Key agreement and transport protocol with implicit signatures

0001The present invention relates generally to key agreement protocols for transfer and authentication of encryption keys.

BACKGROUND

0002To retain privacy during the exchange of information it is well known to encrypt data using a key. The key must be chosen so that the correspondents are able to encrypt and decrypt messages but such that an interceptor cannot determine the contents of the message.

0003In a secret key cryptographic protocol, the correspondents share a common key that is secret to them. This requires the key to be agreed upon between the correspondents and for provision to be made to maintain the secrecy of the key and provide for change of the key should the underlying security be compromised.

0004Public key cryptographic protocols were first proposed in 1976 by Diffie-Hellman and utilized a public key made available to all potential correspondents and a private key known only to the intended recipient. The public and private keys are related such that a message encrypted with the public key of a recipient can be readily decrypted with the private key but the private key cannot be derived from the knowledge of the plaintext, ciphertext and public key.

0005Key establishment is the process by which two (or more) parties establish a shared secret key, called the session key. The session key is subsequently used to achieve some cryptographic goal, such as privacy. There are two kinds of key agreement protocol; key transport protocols in which a key is created by one party and securely transmitted to the second party; and key agreement protocols, in which both parties contribute information which jointly establish the shared secret key. The number of message exchanges required between the parties is called the number of passes. A key establishment protocol is said to provide implicit key authentication (or simply key authentication) if one party is assured that no other party aside from a specially identified second party may learn the value of the session key. The property of implicit key authentication does not necessarily mean that the second party actually possesses the session key. A key establishment protocol is said to provide key confirmation if one party is assured that a specially identified second party actually has possession of a particular session key. If the authentication is provided to both parties involved in the protocol, then the key authentication is said to be mutual; if provided to only one party, the authentication is said to be unilateral.

0006There are various prior proposals which claim to provide implicit key authentication.

0007Examples include the Nyberg-Rueppel one-pass protocol and the Matsumoto-Takashima-Imai (MTI) and the Goss and Yacobi two-pass protocols for key agreement.

0008The prior proposals ensure that transmissions between correspondents to establish a common key are secure and that an interloper cannot retrieve the session key and decrypt the ciphertext. In this way security for sensitive transactions such as transfer of funds is provided.

0009For example, the MTI/AO key agreement protocol establishes a shared secret K, known to the two correspondents, in the following manner: <ol id="ol0001" compact="compact" ol-style=""><li>1. During initial, one-time setup, key generation and publication is undertaken by selecting and publishing an appropriate system prime p and generator <maths id="math0001" num=""><math display="inline"><mi>α</mi><mo>∈</mo><msubsup><mi>Z</mi><mi>p</mi><mo>*</mo></msubsup></math><img file="EP2315390B1_D0001.tif" /></maths> in a manner guaranteeing authenticity. Correspondent A selects as a long-term private key a random integer "a", 1 &lt;a&lt;p-1, and computes a long-term public key z_A = α^a mod p. B generates analogous keys b, z_B. A and B have access to authenticated copies of each other's long-term public key.</li><li>2. The protocol requires the exchange of the following messages. <maths id="math0002" num="(1)"><math display="block"><mi mathvariant="normal">A</mi><mo>→</mo><mi mathvariant="normal">B</mi><mo>:</mo><msup><mi mathvariant="normal">α</mi><mi mathvariant="normal">x</mi></msup><mspace width="1em" /><mi>mod p</mi></math><img file="EP2315390B1_D0002.tif" /></maths><maths id="math0003" num="(2)"><math display="block"><mi mathvariant="normal">A</mi><mo>←</mo><mi mathvariant="normal">B</mi><mo>:</mo><msup><mi mathvariant="normal">α</mi><mi mathvariant="normal">y</mi></msup><mspace width="1em" /><mi>mod p</mi></math><img file="EP2315390B1_D0003.tif" /></maths></li></ol>

0010The values of x and y remain secure during such transmission as it is impractical to determine the exponent even when the value of α and the exponentiation is known provided of course that p is chosen sufficiently large. <ul id="ul0001" list-style="none" compact="compact"><li>3. To implement the protocol the following steps are performed each time a shared key is required. <ol id="ol0002" compact="compact" ol-style=""><li>(a) A chooses a random integer x, I ≤x≤p-2, and sends B message (1) i.e. α^x mod p.</li><li>(b) B chooses a random integer y, 1≤y≤p≤2, and sends A message (2) i.e. α^y mod p.</li><li>(c) A computes the key K = (α^y)^az_B^x mod p</li><li>(d) B computes the key K = (α^x)^bz_A^Y mod p</li><li>(e) Both share the key K - α^bx+ay</li></ol></li></ul>

0011In order to compute the key K, A must use his secret key a and the random integer x, both of which are known only to him. Similarly, B must use her secret key b and random integer y to compute the session key K. Provided the secret keys a, b remain uncompromised, an interloper cannot generate a session key identical to the other correspondent. Accordingly, any ciphertext will not be decipherable by both correspondents.

0012As such this and related protocols have been considered satisfactory for key establishment and resistant to conventional eavesdropping or man-in-the middle attacks.

0013In some circumstances, it may be advantageous for an adversary to mislead one correspondent as to the true identity of the other correspondent.

0014In such an attack an active adversary or interloper E modifies messages exchanged between A and B, with the result that B believes that he shares a key K with E while A believes that she shares the same key K with B. Even though E does not learn the value of K the misinformation as to the identity of the correspondents may be useful.

0015A practical scenario where such an attack may be launched successfully is the following. Suppose that B is a bank branch and A is an account holder. Certificates are issued by the bank headquarters and within the certificate is the account information of the holder. Suppose that the protocol for electronic deposit of funds is to exchange a key with a bank branch via a mutually authenticated key agreement. Once B has authenticated the transmitting entity, encrypted funds are deposited to the account number in the certificate. If no further authentication is done in the encrypted deposit message (which might be the case to save bandwidth) then the deposit will be made to E's account.

0016"<nplcit id="ncit0001" npl-type="s"><text>On seeking Smart Public Key-Distribution Systems", Matsumoto et al, IEICE TRANSACTIONS (1976-1990) Vol.E69-E No.2 pp.99-106 20 February 1986</text></nplcit> relates to the problem of how to distribute the common keys. The paper discusses a sort of scheme (direct schemes) that smartly provides different keys in different communications. Such a property is not attained via the basic scheme for the public key distribution systems proposed by Diffie and Hellman. The paper shows that the recently introduced five direct schemes are classified into three sets (called sequences) of infinite schemes, and points out that there are some tight relations among the sequences. It is clarified which is the best in the three sequences by a systematic evaluation of the complexities for the normal update and for the deliberate forgery of the shared common keys.

0017<patcit id="pcit0001" dnum="EP0639907A"><text>EP 0639907</text></patcit> discloses a digital signature method based on the discrete logarithm problem is provided that allows message recovery. The message x is transformed according to the rule e = x g-r mod p, where r is a secret value generated by the signer. A value y is then calculated according to the rule y = r + s e mod q, where s is the signer's secret key. The signature of x consists of the pair (e,y). The verifier recovers the message x according the the rule x = gy ke e mod p, where k is the signer's public key. The validation of x can be based on some redundancy contained in x. Alternatively, a conventional verification equation can be constructed by using the signature method together with a hash function H. In addition, a key agreement method based on the signature method is provided which establishes with a single transmission pass a shared secret key K between two parties A and B in an authenticated fashion.

GENERAL

0018In one aspect the present invention may provide a protocol in which the above disadvantages are obviated or mitigated.

0019According to one aspect there may be provided a method of establishing a session key as set out in claim 1 of the accompanying claims.

0020According to another aspect there is provided a cryptographic unit as set out in claim 12.

0021According to another aspect there is provided a device as set out in claim 13.

BRIEF DESCRIPTION OF DRAWING

0022Embodiments of the invention will now be described by way of example only with reference to the accompanying drawings in which: <ul id="ul0002" list-style="none" compact="compact"><li><figref idref="f0001">Figure 1</figref> is a schematic representation of a data communication system.</li></ul>

DETAILED DESCRIPTION

0023Referring therefore to <figref idref="f0001">Figure 1</figref>, a pair of correspondents, 10,12, denoted as correspondent A and correspondent B, exchange information over a communication channel 14. A cryptographic unit 16,18 is interposed between each of the correspondents 10, 12 and the channel 14. A key 20 is associated with each of the cryptographic units 16,18 to convert plaintext carried between each unit 16,18 and its respective correspondent 10,12 into ciphertext carried on the channel 14.

0024In operation, a message generated by correspondent A, 10, is encrypted by the unit 16 with the key 20 and transmitted and transmitted as ciphertext over channel 14 to the unit 18.

0025The key 20 operates upon the ciphertext in the unit 18 to generate a plaintext message for the correspondent B, 12. Provided the keys 20 correspond, the message received by correspondent 12 will be that sent by the correspondent 10.

0026In order for the system shown in <figref idref="f0001">Figure 1</figref> to operate, it is necessary for the keys 20 to be identical and therefore a key agreement protocol is established that allows the transfer of information in a public manner to establish the identical keys. A number of protocols are available for such key generation and are variants of the Diffie-Hellman key exchange. Their purpose is for parties A and B to establish a secret session key K.

0027The system parameters for these protocols are a prime number p and a generator α of the multiplicative group Z*_p. Correspondent A has private key a and public key p_A = α^a. Correspondent B has private key b and public key p_B = α^b. In the protocol exemplified below, text_A refers to a string of information that identifies party A. If the other correspondent B possesses and authentic copy of correspondent A's public key, then text_A will contain A's public-key certificate, issued by a trusted center; correspondent B can use his authentic copy of the trusted center's public key to verify correspondent A's certificate, hence obtaining an authentic copy of correspondent A's public key.

0028In each example below it is assumed that, an interloper E wishes to have messages from A identified as having originated from E herself. To accomplish this, E selects a random integer e, 1 ≤e≤p-2, computes p_E=(p_A)^e=α^ae mod p, and gets this certified as her public key. E does not know the exponent ae, although she knows e. By substituting text_E for text_A, the correspondent B will assume that the message originates from E rather than A and use E's public key to generate the session key K. E also intercepts the message from B and uses his secret random integer e to modify its contents. A will then use that information to generate the same session key allowing A to communicate with B.

0029To avoid interloper E convincing B that he is communicating with E, the following protocol is adapted.

0030The purpose of the protocol is for parties A and B to establish a session key K. The protocols exemplified are role-symmetric and non-interactive.

0031The system parameters for this protocol are a prime number p and a generator α of the multiplicative group Z^*p. User A has private key a and public key p_A = α^a. User B has private key b and public key p_B = α^b.

First Protocol

0032<ol id="ol0003" compact="compact" ol-style=""><li>1. A picks a random integer x, 1≤x≤p-2, and computes r_A = α^x and a signature s_A = x - r_Aaα^a mod (p-1). A sends {r_A, s_A, text_A} to B.</li><li>2. B picks a random integer y, 1 ≤y≤p-2, and computes r_B α^y and a signature s_B = y - r_bbα^b mod (p-1). B sends {r_B, s_B, text_B} to A.</li><li>3. A computes <i>α^{s<sub2>B</sub2>}</i> (<i>p_B</i>)<i>^{r<sub2>B</sub2>α<sup2>b</sup2>}</i> and verifies that this is equal to r_B. A computes the session key K = (r_B)^x = α^Xy</li><li>4. B computes <i>α^{s<sub2>A</sub2>}</i>(<i>p_A</i>)<i>^{r<sub2>A</sub2>α<sup2>a</sup2>}</i> and verifies that this is equal to r_A. B computes the session key K = (r_A)^y = α ^Xy.</li></ol>

0033Should E replace text_A with text _E, B will compute α<i>^{s<sub2>A</sub2>}</i>(<i>p_E</i>)^{<i>r_A</i>α<i>^a</i>} which will not correspond with the transmitted value of r_A. B will thus be alerted to the interloper E and will proceed to initiate another session key.

0034One drawback of the first protocol is that it does not offer perfect forward secrecy. That is, if an adversary learns the long-term private key a of party A, then the adversary can deduce all of A's past session keys. The property of perfect forward secrecy can be achieved by modifying Protocol 1 in the following way.

Modified First Protocol

0035In step 1, A also sends α^{x<sub2>1</sub2>} to B, where x₁ is a second random integer generated by A Similarly, in step 2, B also sends α^{y<sub2>1</sub2>} to A, where y₁ is a random integer. A and B now compute the key <i>K</i>= α^xy⊕α^{x<sub2>1</sub2>y<sub2>1</sub2>}<i>.</i>

0036Another drawback of the first protocol is that if an adversary learns the private random integer x of A, then the adversary can deduce the long-term private key a of party A from the equation s_A = x - r_Aaα^a mod p-1. This drawback is primarily theoretical in nature since a well designed implementation of the protocol will prevent the private integers from being disclosed.

Second Protocol

0037A second protocol set out below addresses these two drawbacks. <ol id="ol0004" compact="compact" ol-style=""><li>1. A picks a random integer x, 1≤x≤p-2, and computes (<i>p_B</i>)^x, α^x and a signature <i>s_A</i> = <i>x</i> + <i>a</i>α^a(<i>p_B</i>)^x mod(<i>p-1</i>). A sends {α^x, s_A, text_A} to B.</li><li>2. B picks a random integer y, 1≤y≤p-2, and computes (<i>p_A</i>)<i>^y,</i> α^y and a signature <i>s_B</i> = <i>y</i> + <i>bα^b</i>(<i>p_A</i>)<i>^y</i> mod(<i>p</i>-1). B sends {α^y, s_B, text_B } to A.</li><li>3. A computes (α^y)^a and verifies that α<i>^{s<sub2>B</sub2>}</i> (<i>pB</i>)^{-α<sup2>c</sup2>α<sup2>ay</sup2>}<i>= α^y</i>. A then computes session key <i>K</i> = α<i>^ay</i>(<i>p_B)^x</i>.</li><li>4. B computes (α^x)^b and verifies that α<i>^{s<sub2>A</sub2>}</i> (<i>p_A)^{-α<sup2>a</sup2>α<sup2>bx</sup2>}</i> = <i>α^x.</i> A then computes session key <i>K</i> = α<i>^bx</i> (<i>p_A</i>)<i>^y</i>.</li></ol>

0038The second protocol improves upon the first protocol in the sense that it offers perfect forward secrecy. While it is still the case that disclosure of a private random integer x allows an adversary to learn the private key a, this will not be a problem in practice because A can destroy x as soon as she uses it in step 1 of the protocol.

0039If A does not have an authenticated copy of B's public key then B has to transmit a certified copy of his key to B at the beginning of the protocol. In this case, the second protocol is a three-pass protocol.

0040The quantity s_A serves as A's signature on the value α^x. This signature has the novel property that it can only be verified by party B. This idea can be generalized to all ElGamal-like signatures schemes.

0041The first and second protocols above can be modified to improve the bandwidth requirements and computational efficiency of the key agreement. The modified protocols are described below as Protocol 1' and Protocol 2'. In each case, A and B will share the common key α<i>^{s<sub2>A</sub2>s<sub2>B</sub2>}</i>.

Protocol 1'

0042<ol id="ol0005" compact="compact" ol-style=""><li>1. A picks a random integer x, 1≤x≤p-2, and computes r_A = α^a and s_A = x + r_Aaα^a mod (p-1). A sends {r_A, text_A to B.</li><li>2. B picks a random integer y, 1≤y≤p-2, and computes r_B = α^y and s_B = y + r_Bbα^b mod (p-1). B sends {r_B, text_B} to A.</li><li>3. A computes <i>K</i> = (<i>r_B</i>(<i>p_B</i>)^{<i>r_B</i>α<i>^b</i>})<i>^sA</i> which is equivalent to α<i>^{s<sub2>A</sub2>s<sub2>B</sub2>}.</i></li><li>4. B computes <i>K</i> = (<i>r_A</i>(<i>p_A</i>)^{<i>r_A</i>α<i>^a</i>})<i>^sB</i> which is also equivalent to α<i>^sAsB.</i></li></ol>

0043A and B thus share the common key but it will be noted that the signatures s_A and s_B need not be transmitted.

Protocol 2'

0044<ol id="ol0006" compact="compact" ol-style=""><li>1. A picks a random integer x, 1≤x≤p-2, and computes (p_B)^x, α^x and s_A = x + a(p_B)^x mod (p-1). A sends {α^x, text _A} to B.</li><li>2. B picks a random integer y, 1≤y≤p-2, and computes (p_A)^y, α^y and s_B = y + b(p_A)^y mod (p-1). B sends {α^y, text _B} to A.</li><li>3. A computes (α^y)^a and <i>K</i> = (α^y(<i>p</i>_B)^{α<sup2>b</sup2>α<sup2>ay</sup2>})^{s<sub2>A</sub2>}, i.e. α<i>^{s<sub2>A</sub2>s<sub2>B</sub2>}</i>.</li><li>4. B computes (α^x)^b and <i>K</i> = (α^x(<i>p</i>_A)^{α<sup2>a</sup2><i>α^bx</i>})^{s<sub2>B</sub2>} i.e. α<i>^{s<sub2>A</sub2>s<sub2>B</sub2>}</i>.</li></ol>

0045Again therefore the transmission of s_A and s_B is avoided.

0046A further protocol is available for parties A and B to establish a session key K.

Third Protocol

0047The system parameters for this protocol are a prime number p and a generator α for the multiplicative group Z*<i>p</i>. User A has private key a and public key p_A = α^a. User B has private key b and public key p_B = α^b. <ol id="ol0007" compact="compact" ol-style=""><li>1. A picks two random integers x, x₁, 1≤x,x₁&lt;p-2, and computes r_x1 = α^x1, r_A = α^x and (<i>r_A</i>)^{r<sub2>x1</sub2>}, then computes a signature s_A = xr_x1-(r_A)^rx1aα^a mod (p-1). A sends {r_A, s_A, α^x1, text_A} to B.</li><li>2. B picks two random integers y, y₁ 1≤y,y₁≤p-2, and computes r_y1 = α^y1, r_B= α^y and (<i>r_B</i>)<i>^{r<sub2>y1</sub2>}</i>, then computes a signature s_B= yr_y1 - (r_B)^ry1b mod (p-1). A sends {r_B, s_B, α^y1, text_B} to A.</li><li>3. A computes α^{s<sub2>B</sub2>}(<i>p_B</i>)^{(<i>r_B)r_y1</i>} and verifies that this is equal to (<i>r_B</i>)<i>^{r<sub2>y1</sub2>}</i>. A computes session key <i>K=</i>(α^<i>y</i>1)^<i>x</i>1= α<i>^x1,y1</i></li><li>4. B computes <i>a^sA(p_A)</i>^{(<i>r_A</i>)<i>rx1</i>} and verifies that this is equal to (<i>r_A</i>)<i>^{r<sub2>x1</sub2>}.</i> B computes session key K=(α<i>^x1</i>)^y1 = α^x1y1</li></ol>

0048In these protocols, (r_A, s_A) can be thought of as the signature of r_x1 with the property that only A can sign the message r_x1.

Key Transport Protocol

0049The protocols described above permit the establishment and authentication of a session key K. It is also desirable to establish a protocol in which permits A to transport a session key K to party B. Such a protocol is exemplified below. <ol id="ol0008" compact="compact" ol-style=""><li>1. A picks a random integer x, 1≤x≤p-2 and computes r_A = α^x and a signature s_A = x-r_Aaα^a mod (p-1). A computes session key K = (p_B)^x and sends {r_A, s_A, text_A} to B.</li><li>2. B computes <i>a^{s<sub2>A</sub2>} (p</i>_<i>A</i>)^{<i>r_A</i>α<sup2>a</sup2>} and verifies that this quantity is equal to r_A. B computes session key K = (r_A)^b.</li></ol>

Modified Key Transport Protocol

0050The above protocol may be modified to reduce the bandwidth by avoiding the need to transmit the signature s_A as follows: <ol id="ol0009" compact="compact" ol-style=""><li>1. A picks a random integer x, 1≤x≤p-2, and computes r_A = α^x and s_A = x - r_Aaα^a mod (p-1). A computes <i>K</i>=<i>(p_B)</i>^sA and sends (r_A, text _A) to B.</li><li>2. B computes <i>K</i> = (α<i>^x</i>(<i>p_A</i>)^{-<i>r_A</i>α})<i>^b</i>= α<i>^{bs<sub2>A</sub2>}</i></li></ol>

0051All one-pass key transport protocols have the following problem of replay. Suppose that a one-pass key transport protocol is used to transmit a session key K from A to B as well as some text encrypted with the session key K. Suppose that E records the transmission from A to B. If E can at a later time gain access to B's decryption machine (but not the internal contents of the machine, such as B's private key), then, by replaying the transmission to the machine, E can recover the original text. (In this scenario, E does not learn the session key K).

0052This replay attack can be foiled by usual methods, such as the use of timestamps. There are, however, some practical situations when B has limited computational resources, in which it is more suitable at the beginning of each session, for B to transmit a random bit string k to A. The session key that is used to encrypt the text is then k ⊕ K, i.e. k XOR'd with K.

0053The signing equation s_A = x - r_Aaα^a where r_a = a^x in protocol 1, and the key transportation protocols; r_A = α^xb in protocol 2, can be replaced with several variants. Some of them are: <maths id="math0004" num=""><math display="block"><msub><mi mathvariant="normal">r</mi><mi mathvariant="normal">A</mi></msub><mo>=</mo><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo>⁢</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">z</mi></math><img file="EP2315390B1_D0004.tif" /></maths><maths id="math0005" num=""><math display="block"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo>=</mo><msup><mi mathvariant="normal">xα</mi><mi mathvariant="normal">a</mi></msup><mo>+</mo><msub><mi>ar</mi><mi mathvariant="normal">A</mi></msub></math><img file="EP2315390B1_D0005.tif" /></maths><maths id="math0006" num=""><math display="block"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo>=</mo><msub><mi>xr</mi><mi mathvariant="normal">A</mi></msub><mo>+</mo><msup><mi mathvariant="normal">aα</mi><mi mathvariant="normal">a</mi></msup></math><img file="EP2315390B1_D0006.tif" /></maths><maths id="math0007" num=""><math display="block"><mi mathvariant="normal">l</mi><mo>=</mo><msub><mi>ar</mi><mi mathvariant="normal">A</mi></msub><mo>+</mo><msub><mi>xs</mi><mi mathvariant="normal">a</mi></msub></math><img file="EP2315390B1_D0007.tif" /></maths>

0054All the protocols discussed above have been described in the setting of the multiplicative group Z*<i>_p</i>. However, they can all be easily modified to work in any finite group in which the discrete logarithm problem appears intractable. Suitable choices include the multiplicative group of a finite filed (in particular the finite field GF(2ⁿ), subgroups of Z*<i>_p</i> of order q, and the group of points on an elliptic curve defined over a finite field. In each case, an appropriate generator a will be used to define the public keys.

0055The protocols discussed above can also be modified in a straightforward way to handle the situation when each user picks their own system parameters p and α (or analogous parameters if a group other than <i>Z*_p</i> is used).

0056In the above protocols, a signature component of the general form s_A = x + r_a.a.α^a has been used.

0057The protocols may be modified to use a simpler signature component of the general form s_A = x + r_a.a without jeopardizing the security.

0058Examples of such protocols will be described below using the same notation although it will be understood that the protocols could be expressed in alternative notation if preferred.

Protocol 1"

0059This protocol will be described using an implementation in the multiplicative group Z*<i>_p</i> with the following notation: <ul id="ul0003" list-style="none" compact="compact"><li>p is a prime number,</li><li>α is a generator of Z*<i>_p</i>,</li><li>a and b are party A's and B's respective long-term private key,</li><li>α^a mod p is party A's long-term private key,</li><li>α^b mod p is party B's long-term private key,</li><li>x is a random integer selected by A as a short-term private key,</li><li>r_a = α^x mod p is party A's short-term public key,</li><li>y is a random integer selected by B as a short-term private key,</li><li>r_b = a^y mod p is party B's short-term public key,</li><li><i><o ostyle="single">r_a</o></i> is an integer derived from r_a and is typically the 80 least significant bits of r_a,</li><li><i><o ostyle="single">r_b</o></i> is an integer derived from r_b and is typically the 80 least significant bits of r_b.</li></ul>

0060To implement the protocol, <ol id="ol0010" ol-style=""><li>1. A sends r_a to B.</li><li>2. B sends r_B to A.</li><li>3. A computes s_A = x + <i><o ostyle="single">r_a</o></i>.a mod (p-1).</li><li>4. A computes the session key K where <i>K</i> (αy(α<i>^b</i>)<i>^{<o ostyle="single">r_b</o>}</i>)^{s<sub2>A</sub2>} mod p.</li><li>5. B computes s_B = y + <i><o ostyle="single">r_b</o></i> .b mod (p-1).</li><li>6. B computes the session key K where <i>K =</i> (α<i>^x</i>(α<i>^a</i>)<i>^{<o ostyle="single">ya</o>}</i>)<i>^{s<sub2>B</sub2>}</i> mod p.</li><li>7. The shared secret is α^{s<sub2>B</sub2>s<sub2>A</sub2>} mod p.</li></ol>

0061In this protocol, the bandwidth requirements are again reduced by the signature components combine the short and long-term keys of the correspondent to inhibit an attack by an interloper.

0062The above protocol may also be implemented using a subgroup of Z*<i>_p</i>. In this case, q will be a prime divisor of (p-1) and g will be an element of order p in Z*<i>_p</i>.

0063A's and B's public keys will be of the form g^a, g^b respectively and the short-term keys r_a, r_b will be of the form g^x, g^y.

0064The signature components s_A, s_B are computed mod q and the session key K computed mod q as before. The shared secret is then g^{s<sub2>B</sub2>s<sub2>A</sub2>} mod p.

0065As noted above, the protocols may be implemented in groups other than Z*<i>_p</i> and a particularly robust group is the group of points on an elliptic curve over a finite field. An example of such an implementation is set out below as protocol 1"'.

Protocol 1'''

0066The following notation is used: <ul id="ul0004" list-style="none" compact="compact"><li>E is an elliptic curve defined over Fq,</li><li>P is a point of prime order n in E(Fq),</li><li>d_a (1&lt;d_a&lt;n-1) is party A's long-term private key,</li><li>d_b (1&lt;d_b&lt;n-1) is party B's long-term private key,</li><li>Q_a = d_aP is party A's long-term public key,</li><li>Q_b = d_bP is party B's long-term public key,</li><li>k(1&lt;k&lt;n-1) is party A's short-term private key,</li><li>r_a = kP is party A's short-term public key,</li><li>m (1&lt;m&lt;n-1) is party B's short-term private key,</li><li>r_b = mP is party B's short-term public key,</li><li><i><o ostyle="single">r_a</o></i> and <i><o ostyle="single">r_b</o></i> respectively are bit strings, for example the 80 least significant bits of the x co-ordinate of r_a and r_b.</li></ul>

0067To implement the protocol: <ol id="ol0011" ol-style=""><li>1. A sends r_a to B.</li><li>2. B sends r_b to A</li><li>3. A computes s_A = (k + <i><o ostyle="single">r_a</o></i> .d_a) mod n.</li><li>4. A computes the session key K where <i>K</i>=<i>s</i>_b(r_b + <i><o ostyle="single">r_b</o> Q</i>_b)</li><li>5. B computes s_b = (m + <i><o ostyle="single">r_b</o></i> d_b) mod n.</li><li>6. B computes the session key K where <i>K</i>=<i>s_b</i>(<i>r_a</i> + <i><o ostyle="single">r_a</o> Q_a</i>).</li><li>7. The shared secret is s_as_bP.</li></ol>

0068Again, it will be noted that it is not necessary to send the signature components s_A, s_B between the correspondent but the short and long-term keys of the correspondents are combined by the form of the components. (It will be appreciated that the notation m has been substituted for x,y in the previous examples to avoid confusion with the co-ordinate (x,y) of the points on the curve).