to FIG. 2, after Bob validates the key R_A (300) and randomly selects an ephemeral private key, k_B (224), he continues to compute the corresponding ephemeral public key, Rb, according to R_B = k_BP (226). This is used to compute intermediate value s_B = (k_B +d_BRg)modq (228). R_B is a truncated version of R_B in order to reduce the bit length, thereby increasing computing efficiency. The shared secret key, K, is dependent on(7?_z + RaQ_a) . As exemplified, the key K is computed according to

K = hs_B (R_a +RaQ_a) (230), where h is the cofactor of the group. The x-coordinate of K is used to derive two other secret keys k{ and through a suitable Key Derivation Function (KDF) (232). These secret keys ki, k? are used to compute authentication tags using a Message Authentication Code (MAC). Bob uses secret key k\ to compute authentication tag t_B according to t_B = MAC_k (2,Q_b,Q_a,R_b,R_a) (234). The string “2” is included in the MAC input to indicate that the tag originates from the responder. He then passes parameters Q_B, Rb, and t_B to Alice (236).

[0049] Upon receiving the parameters from Bob, Alice steps through another set of instructions (240). Alice performs embedded public key validation (300) on R_B and then computes intermediate value s_A = (k. + A_aRa)modq (244). Shared secret key Kis dependent onof (R_B + RbQ_b) , in this example according to K = hs_A(R_B + RbQ_b) (246). The same KDF is applied to the x-coordinate of K, i.e. x_K, such that KDF(xfe gives secret keys fe, fe (232). Alice computes authentication tag t = MAC_k (2,Q_B,Q_A,R_B,R_A) (250) and verifies t -t_B (252). If equality of the tags is verified, Alice computes a secondary authentication tag to send to Bob (256), where t_A - MAC_ki(3,Q_B,Q_A,R_B,R_A) (254). The string “3” is included in the MAC input to indicate that the tag originates from the initiator.

[0050] In a following set of instructions (260), Bob then computes a corresponding tag t = MAC_ki (3, Q_b , Q_a , R_B, R_A ) (262) and verifies that t = t_A (264). In both Alice’s and Bob’s devices, if condition t = t_A is true, then both secret keys ki and fe have been shared. Successful verification of the authentication tags t_A and t_B convinces each entity that the other entity has indeed computed the shared secret key K (since computing the tags requires knowledge of ki and therefore also of K), that the communications have not been tampered with (assuming that the MAC is secure), and that the other entity knows the identity of the

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WO 2010/069063 PCT/CA2009/001846 entity with which it is communicating (since the identities are included in the messages that are MACed). After verifying the above conditions, the ECMQV protocol returns the agreed secret session key k₂ (270) which was derived from the KDF. The session key k₂ may then be used for an exchange of information over the communication channel 16, using a symmetric encryption under the k₂.

[0051] At any step within the ECMQV protocol, if a verification process does not return “true”, the protocol will be terminated.

[0052] The most computationally intensive aspect of the ECMQV protocol described above is the scalar multiplication operation, more commonly referred to as pointmultiplication. The efficiency of a ECMQV protocol is measured using the number of scalar multiplications required to compute the shared secret key. In the protocol described in Figures 2 and 3, Alice performs scalar multiplication to compute K = hs_A(R_B + RbQ_b) (246), and Bob performs scalar multiplication to compute K = hs_B(R_A + RaQ_a) (230). Specifically, both Alice and Bob each perform 1.5 scalar multiplications to compute K in the above format. In the case of Alice, the ALU (28) computes K - hs_A(R_B + RbQ_b) (246). Within the bracket term (R_B + RbQ_b ) , the multiplication RbQ_b is performed, where Rb is an integer and Q_B is a point on the elliptic curve. Since Rb has been truncated and has a shorter bit length, the operation RbQ_b costs half of a scalar multiplication, or 0.5 scalar multiplications. The next scalar multiplication occurs when calculating the product between integer hs_A and elliptic curve point (R_B + RbQ_b), which costs 1 full scalar multiplication. Thus, calculating K costs a total of (1+0.5)=1.5 scalar multiplications.

[0053] To reduce the number of scalar multiplication, and thereby accelerate the key agreement protocol, the framework of Figure 3 is adapted as shown in FIG. 4 based on the recognition that scalar multiplication costs may be further reduced when the pair of public keys for a correspondent are equal, such that R_B = Q_B and/or R_A=Q_A. For Bob, the accelerated computation occurs when R_A = Q_A, which allows the shared secret key computation to simplify and become K = uQ,, where the simplified factor u is dependent on (1 + R a ). In the particular example, u = hs_B (1 + R 7 ). Similarly, for Alice, the accelerated

- 11 CA 02746830 2011-06-14 computation occurs when R_B=Q_B, which allows the shared secret key computation to simplify and become K = vQ_B, where the simplified factor v is dependent on (1 + Rb) . In this example, v = hs_A(\ +RO). A scalar multiplication cost of 1.0 is required to compute the shared secret key. Therefore the protocol described in Figure 4 can be implemented as much as 33% faster than that described in Figure 3.

[0054] Referring to FIG. 4, an embodiment of accelerating an ECMQV protocol by reducing the number of scalar multiplications is shown when it is identified that a pair of identical public keys is used by a correspondent. In this embodiment, both Alice and Bob intentionally set their respective ephemeral public key to be identical to their long-term public key. The accelerated algorithm begins with Alice performing a block of instructions (211), including randomly selecting an ephemeral private key A/ (212), intentionally selecting ephemeral public key Ra to be equal to long-term key Qa (215), and sending Qa and Ra to Bob (216).

[0055] Upon receiving the pair of public keys, Bob determines whether the keys are identical. As shown in Figure 4, Bob begins to perform a series of instructions (610) by first comparing Q_A and Ra (612). If Q_A does not equal Ra, Bob continues with the set of steps 221. Referring to FIG. 5, the set 221 is shown in more detail. Although set 221 is similar to the series of steps explained previously in the set 220, set 221 does not compute Rp using the ephemeral private key kp. Instead, in the set 221, Bob intentionally selects Rp to equal Qp, as per step 227.

[0056] However, if Q_A and Ra are identified as equal in the decision process (612) in

FIG. 4, then Bob employs an accelerated scalar multiplication format (701). Referring to FIG. 6, first Bob performs embedded public key validation (300) of R_A, randomly selects kp between 1 to (g-1) (712), and then intentionally selects Rp to equal Q_B (715). The intermediate component sp (716), followed by the simplified factor u = fo_g(l + Ra) (718), are computed. This is used to compute the shared secret key K = uQ_A using the simplified scalar multiplication format (720). Two keys, k\ and A₂, are then derived using a KDF(x^) (722). Bob then computes the authentication tag t_B using a MAC algorithm (724). He then sends Op, Rp, and tp to Alice (726).

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[0057] It is appreciated that Bob may carry out a public key comparison (612) if he is not certain that Alice may intentionally set her ephemeral public key Ra to equal her longterm public key Qa- If Bob has determined that Alice’s ephemeral and long-term public keys are identical, for example through some agreement established beforehand, then Bob may choose not to implement a public key comparison (612), thereby reducing the computational load. Therefore, upon receiving Alice’s pair of public keys, Bob may directly employ an accelerated scalar multiplication format (701). That is, step 216 may proceed directly to step 701 if it is understood beforehand that the public keys are identical. It is understood that Alice may also choose not to perform a comparison (616), given that Alice knows beforehand that Bob has intentionally set his ephemeral and long-term public keys to be identical. Similarly, from steps 236 or 726 the algorithm may proceed directly to step 800. In the specific example referred to in Figure 4, however, the determination is made by a comparison of both Alice and Bob.

[0058] Returning back to FIG. 4, upon receiving the pair of public keys from Bob,

Alice begins to perform a series of instructions (614) by first comparing Qb and Rb (616). If Qb does not equal Rb, Alice continues with the original set of steps 240, shown in detail in FIG. 2. However, if Qb and Rb are equal, then Alice employs an accelerated scalar multiplication format (800). Referring to FIG. 7, first Alice performs embedded public key validation (300) of Rb. The intermediate values Sa (812), followed by the simplified factor v = hs_A(1 + Rb) (814), are computed. This is used to compute the shared secret key K = vQ_b using the simplified scalar multiplication format (816). Two keys, k\ and fy, are then derived using a KDF(xfy (818). Alice then computes the authentication tag t using a MAC algorithm (820) and verifies that t=t_B (822). Alice computes îa (824) and sends it to Bob (826).

[0059] Referring back to FIG. 4, after Alice completes her algorithm, (240) or (800),

Bob performs a set of steps (260), in which he computes an authentication tag t in step 262 and verifies that t=tA (264). If all tags have been verified in the accelerated ECMQV process, then for both Alice and Bob the agreed session key is ki (270).

[0060] With regard to the embodiment described in FIG. 4, it is optional whether the ephemeral public keys (i.e. Ra and/or Rb) are intentionally selected to be identical to the

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WO 2010/069063 PCT/CA2009/001846 corresponding long-term public key. If not intentionally selected to be identical, then, for example, Alice’s ephemeral public key may be determined by RaHcaP. Similarly, Bob may choose not to intentionally select a pair of identical public keys and, thus, may compute R/j= kpP. However, it is noted that when a correspondent intentionally selects the ephemeral public key to be identical to the long-term public key, then the computation to obtain the ephemeral public key from the ephemeral private key is not required. Therefore, intentionally selecting identical public keys at a correspondent reduces the computational load in determining the ephemeral public key.

[0061] The condition of R_r=Q_r and/or R_A = Q_A may be detected during run-time and stored in advance in a fat certificate containing the long-term, public key Q and precomputed scalar multiples of Q. This pre-identified condition may be used to create a precomputed table of multiples of the public key 0,, which are combined to provide the scalar uQ_a . This will further accelerate the computation of the shared secret key K.

[0062] As may be appreciated from a consideration of Figure 4, the ability to determine whether or not the public keys are the same offers versatility in communication between the correspondents. For example, Alice may use two different public keys but Bob may use identical public keys if more convenient. For example, if Bob is a computing device with limited computing power he may choose not to perform the scalar multiplication associated with the ephemeral public key.

[0063] Alice will ascertain this upon receipt of the data from Bob and can expedite the computation of the shared key.

[0064] Similarly, Alice may choose to use identical public keys, allowing Bob to expedite computation of the shared key, but Bob may choose to use different public keys for communicating with Alice.

[0065] Finally, both Alice and Bob may each choose to use respective identical public keys and each will expedite computation of the shared key.

[0066] It is possible for Alice and Bob to use their long-term private key Ba, dp respectively, in place of the ephemeral private key kp in computing their interim value Sa,

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Sb- However, this would result in identical keys being used in successive sessions, which generally is considered undesirable.

[0067] In another preferred embodiment of the method, tables may be used to farther accelerate the MQV process. The cost savings gained by identifying pairs of identical public keys may be applied to the methods described for implementing simultaneous multiple pointmultiplication. As shown in FIG. 8a, a method conventionally used to speed the computation is to employ multiple point-multiplication, also referred to as Shamir’s trick. First, the original equation of the shared secret key (i.e. K - hs_A (R_B + RbQ_bY) must be rearranged to match the general linear combination form mP+lQ. The brackets are eliminated, giving K = hs_AR_B + hs_A RbQ_b, so that m= hs_A, P=R_B, l⁼ hs_A R b , and Q= Q_B in the representation of mP+lQ. As R_b and Q_B are both points, the expression for the key K is of the general form mP+lQ, that is the addition of two scalar multiples.

[0068] In FIG. 8a, a window width of a predetermined number of bits w is first established (902). Then a table (920) of small multiples of a of R_b is established (904) and a table (922) of small multiples of β of Q_B is established (906). The table entries consist of a column of possible bit combinations (e.g. a =10012), and a column of corresponding scalar multiples (e.g. 1OOI2 ). Then, the scalars hs_A and hs .Rn are examined (908) using windows of the window width w. The scalar multiples of R_B and Q_B corresponding to each window are retrieved from each respective table (910). The sum of the table entries from the two windows is added into an accumulator (912). The accumulator is then doubled w times in accordance with the width w of the window (914), and then the next window is examined (916). The scalars are repeatedly examined and table entries added into the accumulator and the accumulator doubled w times for each repetition as described above (924) until the shared secret Ais computed (918). This format (i.e. K = hs_AR_B + Rs_aRbQ_b) for simultaneous multiple point-multiplication provides a computational cost saving of 0.25 scalar multiplications over the computations shown in Figure 3, thereby requiring a total of 1.25 scalar multiplications.

[0069] The above simultaneous multiple point-multiplication method may be farther accelerated when a correspondent’s pair of public keys are identical. In the computation

- 15CA 02746830 2011-06-14

WO 2010/069063 PCT/CA2009/001846 shown in FIG. 8a, where different public keys are used, two tables (920, 922) are required to perform simultaneous multiplication. Referring to FIG. 8b, when the ephemeral public key and long-term public key identical, for example R_B⁼Q_B, only one table (942) is required by the correspondent to perform the window methods. Therefore, the accelerated simultaneous scalar multiplication (720, 816) begins by establishing a window width of w bits (926), then establishing a table of scalar multiples of R_B (928). The scalars hs_A and hs.Rn are examined using windows of the window width w (930). The scalar multiples of R_B that correspond to each window are retrieved (932) and stored into an accumulator (934). The accumulator is then doubled w times in accordance with the width w of the window (936), and then the next window is examined (938). The scalars are repeatedly examined and table entries added into the accumulator and the accumulator is doubled w times for each repetition as described above (944) until the shared secret K is computed (940).

[0070] It should be appreciated that the cost saving methods resulting from the use of identical public keys and applied to the tables used for simultaneous point-multiplication may also be applied to jointly constructed tables. Eliminating the need for constructing two separate tables or constructing a joint table for pairs of public keys R_A , Q_A and R_B , Q_Bconsequently reduces the run-time and may achieve cost savings considerably greater than 20%.

[0071] It should also be appreciated that the acceleration of the key agreement protocol using pairs of identical public keys is also applicable to implementations using multiplicative groups. The implementations described above utilise elliptic curve groups in which group operations are referred to by an additive notation. Other groups, such as the group Fp, which is the set of non-zero integers module p, use multiplicative notation. The additive notation in an elliptic curve setting is replaced by multiplicative notation in a multiplicative group setting.

[0072] Referring therefore to FIG. 9, by way of background, the method for generally computing a shared secret key K using MQV in a multiplicative group setting is shown by the numeral 1000. In a set of actions by Alice (1010), an ephemeral private key kA is selected at random from the interval 1 to (g-1) (1012), where q is the order of the group. Alice computes the corresponding ephemeral public key R_A = g^k4 (1014), where g is the group generator,

- 16CA 02746830 2011-06-14

WO 2010/069063 PCT/CA2009/001846 and sends R_A and long-term public key Q_A to Bob (1016). In a similar set of actions by Bob (1020), an ephemeral private key ke is selected at random from the interval 1 to (q-1) (1022). Bob computes the corresponding ephemeral public key R_B = g^ke (1024), and then sends RB and long-term public key QB to Alice (1026). Alice then uses these keys in set of actions (1030) to compute interim value sA = (k, +dAR//)modq (1032), followed by the exponentiation formula of the shared secret key K = (Rb(Qb)^Rb )^h'’ (1034), where h is the cofactor of the group. In numeral 1040, Bob computes interim value sB = (k_B + d_BR_B) mod q ( 1042), which is used to compute the exponentiation formula of the shared secret key K = (R_A (Q_A )^ (1044).

[0073] Referring to FIG. 10, an embodiment for accelerating the implementation of

Figure 9 for multiplicative groups is shown. Similar to the accelerated ECMQV described above, the number of intensive computations is reduced when the pair of public keys have the same value. In numeral 1012, Alice selects an ephemeral private key k_A at random from the interval 1 to (q-l). Alice intentionally selects the ephemeral public key R_A to equal longterm public key Q_a (1015), and sends R_A and Q_A to Bob (1016). Similarly in numeral 1022, Bob selects an ephemeral private key kg at random from the interval 1 to 0/-1). Bob intentionally selects the ephemeral public key R_B to equal the long-term public key Q_B(1025), and then sends R_B and Q_B to Alice (1026). At step 1110, Alice computes interim value s_A = (k_A + d,R A)modq (1112) and then compares R_B and Q_B (1114). If Bob’s pair of public keys are identical, such that R_B = Q_B, then Alice computes the accelerated exponentiation formula for the shared secret key by calculating the simplified exponent y = hs_ARB +hs_A (1118) or y - hs_A(l + R_B), i.e. y is dependent on (1 + 7? λ ), then calculating the exponent K = R_B^y (1120). Otherwise, if RBLQB, then Alice computes the non-accelerated exponentiation formula of the shared secret key K = (Rb(Qb)^Rs (1116). Similarly in numeral 1122, Bob computes interim value sB =(k_B +d_BRs)modq (1124) and compares R_A and Q_A (1126). If Alice’s pair of public keys are identical, such that R_A=Q_A, then Bob computes the accelerated exponentiation formula for the shared secret key by calculating the simplified exponent z = hs_BRA+hs_B (1130) or z = hs_B (1 + R _A ), then

- 17CA 02746830 2015-02-18 calculating the exponent K = Rf (1132). Otherwise, if R_AfQ_A> then Bob computes the noil-accelerated exponentiation formula of the shared secret key K. ~(R_A (QÀ)** f* (1128).

[0074] In the embodiment shown in FIG. 10, the comparison process (1114,1126) may be required if Alice and Bob are uncertain if the other will use identical public keys. In the case when. Alice andBob are certain that pairs of identical public keys may be employed, then the comparison process (1114,1126) may be avoided and the accelerated MQV protocol may be applied directly.

[0075] Another embodiment of the accelerated MQV method uses simultaneous

<img file="CA2746830C_D0001.tif" />

The formula K ~(R₃(fia)**)¹'¹ that is used to determine the key can be rearranged as

K = (R_B (Qb)*’ Y* = Rb¹' This rearrangement allows the key to be computed by using a technique known as simultaneous multiple exponentiation, which uses only one set of squares.

[0076] Referring to FIG. 11 a, a method of simultaneous multiple exponentiation, is shown for two public keys of non-equal value. A window width of a predetermined number of bits w is first established (1202). Then a table (1220) of small exponents a ofA^is established (1204) and atable (1222) of small exponents β ofQg is established (1206). The table comprises a column ofpossible bit combinations (e.g. a =100^) and a column of the values resulting from the corresponding exponentiations (e.g. A/⁰⁰’³ ). The exponents hsA and Rs _AR _B are then examined using windows of the window width w (1208). The results of the exponentiation of Rg and Qg by an exponent corresponding to each window are retrieved from each respective table (1210). The product of the table entries from the two windows is multiplied into an accumulator (1212). The value in the accumulator is then squared w times in accordance with the width w of the window (1214), after which the next window is examined (1216). The scalars are repeatedly examined, table entries multiplied and stored in the accumulator, and the accumulator squared w times for each repetition as described above (1224) until the shared secret AT is computed (1218).

[0077] Referring to FIG. 1 lb, a preferred embodiment of the accelerated MQV method eliminates one of the exponentiation tables (1222) if the pai r of public keys are

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PCT/CA2009/001846 identical. In the situation of pairs of identical public keys, referring briefly to FIG. 10, steps 1120 and 1132 may use a corresponding accelerated algorithm described herein. In FIG. 1 lb the accelerated simultaneous exponentiation begins by establishing a window width of w bits (1226), followed by establishing a table of the result of exponentiation of R_B by small exponents (1228, 1242). The exponents hs_A and hs_ARn are examined using windows of the window width w (1230). The powers of R_B that correspond to each window are retrieved (1232) and stored into an accumulator (1234). The accumulator is then squared w times in accordance with the width w of the window (1236), and then the next window is examined (1238). The scalars are repeatedly examined and table entries added into the accumulator and the accumulator squared w times for each repetition as described above (1244) until the shared secret K is computed (1240).

[0078] It should be appreciated that different architectures of the accelerated MQV algorithm may be formed, which includes, for example, organizing the timing of the computations in order for the correspondents to perform sections of the algorithm in parallel. Referring to FIG. 12, one alternative embodiment of the parallel computation is shown for the case of accelerated MQV over multiplicative groups. Both Alice and Bob carry out a set of initial steps (1310, 1320) in parallel. Alice, in numeral 1311, randomly selects ephemeral private key kA , selects ephemeral public key Ra to equal long-term public key Qa, and sends Ra and Qa to Bob. Simultaneously, as Alice performs step 1311, Bob performs step 1321, such that ke is generated, Rb is intentionally selected to equal Qb, and then R_B and Qb are sent to Alice.

[0079] In parallel, Bob and Alice both receive the other’s pair of identical public keys. Then Alice determines if Qb=R_b (1330) in parallel to Bob, who also determines if Qa=Ra (1340). For Alice, after determining equivalency (1330), if R_B/Q_B, then Alice computes a non-accelerated exponentiation formula (1350) by first computing interim value sa, then shared secret key K = (R_B (Q_b)^Rb )'^j. However, if R_B = Q_B, Alice computes an accelerated exponentiation formula (1370), by first computing sa, then y = hs_ARn + hs_A, and finally K = R_B . In parallel to Alice, after Bob determines equivalency (1340), if R_A/Q_A, then Bob computes a non-accelerated exponentiation formula (1360) by first computing

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WO 2010/069063

PCT/CA2009/001846 interim value Sb, then shared secret key K = (R_A (Q_a)^Ra )' . However, if R_A-Q_A, Bob computes an accelerated formula (1380), by first computing s_B, then simplified exponent z - hs_BRA + hs_B, and finally K-R_A'. It should further be appreciated that the parallel computational arrangement also applies to accelerated MQV algorithms over elliptic curve groups.

[0080] It is noted that the comparison process in steps 1330 and 1340 may not be necessary if Alice and Bob know that an accelerated algorithm using pairs of identical keys is being used. In other words, Alice does not need to verify if Bob’s keys pairs are identical (1330), if she already knows beforehand. Therefore, in the above case, from step 1320, the algorithm may proceed directly to step 1370. Similarly, from step 1310, the algorithm may proceed directly to step 1380.

[0081] In an alternative approach to the accelerated MQV protocols, a correspondent would be only required to send a single public key since the ephemeral and long-term public keys are identical. This would reduce the amount of data transferred between correspondents, allowing larger keys to be used.

[0082] Referring to FIG. 13, an embodiment of an accelerated MQV protocol over multiplicative groups that sends only a single public key is illustrated. It is understood that both Alice and Bob know that an accelerated algorithm is implemented using identical public keys. Alice begins by randomly selecting kA (1402) and intentionally selecting R,\ = Qa (1405). Alice then sends Qa to Bob (1411). This reduces the amount of data sent over the network compared to the data size of two public keys.

[0083] Continuing with FIG. 13, after the data from step 1411 has been sent by Alice,

Bob proceeds to randomly select k% (1412) and intentionally selects Rb = Qb (1415). Similarly, Bob then sends Q_B to Alice (1420). Alice then uses an accelerated MQV algorithm and calculates the intermediate component sa, simplified exponent y = hs_ARu + hs_A, and finally the shared secret key K = Rf (1426). It is appreciated that Rb = Qb- In an analogous fashion, Bob uses the received data from step 1411 to compute the accelerated MQV algorithm. Bob calculates intermediate component sb, simplified exponent z = Hs_bRa + hs_s, and finally the shared secret key K = R_A‘ (1432). It is noted that R_A = Qa- It should be

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WO 2010/069063 PCT/CA2009/001846 appreciated that this alternative approach described herein may also be applied over elliptic curve groups.

[0084] The data size savings may also be applied in a parallel architecture for accelerated MQV algorithms. In yet another embodiment of the accelerated MQV algorithm, referring to FIG. 14, the parallel architecture is illustrated. Similar to the previously described embodiment, it is understood that both Alice and Bob know that an accelerated algorithm is implemented using identical public keys. Alice performs a set of steps (1503) in which she selects / and intentionally selects Ra = Qa- In parallel, Bob performs a set of steps (1505) in which he selects kg and intentionally selects Rb = Qb- Then Alice sends Qa to Bob (1512), and Bob sends Qb to Alice (1516). Upon receiving the data, Alice computes shared secret key K using an accelerated algorithm (1524). In parallel, Bob computes shared secret key K using an accelerated algorithm (1528). It will be appreciated that the above single public key transfer for a parallel architecture in an accelerated MQV algorithm also applies to elliptic curve groups.

[0085] In an alternative approach to the accelerated MQV protocols, the verification of the correspondent’s identical pair of public keys can take place on the same ALU that has generated the ephemeral public key. For example, after Alice computes R,\. she then proceeds to verify if Ra=Qa- Benefits of this approach include the requirement of sending only a single public key, instead of two keys, to the correspondent if the pair of public keys are identical, which consequently reduces the required bandwidth of the network.

[0086] Referring to FIG. 15, an embodiment of an accelerated MQV protocol over multiplicative groups that sends only a single public key is illustrated. In this embodiment, each correspondent has the option of intentionally selecting the ephemeral public key to be identical with the long-term public key. Alice begins by randomly selecting kA (1402) and determining R_f\ (1404), either according to Ra =Qa or by computing R_A = g^k'. Alice then determines if Qa = Ra (1406). If Qa # Ra, then Alice sends Ra, Qa and Equal=iahe to Bob (1408); otherwise if Qa = Ra, then Alice sends Qa and Equal=tme to Bob (1410). The variable Equal may be a Boolean parameter of a small bit size, for example 1 bit, that indicates whether the pair of public keys are identical or not identical. In the situation when the pair of public keys are identical (i.e. Equaldme) only a single public key, such as the

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WO 2010/069063 PCT/CA2009/001846 long-term public key, and the small bit-size parameter Equal is sent to the correspondent. This reduces the amount of data sent over the network compared to the data size of two public keys.

[0087] Continuing with FIG. 15, after the data from step 1408 or step 1410 has been sent by Alice, Bob proceeds to randomly select k_B (1412) and determining R_A (1414), either according to R_B =Q_b or by computing R_b = g^ki‘. Similarly, Bob then determines if QB = RB (1416). If Qb RB, then Bob sends RB, QB and Equal=fafx to Alice (1418); otherwise if QB = Rg, then Bob sends Qb and Equal=iïü&amp; to Alice (1420). Alice then uses the received data from step 1418 or step 1420 to determine if Equal=true (1422). If Equal=false, Alice computes the shared key K using the non-accelerated MQV method (1424). if Equal=tnie. Alice uses an accelerated MQV algorithm and calculates the intermediate component sA, simplified exponent y = hsA Rb + hsA, and finally K - RB^y (1426). In an analogous fashion, Bob uses the received data from step 1408 or step 1410 to determine if Equal=true (1428). If Equal=fa\se, Bob computes the shared key K using the non-accelerated MQV method (1430). If Equal=true, Alice uses an accelerated MQV algorithm and calculates sB, interim value z = hs_B Ra + hs_B, and finally K = R_A‘ (1432). It should be appreciated that this alternative approach described herein may also be applied over elliptic curve groups.

[0088] The data-size savings, resulting from the computation of an ephemeral public key and the determination of the respective identical key pair status by the same correspondent, may also be applied in a parallel architecture for accelerated MQV algorithms. In yet another embodiment of the accelerated MQV algorithm, referring to FIG. 16, the parallel architecture is illustrated. Alice performs a set of steps (1502) in which she selects k_Aand determines R_A. Alice may intentionally set R_A =Q_A or compute R. = g^k·'. In parallel, Bob performs a set of steps (1504) in which he selects kB and determines RB. Bob may intentionally set RB =QB or compute R,, - g^kB. Then both Alice and Bob simultaneously determine if their respective pair of public keys are equal (1506, 1508). If QA Ψ Ry, Alice sends Ry, Q_A and Equal-fake to Bob (1510); otherwise if Q_A = Ry, then Alice sends Q_A and Equal-lrue to Bob (1512). Similarly and in parallel, if Q_B # R_B, then Bob sends R_B, Q_B and £&lt;7Mrt/=false to Alice (1418); otherwise if Qb = Rb, then Bob sends Qb and Equal=trae to Alice (1420). Upon receiving the data, both Alice and Bob determine in parallel if

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WO 2010/069063 PCT/CA2009/001846

Equal=tme (1518, 1520). For Alice, if E^waMalse, then she computes shared secret key K using a non-accelerated algorithm (1522). Otherwise, if Equal=Vme, she computes shared secret key K uses an accelerated algorithm (1524). In parallel, continuing from step 1520 for Bob, if EqualMa/e, he computes shared secret key K using a non-accelerated algorithm (1526). Otherwise, if Equal=\.mo, he computes shared secret key K using an accelerated algorithm (1528). It will be appreciated that the above single public key transfer for a parallel architecture in an accelerated MQV algorithm also applies to elliptic curve groups.

[0089] Each of the above examples indicates that the public keys are sent from one correspondent to the other. The long-term public keys may be obtained alternatively from a certificate authority, from directories or stored within the memory 30 from previous exchanges. Where only long-term public keys are used, therefore, the direct transfer of information between the correspondents may originate with the exchange of the MAC, tp.

[0090] It is appreciated that the acceleration based on identical public keys is not limited to MQV algorithms. Other algorithms comprising a linear combination of an ephemeral public key and a long-term public key may also be suitable.

[0091] One class of such protocols are the MTI protocols, as described at page 518 of the Handbook of Applied Cryptology, Menzes et al. ISBN 0-8493-8523-7. As shown in Figure 17, the MTI AO protocol may be implemented with Alice and Bob having respective long-term private/public keys a,Z.\ and 6,Zu and the long-term public keys Za, Zb are exchanged with or made available to the other correspondent. Alice generates an ephemeral private key x, and computes a corresponding ephemeral public key, a ^x, which is sent to Bob.

[0092] Bob similarly would normally generate an ephemeral private key y and send the ephemeral public key a ^y to Alice.

[0093] The shared key K is computed at Alice from ( a ^y)^a Zf and at Bob from ( a ^x)^b

ZZ

L

[0094] As shown in Figure 17, Bob uses the long-term keys b, a as short term keys so that y=b and a ^y=Z_li; and notifies Alice either by sending Zb to Alice who can compare the values, or by sending an indication in the form of message Equal=true, or by prior agreement, such as during exchange of the long-term public keys.

-23 CA 02746830 2015-02-18

[0095] Alice may the compute the shared key K= (Zb) æ** as Alice knows both a and x, and Bob computes ( a ^x Za)¹³, as he knows b. One exponentiation is performed at both Bob and Alice to reduce the overall computational load.

[0096] Although the invention has been described with reference to certain specific embodiments, various modifications thereof will be apparent to those skilled in the art. In addition although the various methods described are conveniently implemented in a general purpose computer selectively activated or reconfigured by software, one of ordinary skill in the art would also recognize that such embodiments may be carried out in hardware.

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