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/A0 key agreement protocol establishes a shared secret K, known to the two correspondents, in the following manner: <ol id="ol0001" compact="compact"><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"><math display="inline"><mi>α</mi><mo>∈</mo><msubsup><mi>Z</mi><mi>p</mi><mo>*</mo></msubsup></math><img file="EP2315391A2_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-l, 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 mathvariant="normal">→</mo><mi mathvariant="normal">B</mi><mo mathvariant="normal">:</mo><msup><mi mathvariant="normal">α</mi><mi mathvariant="normal">x</mi></msup><mspace width="1em" /><mi>mod p</mi></math><img file="EP2315391A2_D0002.tif" /></maths><maths id="math0003" num="(2)"><math display="block"><mi mathvariant="normal">A</mi><mo mathvariant="normal">←</mo><mi mathvariant="normal">B</mi><mo mathvariant="normal">:</mo><msup><mi mathvariant="normal">α</mi><mi mathvariant="normal">y</mi></msup><mspace width="1em" /><mi>mod p</mi></math><img file="EP2315391A2_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"><li>(a) A chooses a random integer x, 1 &lt;x&lt;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.

GENERAL

0016In one aspect the present invention may provide a protocol in which the above disadvantages are obviated or mitigated.

0017According to one aspect there may be provided a method of authenticating a pair of correspondents A,B to permit exchange of information therebetween, each of said correspondents having a respective private key a,b and a public key P_A, P_B derived from a generator α and a respective ones of said private keys a,b, said method including the steps of: <ul id="ul0002" list-style="none" compact="compact"><li>i) a first of said correspondents A selecting a first random integer x and exponentiating a function f(α) including said generator to a power g^(x) to provide a first exponentiated function f(α)^g(x);</li><li>ii) said first correspondent A generating a first signature s_A from said random integer x and said first exponentiated function f(α)^g(x);</li><li>iii) said first correspondent A forwarding to a second correspondent B a message including said first exponentiated function f(α)^g(x) and the signature s_A;</li><li>iv) said correspondent B selecting a second random integer y and exponentiating a function f'(α) including said generator to a power g^(y) to provide a second exponentiated function f'(α)^g(y) and a signature s_B obtained from said second integer y and said second exponentiated function f(α)^g(y);</li><li>v) said second correspondent B forwarding a message to said first correspondent A including said second exponentiated function f'(α)^g(y) and said signature S_B;</li><li>vi) each of said correspondents verifying the integrity of messages received by them by computing from said signature and said exponentiated function in such a received message a value equivalent to said exponentiated function and comparing said computed value and said transmitted value;</li><li>vii) each of said correspondents A and B constructing a session key K by exponentiating information made public by said other correspondent with said random integer that is private to themselves.</li></ul>

0018According to another aspect there may be provided a method of authenticating a pair of correspondents A,B to permit exchange of information therebetween, each of said correspondents having a respective private key a,b and a public key p_A, p_B derived from a generator α and respective ones of said private keys a,b, said method including the steps of: <ul id="ul0003" list-style="none" compact="compact"><li>i) a first of said correspondents A selecting a first random integer x and exponentiating a function f(α) including said generator to a power g^(x) to provide a first exponentiated function f(α)^g(x);</li><li>ii) said first correspondent A generating a first signature s_A from said random integer x and said exponentiated function f(α)^g(x);</li><li>iii) said first correspondent A forwarding to a second correspondent B a message including said first exponentiated function f(α)^g(x);</li><li>iv) said correspondent B selecting a second random integer y and exponentiating a function f'(α) including said generator to a power g^(y) to provide a second exponentiated function f'(α)^g(y) and generating a signature s_B obtained from said second integer y and said second exponentiated function f'(α)^g(y);</li><li>v) said second correspondent B forwarding a message to said first correspondent A including said second exponential function f'(α)^g(y);</li><li>vi) each of said correspondents constructing a session key K by exponentiating information made public by another correspondent with information that is private to themselves.</li></ul>

0019preferably wherein said message forwarded by said first correspondent includes an identification of the first correspondent;

0020preferably wherein said message forwarded by said second correspondent includes an identification of said second correspondent,

0021more preferably wherein said message forwarded by said first correspondent includes an identification of the first correspondent;

0022preferably wherein said first function including said generator f(α) is said generator itself;

0023preferably wherein said second function f(α) including said generator is said generator itself,

0024more preferably wherein said first function f(α) including said generator is said generator itself;

0025preferably wherein said first function including said generator f(α) includes the public key p_B of said second correspondent;

0026preferably wherein said second function including said generator f'α includes the public key p_A of said first correspondent;

0027preferably wherein said signature generated by a respective one of the correspondents combine the random integer, exponentiated function and private key of that one correspondent,

0028more preferably wherein said signature of correspondent A is of the form x-r_Aaα^a mod (p-1), or wherein said signature of correspondent A is of the form x + aα^a(p_B)^x mod (p-1), or wherein said signature of correspondent A is of the form <maths id="math0004"><math display="inline"><msub><mi mathvariant="italic">xr</mi><msub><mi mathvariant="italic">x</mi><mn>1</mn></msub></msub><mo>-</mo><msup><mfenced><msub><mi>r</mi><mi>A</mi></msub></mfenced><msub><mi>r</mi><msub><mi mathvariant="italic">x</mi><mn>1</mn></msub></msub></msup></math><img file="EP2315391A2_D0004.tif" /></maths><i>a</i> mod (p-1) where x₁, is a second random integer selected by A and <maths id="math0005"><math display="inline"><msub><mi>r</mi><msub><mi mathvariant="italic">x</mi><mn>1</mn></msub></msub><mo>=</mo><msup><mi>α</mi><msub><mi mathvariant="italic">x</mi><mn>1</mn></msub></msup><mo>,</mo></math><img file="EP2315391A2_D0005.tif" /></maths> or, wherein said signature of correspondent B is of the form y_B - r_Bbα^b mod (p-1), or, wherein said signature of correspondent B is of the form y + bα^b (p_A)^y mod (p-1), or wherein said signature of correspondent B is of the form <maths id="math0006"><math display="inline"><msub><mi mathvariant="italic">yr</mi><msub><mi>y</mi><mn>1</mn></msub></msub><mo>-</mo><msup><mfenced><msub><mi>r</mi><mi>B</mi></msub></mfenced><msub><mi>r</mi><msub><mi>y</mi><mn>1</mn></msub></msub></msup></math><img file="EP2315391A2_D0006.tif" /></maths><i>b</i> mod (p-1) whereby y₁ is a second integer selected by correspondent B and r<i>_y1</i> = α^y1,

0029preferably wherein said correspondent A selects a second integer x₁, and forwards <i>r</i>_A1, to correspondent B where <i>r</i>_<i>A</i>1 = α^x1 and said correspondent B selects a second random integer y₁ and sends r_B1 to correspondent A, where <i>r_B1</i> = α^y1 each of said correspondents computing a pair of keys k₁, k₂ equivalent to α^xy and α^x1y1 respectively, said session key K being generated by XORing k₁ and k₂.

0030Thus although the interloper E can substitute her public key p_E = α^ae in the transmission as part of the message, B will use p_E rather than p_A when authenticating the message. Accordingly, the computed and transmitted values of the exponential functions will not correspond.

0031According to another aspect there may be provided a method of establishing a session key K at a correspondent A in a public key data communication system, said correspondent A having a private key a and a public key p_A derived from a generator α and said private key a, said method including the steps of: <ul id="ul0004" list-style="none" compact="compact"><li>i) said correspondent A selecting a random integer x and exponentiating a first function f(α) including said generator to a power g(x) to provide a first exponentiated function f(α)^g(x) ;</li><li>ii) said correspondent A generating a signature s_A from said random integer x, said exponentiated function f(α)^g(x), and said private key a to bind said integer x and said private key a;</li><li>iii) said correspondent A forwarding to a second correspondent B a message including said first exponentiated function f(α)^g(x), whereby said second correspondent B can utilize information in said message and information private to itself to compute said session key K.</li><li>iv) said correspondent A receiving from said correspondent B a message including a second exponentiated function f'(α)^g(y), wherein y is a random integer selected by said correspondent B, g(y) is a function of said random integer y, and f'(α) is a second function of said generator; and</li><li>vi) said correspondent A constructing a session key K by exponentiating information made public by said correspondent B with information that is private to said correspondent A.</li></ul>

0032The first correspondent A may maintain said signature s_A private to itself.

0033According to another aspect there may be provided a method of establishing a session key K at a correspondent B in a public key data communication system, said correspondent B having a private key b and a public key p_B derived from a generator α and said private key b, said method including the steps of: <ul id="ul0005" list-style="none" compact="compact"><li>i) said correspondent B selecting a random integer y and exponentiating a first function f'(α) including said generator to a power g(y) to provide a first exponentiated function f'(α)^g(y) ;</li><li>ii) said correspondent B generating a signature s_B from said random integer y, said exponentiated function f'(α)^g(y), and said private key b to bind said integer y and said private key b;</li><li>iii) said correspondent B forwarding to another correspondent A a message including said first exponentiated function f'(α)^g(y), whereby said correspondent A can utilize information in said message and information private to itself to compute said session key K.</li><li>iv) said correspondent B receiving from said correspondent A a message including a second exponentiated function f(α)^g(x), wherein x is a random integer selected by said correspondent A, g(x) is a function of said random integer x, and f(α) is a second function of said generator; and</li><li>vi) said correspondent B constructing a session key K by exponentiating information made public by said correspondent A with information that is private to said correspondent B.</li></ul>

0034The first correspondent B may maintain said signature s_B private to itself.

0035According to another aspect there may be provided a method of transporting a session key K from a first correspondent A to a second correspondent B in a public key data communication system to establish a common key to permit exchange of information therebetween over a communication channel, said correspondent A having a private key a and a public key p_A derived from a generator α and said private key a, said method including the steps of: <ul id="ul0006" list-style="none" compact="compact"><li>i) said correspondent A selecting a first random integer x and exponentiating a first function f(α) including said generator to a power g(x) to provide a first exponentiated function f(α)^g(x) ;</li><li>ii) said first correspondent A generating a signature s_A from said random integer x, said exponentiated function f(α)^g(x), and said private key a to bind said integer x and said private key a;</li><li>iii) said first correspondent A forwarding to said second correspondent B a message including said first exponentiated function f(α)^g(x) , whereby said second correspondent B can utilize information in said message to compute said session key K; and</li><li>iv) said first correspondent computing said session key K from a public key p_B of said second correspondent B and said signature s_A.,</li></ul>

0036The first correspondent A may maintains said signature s_A private to itself.

0037According to another aspect there may be provided a method of establishing a session key K at a correspondent B in a public key data communication system to establish a common key with a correspondent A to permit exchange of information therebetween over a communication channel, said correspondent B having a private key b and a public key p_B derived from a generator α and said private key b, said method including the steps of: <ul id="ul0007" list-style="none" compact="compact"><li>i) said correspondent B receiving from said correspondent A a message including an exponentiated function f(α)^g(x) , wherein x is a random integer selected by said correspondent A, g(x) is a function of said random integer x, and f(α) is a function of said generator; and</li><li>ii) said correspondent B utilizing a public key p_A of said correspondent A and information in said message to compute a session key K, said session key K corresponding to a session key computable by said correspondent A.</li></ul>

0038According to another aspect there may be provided a method of establishing a session key K at a correspondent A in a public key data communication system to permit exchange of information with a correspondent B over a communication channel, said correspondent A having a respective private key and a public key derived from a generator α and said private key, said method including the steps of: <ul id="ul0008" list-style="none" compact="compact"><li>i) said correspondent A selecting a first random integer and exponentiating a function including said generator to a power to provide a first exponentiated function r_a;</li><li>ii) said correspondent A generating a first signature <i>s_A</i> from said random integer and an integer <maths id="math0007"><math display="inline"><mover><msub><mi>r</mi><mi>a</mi></msub><mo>‾</mo></mover></math><img file="EP2315391A2_D0007.tif" /></maths> derived from said exponentiated function r_a;</li><li>iii) said correspondent A forwarding to a second correspondent B a message derived from said first exponentiated function r_a, whereby said correspondent B can use information in said message along with information private to itself to construct said session key K;</li><li>iv) said correspondent A receiving from said second correspondent B a message derived from a second exponential function r_b, wherein r_b is a function derived from said generator and a random integer selected by said correspondent B; and</li><li>v) said correspondent A constructing said session key K by exponentiating information made public by said correspondent B with information that is private to said correspondent A.</li></ul>

0039According to another aspect there may be provided a method of establishing a session key K at a correspondent B in a public key data communication system to permit exchange of information with a correspondent A over a communication channel, said correspondent B having a respective private key and a public key derived from a generator α and said private key, said method including the steps of: <ul id="ul0009" list-style="none" compact="compact"><li>i) said correspondent B selecting a first random integer and exponentiating a function including said generator to a power to provide a first exponentiated function r_b;</li><li>ii) said correspondent B generating a signature <i>s_B</i> from said random integer and an integer <img file="EP2315391A2_D0008.tif" /> derived from said exponentiated function r_b;</li><li>iii) said correspondent B forwarding to a second correspondent A a message derived from said first exponentiated function r_b, whereby said correspondent A can use information in said message along with information private to itself to construct said session key K;</li><li>iv) said correspondent B receiving from said second correspondent A a message derived from a second exponential function r_a, wherein r_a is a function derived from said generator and a random integer selected by said correspondent A; and</li><li>v) said correspondent B constructing said session key K by exponentiating information made public by said correspondent A with information that is private to said correspondent B.</li></ul>

0040The first signature <i>s_A</i> may be computed as <maths id="math0008"><math display="inline"><msub><mi>s</mi><mi mathvariant="italic">A</mi></msub><mo mathvariant="italic">=</mo><mi mathvariant="italic">x</mi><mo mathvariant="italic">+</mo><mover><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">a</mi></msub><mo mathvariant="italic">‾</mo></mover><mo>⁢</mo><mi mathvariant="italic">a</mi></math><img file="EP2315391A2_D0009.tif" /></maths> mod(<i>p</i>-1) , wherein x is said first random integer and <i>a</i> is said private key of said correspondent A.

0041The signature s_A may be computed as <maths id="math0009"><math display="inline"><msub><mi>s</mi><mi mathvariant="italic">A</mi></msub><mo mathvariant="italic">=</mo><mi mathvariant="italic">x</mi><mo mathvariant="italic">+</mo><mover><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">a</mi></msub><mo mathvariant="italic">‾</mo></mover><mo>⁢</mo><mi mathvariant="italic">a</mi></math><img file="EP2315391A2_D0010.tif" /></maths> mod(<i>p</i>-1).

0042The signature s_A may be of the form <maths id="math0010"><math display="inline"><msub><mi>s</mi><mi mathvariant="italic">A</mi></msub><mo mathvariant="italic">=</mo><mi mathvariant="italic">x</mi><mo>-</mo><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">A</mi></msub><mo>⁢</mo><msup><mi mathvariant="italic">aα</mi><mi>a</mi></msup></math><img file="EP2315391A2_D0011.tif" /></maths> mod(<i>p</i>-1).

0043According to another aspect f(α) = α.

0044According to another aspect there may be provided a system for performing the method described herein.

0045According to another aspect there may be provided a computer readable medium having stored thereon computer readable instructions for performing the method described herein.

BRIEF DESCRIPTION OF DRAWING

0046Embodiments of the invention will now be described by way of example only with reference to the accompanying drawings in which: <ul id="ul0010" 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

0047Referring 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.

0048In 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.

0049The 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.

0050In 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.

0051The 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.

0052In 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.

0053To avoid interloper E convincing B that he is communicating with E, the following protocol is adapted.

0054The 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.

0055The system parameters for this protocol are a prime number p and a generator α of the multiplicative group <i>Z</i>^*_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

0056<ol id="ol0003" compact="compact"><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 <maths id="math0011"><math display="inline"><msup><mi>α</mi><msub><mi>s</mi><mi mathvariant="italic">B</mi></msub></msup><mo>⁢</mo><msup><mfenced><msub><mi mathvariant="italic">p</mi><mi mathvariant="italic">B</mi></msub></mfenced><mrow><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">B</mi></msub><mo>⁢</mo><msup><mi>α</mi><mi mathvariant="italic">b</mi></msup></mrow></msup></math><img file="EP2315391A2_D0012.tif" /></maths>and verifies that this is equal to r_B. A computes the session key <maths id="math0012"><math display="inline"><mi mathvariant="normal">K</mi><mo>=</mo><msup><mfenced><msub><mi mathvariant="normal">r</mi><mi mathvariant="normal">B</mi></msub></mfenced><mi mathvariant="normal">x</mi></msup><mo>=</mo><msup><mi mathvariant="normal">α</mi><msub><mi mathvariant="normal">x</mi><mi mathvariant="normal">y</mi></msub></msup></math><img file="EP2315391A2_D0013.tif" /></maths></li><li>4. B computes <maths id="math0013"><math display="inline"><msup><mi>α</mi><msub><mi>s</mi><mi mathvariant="italic">A</mi></msub></msup><mo>⁢</mo><msup><mfenced><msub><mi mathvariant="italic">p</mi><mi mathvariant="italic">A</mi></msub></mfenced><mrow><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">A</mi></msub><mo>⁢</mo><msup><mi>α</mi><mi mathvariant="italic">a</mi></msup></mrow></msup></math><img file="EP2315391A2_D0014.tif" /></maths> and verifies that this is equal to r_A. B computes the session key <maths id="math0014"><math display="inline"><mi mathvariant="normal">K</mi><mo>=</mo><msup><mfenced><msub><mi mathvariant="normal">r</mi><mi mathvariant="normal">A</mi></msub></mfenced><mi mathvariant="normal">y</mi></msup><mo>=</mo><msup><mi mathvariant="normal">α</mi><msub><mi mathvariant="normal">x</mi><mi mathvariant="normal">y</mi></msub></msup><mn>.</mn></math><img file="EP2315391A2_D0015.tif" /></maths>.</li></ol>

0057Should E replace text_A with text _E, B will compute <maths id="math0015"><math display="inline"><msup><mi mathvariant="italic">α</mi><msub><mi mathvariant="italic">S</mi><mi mathvariant="italic">A</mi></msub></msup><mo>⁢</mo><msup><mfenced><msub><mi mathvariant="italic">p</mi><mi mathvariant="italic">E</mi></msub></mfenced><mrow><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">A</mi></msub><mo>⁢</mo><msup><mi mathvariant="italic">α</mi><mi mathvariant="italic">a</mi></msup></mrow></msup></math><img file="EP2315391A2_D0016.tif" /></maths> 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.

0058One 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

0059In step 1, A also sends <maths id="math0016"><math display="inline"><msup><mi>α</mi><msub><mi>x</mi><mn>1</mn></msub></msup></math><img file="EP2315391A2_D0017.tif" /></maths> to B, where x₁ is a second random integer generated by A Similarly, in step 2, B also sends <maths id="math0017"><math display="inline"><msup><mi>α</mi><msub><mi>y</mi><mn>1</mn></msub></msup></math><img file="EP2315391A2_D0018.tif" /></maths> to A, where y₁ is a random integer. A and B now compute the key <maths id="math0018"><math display="inline"><mi>K</mi><mo>=</mo><msup><mi mathvariant="normal">α</mi><mi>xy</mi></msup><mo /><mo>⊕</mo><mo /><msup><mi>α</mi><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>y</mi><mn>1</mn></msub></mrow></msup><mn>.</mn></math><img file="EP2315391A2_D0019.tif" /></maths>.

0060Another 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 <maths id="math0019"><math display="inline"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo mathvariant="italic">=</mo><mi mathvariant="normal">x</mi><mo>-</mo><msub><mi mathvariant="normal">r</mi><mi mathvariant="normal">A</mi></msub><mo>⁢</mo><msup><mi mathvariant="normal">aα</mi><mi mathvariant="normal">a</mi></msup></math><img file="EP2315391A2_D0020.tif" /></maths> 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

0061A second protocol set out below addresses these two drawbacks. <ol id="ol0004" compact="compact"><li>1. A picks a random integer x, 1≤x≤p-2, and computes (<i>p_B</i>)<i>^x</i> , α^x and a signature <i>s_A</i> = <i>x</i> + <i>a</i>α<i>^a</i>(<i>p_B</i>)<i>^x</i> mod(<i>p</i>-1)<i>.</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>b</i>α<i>^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 <maths id="math0020"><math display="inline"><msup><mi mathvariant="italic">α</mi><msub><mi>s</mi><mi mathvariant="italic">B</mi></msub></msup><mo>⁢</mo><msup><mfenced><msub><mi mathvariant="italic">p</mi><mi mathvariant="italic">B</mi></msub></mfenced><mrow><mo>-</mo><msup><mrow><msup><mi>α</mi><mi>b</mi></msup><mo>⁢</mo><mi>α</mi></mrow><mi mathvariant="italic">ay</mi></msup></mrow></msup><mo>=</mo><msup><mi>α</mi><mi>y</mi></msup><mn>.</mn></math><img file="EP2315391A2_D0021.tif" /></maths> A then computes session key <i>K =</i> α<i>^ay</i>(<i>p_B</i>)<i>^x.</i></li><li>4. B computes (α^x)^b and verifies that <maths id="math0021"><math display="inline"><msup><mi mathvariant="italic">α</mi><msub><mi>s</mi><mi>A</mi></msub></msup><mo>⁢</mo><msup><mfenced><msub><mi mathvariant="italic">p</mi><mi>A</mi></msub></mfenced><mrow><mo>-</mo><msup><mrow><msup><mi>α</mi><mi>a</mi></msup><mo>⁢</mo><mi>α</mi></mrow><mi mathvariant="italic">bx</mi></msup></mrow></msup><mo>=</mo><msup><mi>α</mi><mi>x</mi></msup><mn>.</mn></math><img file="EP2315391A2_D0022.tif" /></maths> A then computes session key <i>K</i> = α<i>^bx</i>(<i>p_A</i>)<i>^y.</i></li></ol>

0062The 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.

0063If 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.

0064The 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.

0065The 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 <maths id="math0022"><math display="inline"><msup><mi mathvariant="normal">α</mi><mrow><msub><mi>s</mi><mi>A</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>B</mi></msub></mrow></msup><mn>.</mn></math><img file="EP2315391A2_D0023.tif" /></maths>

Protocol 1'

0066<ol id="ol0005" compact="compact"><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 <maths id="math0023"><math display="inline"><mi>K</mi><mo>=</mo><msup><mfenced><msub><mi>r</mi><mi>B</mi></msub><mo>⁢</mo><msup><mfenced><msub><mi>p</mi><mi>B</mi></msub></mfenced><mrow><msub><mi>r</mi><mi>B</mi></msub><mo>⁢</mo><msup><mi>α</mi><mi>b</mi></msup></mrow></msup></mfenced><msub><mi>s</mi><mi>A</mi></msub></msup></math><img file="EP2315391A2_D0024.tif" /></maths> which is equivalent to <maths id="math0024"><math display="inline"><msup><mi>α</mi><mrow><msub><mi>s</mi><mi>A</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>B</mi></msub></mrow></msup><mn>.</mn></math><img file="EP2315391A2_D0025.tif" /></maths></li><li>4. B computes <maths id="math0025"><math display="inline"><mi>K</mi><mo>=</mo><msup><mfenced><msub><mi>r</mi><mi>A</mi></msub><mo>⁢</mo><msup><mfenced><msub><mi>p</mi><mi>A</mi></msub></mfenced><mrow><msub><mi>r</mi><mi>A</mi></msub><mo>⁢</mo><msup><mi>α</mi><mi>a</mi></msup></mrow></msup></mfenced><msub><mi>s</mi><mi>B</mi></msub></msup></math><img file="EP2315391A2_D0026.tif" /></maths> which is also equivalent to <maths id="math0026"><math display="inline"><msup><mi>α</mi><mrow><msub><mi>s</mi><mi>A</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>B</mi></msub></mrow></msup><mn>.</mn></math><img file="EP2315391A2_D0027.tif" /></maths></li></ol>

0067A 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'

0068<ol id="ol0006" compact="compact"><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 <maths id="math0027"><math display="inline"><mi>K</mi><mo>=</mo><msup><mfenced><msup><mi>α</mi><mi>y</mi></msup><mo>⁢</mo><msup><mfenced><msub><mi>p</mi><mi>B</mi></msub></mfenced><mrow><msup><mi>α</mi><mi>b</mi></msup><mo>⁢</mo><msup><mi>α</mi><mi mathvariant="italic">ay</mi></msup></mrow></msup></mfenced><msub><mi>s</mi><mi>A</mi></msub></msup><mo>,</mo></math><img file="EP2315391A2_D0028.tif" /></maths> i.e. <maths id="math0028"><math display="inline"><msup><mi>α</mi><mrow><msub><mi>s</mi><mi>A</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>B</mi></msub></mrow></msup><mn>.</mn></math><img file="EP2315391A2_D0029.tif" /></maths></li><li>4. B computes (α^x)^b and <maths id="math0029"><math display="inline"><mi>K</mi><mo>=</mo><msup><mfenced><msup><mi>α</mi><mi>x</mi></msup><mo>⁢</mo><msup><mfenced><msub><mi>p</mi><mi>A</mi></msub></mfenced><mrow><msup><mi>α</mi><mi>a</mi></msup><mo>⁢</mo><msup><mi>α</mi><mi mathvariant="italic">bx</mi></msup></mrow></msup></mfenced><msub><mi>s</mi><mi>B</mi></msub></msup></math><img file="EP2315391A2_D0030.tif" /></maths> i.e. <maths id="math0030"><math display="inline"><msup><mi>α</mi><mrow><msub><mi>s</mi><mi>A</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>B</mi></msub></mrow></msup><mn>.</mn></math><img file="EP2315391A2_D0031.tif" /></maths></li></ol>

0069Again therefore the transmission of s_A and s_B is avoided.

0070A further protocol is available for parties A and B to establish a session key K.

Third Protocol

0071The 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"><li>1. A picks two random integers x, x₁, 1≤x,x₁≤p-2, and computes r_x1 = α^x1, r_A = α^x and <maths id="math0031"><math display="inline"><msup><mfenced><msub><mi>r</mi><mi>A</mi></msub></mfenced><msub><mi>r</mi><mrow><mi>x</mi><mo>⁢</mo><mn>1</mn></mrow></msub></msup><mo>,</mo></math><img file="EP2315391A2_D0032.tif" /></maths> 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 <maths id="math0032"><math display="inline"><msup><mfenced><msub><mi>r</mi><mi>B</mi></msub></mfenced><msub><mi>r</mi><mrow><mi mathvariant="italic">y</mi><mo>⁢</mo><mn>1</mn></mrow></msub></msup><mo>,</mo></math><img file="EP2315391A2_D0033.tif" /></maths> 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 <maths id="math0033"><math display="inline"><msup><mi>α</mi><msub><mi>s</mi><mi>B</mi></msub></msup><mo>⁢</mo><msup><mfenced><msub><mi>p</mi><mi>B</mi></msub></mfenced><msup><mfenced><msub><mi>r</mi><mi>B</mi></msub></mfenced><msub><mi>r</mi><mrow><mi>y</mi><mo>⁢</mo><mn>1</mn></mrow></msub></msup></msup></math><img file="EP2315391A2_D0034.tif" /></maths> and verifies that this is equal to <maths id="math0034"><math display="inline"><msup><mfenced><msub><mi>r</mi><mi>B</mi></msub></mfenced><msub><mi>r</mi><mrow><mi>y</mi><mo>⁢</mo><mn>1</mn></mrow></msub></msup><mn>.</mn></math><img file="EP2315391A2_D0035.tif" /></maths> A computes session key <i>K</i>=<i>(</i>α<i>^y1)^x1</i> = α<i>^x1.y1</i></li><li>4. B computes <maths id="math0035"><math display="inline"><msup><mi>α</mi><msub><mi>s</mi><mi>A</mi></msub></msup><mo>⁢</mo><msup><mfenced><msub><mi>p</mi><mi>A</mi></msub></mfenced><msup><mfenced><msub><mi>r</mi><mi>A</mi></msub></mfenced><msub><mi>r</mi><mrow><mi>x</mi><mo>⁢</mo><mn>1</mn></mrow></msub></msup></msup></math><img file="EP2315391A2_D0036.tif" /></maths> and verifies that this is equal to <maths id="math0036"><math display="inline"><msup><mfenced><msub><mi>r</mi><mi>A</mi></msub></mfenced><msub><mi>r</mi><mrow><mi>x</mi><mo>⁢</mo><mn>1</mn></mrow></msub></msup><mo>•</mo></math><img file="EP2315391A2_D0037.tif" /></maths> B computes session key <i>K=(</i>α<i>^x1)^y1</i> = α<i>^x1y1</i></li></ol>

0072In 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

0073The 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"><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 <maths id="math0037"><math display="inline"><msup><mi>α</mi><msub><mi>s</mi><mi>A</mi></msub></msup><mo>⁢</mo><msup><mfenced><msub><mi>p</mi><mi>A</mi></msub></mfenced><mrow><msub><mi>r</mi><mi>A</mi></msub><mo>⁢</mo><msup><mi>α</mi><mi>a</mi></msup></mrow></msup></math><img file="EP2315391A2_D0038.tif" /></maths>and verifies that this quantity is equal to r_A. B computes session key K = (r_A)^b.</li></ol>

Modified Key Transport Protocol

0074The 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"><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</i>_B<i>)</i>^sA and sends (r_A, text_A) to B.</li><li>2. B computes <maths id="math0038"><math display="inline"><mi>K</mi><mo>=</mo><msup><mfenced><msup><mi>α</mi><mi>x</mi></msup><mo>⁢</mo><msup><mfenced><msub><mi>p</mi><mi>A</mi></msub></mfenced><mrow><mo>-</mo><msub><mi>r</mi><mi>A</mi></msub><mo>⁢</mo><msup><mi>α</mi><mi>a</mi></msup></mrow></msup></mfenced><mi>b</mi></msup><mo>=</mo><msup><mi>α</mi><msub><mrow><mi mathvariant="italic">b</mi><mo>⁢</mo><mi mathvariant="italic">s</mi></mrow><mi>A</mi></msub></msup></math><img file="EP2315391A2_D0039.tif" /></maths></li></ol>

0075All 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).

0076This 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.

0077The signing equation s_A = x - r_Aaα^a where r_a = α^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="math0039"><math display="block"><msub><mi mathvariant="normal">r</mi><mi mathvariant="normal">A</mi></msub><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo>⁢</mo><mi mathvariant="normal">x</mi><mo mathvariant="normal">+</mo><mi mathvariant="normal">z</mi></math><img file="EP2315391A2_D0040.tif" /></maths><maths id="math0040"><math display="block"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo mathvariant="normal">=</mo><msup><mi mathvariant="normal">xα</mi><mi mathvariant="normal">a</mi></msup><mo mathvariant="normal">+</mo><msub><mi>ar</mi><mi mathvariant="normal">A</mi></msub></math><img file="EP2315391A2_D0041.tif" /></maths><maths id="math0041"><math display="block"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo mathvariant="normal">=</mo><msub><mi>xr</mi><mi mathvariant="normal">A</mi></msub><mo mathvariant="normal">+</mo><msup><mi mathvariant="normal">aα</mi><mi mathvariant="normal">a</mi></msup></math><img file="EP2315391A2_D0042.tif" /></maths><maths id="math0042"><math display="block"><mn>1</mn><mo mathvariant="normal">=</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="EP2315391A2_D0043.tif" /></maths>

0078All 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.

0079The 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 Z^*<i>_p</i> is used).

0080In the above protocols, a signature component of the general form s_A = x + r_a.a.α^a has been used.

0081The protocols may be modified to use a simpler signature component of the general form s_A = x + r_a.a without jeopardizing the security.

0082Examples 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"

0083This protocol will be described using an implementation in the multiplicative group Z^*<i>_p</i> with the following notation: <ul id="ul0011" 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 = α^y mod p is party B's short-term public key,</li><li><img file="EP2315391A2_D0044.tif" /> is an integer derived from r_a and is typically the 80 least significant bits of r_a,</li><li><img file="EP2315391A2_D0045.tif" /> is an integer derived from r_b and is typically the 80 least significant bits of r_b.</li></ul>

0084To implement the protocol, <ol id="ol0010"><li>1. A sends r_a to B.</li><li>2. B sends r_B to A.</li><li>3. A computes <maths id="math0043"><math display="inline"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo mathvariant="italic">=</mo><mi mathvariant="normal">x</mi><mo>+</mo><mover><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">a</mi></msub><mo>‾</mo></mover></math><img file="EP2315391A2_D0046.tif" /></maths>a mod (p-1).</li><li>4. A computes the session key K where <maths id="math0044"><math display="inline"><msup><mrow><mi>K</mi><mo>=</mo><mfenced><msup><mi>α</mi><mi>y</mi></msup><mo>⁢</mo><msup><mfenced><msup><mi>α</mi><mi>b</mi></msup></mfenced><mover><msub><mi>r</mi><mi>b</mi></msub><mo>‾</mo></mover></msup></mfenced></mrow><msub><mi>s</mi><mi>A</mi></msub></msup></math><img file="EP2315391A2_D0047.tif" /></maths> mod p.</li><li>5. B computes <maths id="math0045"><math display="inline"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">B</mi></msub><mo mathvariant="italic">=</mo><mi mathvariant="normal">y</mi><mo>+</mo><mover><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">b</mi></msub><mo>‾</mo></mover></math><img file="EP2315391A2_D0048.tif" /></maths> .b mod (p-1).</li><li>6. B computes the session key K where <maths id="math0046"><math display="inline"><msup><mrow><mi>K</mi><mo>=</mo><mfenced><msup><mi>α</mi><mi>x</mi></msup><mo>⁢</mo><msup><mfenced><msup><mi>α</mi><mi>a</mi></msup></mfenced><mover><msub><mi>r</mi><mi>a</mi></msub><mo>‾</mo></mover></msup></mfenced></mrow><msub><mi>s</mi><mi>B</mi></msub></msup></math><img file="EP2315391A2_D0049.tif" /></maths> mod p.</li><li>7. The shared secret is <maths id="math0047"><math display="inline"><msup><mi>α</mi><mrow><msub><mi>s</mi><mi>B</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>A</mi></msub></mrow></msup></math><img file="EP2315391A2_D0050.tif" /></maths> mod p.</li></ol>

0085In 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.

0086The 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>.

0087A'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.

0088The 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 <maths id="math0048"><math display="inline"><msup><mi mathvariant="italic">g</mi><mrow><msub><mi>s</mi><mi>B</mi></msub><mo>⁢</mo><msub><mi>s</mi><mi>A</mi></msub></mrow></msup></math><img file="EP2315391A2_D0051.tif" /></maths>mod p.

0089As 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"'

0090The following notation is used: <ul id="ul0012" 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>da (1&lt;da&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 = daP 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><img file="EP2315391A2_D0052.tif" /> and <img file="EP2315391A2_D0053.tif" /> respectively are bit strings, for example the 80 least significant bits of the x co-ordinate of r_a and r_b.</li></ul>

0091To implement the protocol: <ol id="ol0011"><li>1. A sends r_a to B.</li><li>2. B sends r_b to A.</li><li>3. A computes <maths id="math0049"><math display="inline"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">A</mi></msub><mo mathvariant="italic">=</mo><mfenced><mi mathvariant="normal">k</mi><mo>+</mo><mover><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">a</mi></msub><mo>‾</mo></mover><mn>.</mn><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">a</mi></msub></mfenced></math><img file="EP2315391A2_D0054.tif" /></maths> mod n.</li><li>4. A computes the session key K where <maths id="math0050"><math display="inline"><mi>K</mi><mo>=</mo><msub><mi>s</mi><mi mathvariant="normal">b</mi></msub><mo>⁢</mo><mfenced><msub><mi>r</mi><mi>b</mi></msub><mo>+</mo><mover><msub><mi>r</mi><mi>b</mi></msub><mo>‾</mo></mover><mo>⁢</mo><msub><mi>Q</mi><mi mathvariant="normal">b</mi></msub></mfenced></math><img file="EP2315391A2_D0055.tif" /></maths></li><li>5. B computes <maths id="math0051"><math display="inline"><msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">b</mi></msub><mo>=</mo><mfenced><mi mathvariant="normal">m</mi><mo>+</mo><mover><msub><mi mathvariant="italic">r</mi><mi>b</mi></msub><mo>‾</mo></mover><mo>⁢</mo><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">b</mi></msub></mfenced></math><img file="EP2315391A2_D0056.tif" /></maths> mod n.</li><li>6. B computes the session key K where <maths id="math0052"><math display="inline"><mi>K</mi><mo>=</mo><msub><mi>s</mi><mi mathvariant="italic">b</mi></msub><mo>⁢</mo><mfenced><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">a</mi></msub><mo mathvariant="italic">+</mo><mover><msub><mi mathvariant="italic">r</mi><mi mathvariant="italic">a</mi></msub><mo mathvariant="italic">‾</mo></mover><mo>⁢</mo><msub><mi mathvariant="italic">Q</mi><mi mathvariant="italic">a</mi></msub></mfenced><mn>.</mn></math><img file="EP2315391A2_D0057.tif" /></maths></li><li>7. The shared secret is s_as_bP.</li></ol>

0092Again, 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).