U.S. patent application number 10/899303 was filed with the patent office on 2005-08-18 for key agreement and transport protocol with implicit signatures.
This patent application is currently assigned to Cryptech Systems Inc.. Invention is credited to Menezes, Alfred John, Qu, Minghua, Vanstone, Scott.
Application Number | 20050182936 10/899303 |
Document ID | / |
Family ID | 32909096 |
Filed Date | 2005-08-18 |
United States Patent
Application |
20050182936 |
Kind Code |
A1 |
Vanstone, Scott ; et
al. |
August 18, 2005 |
Key agreement and transport protocol with implicit signatures
Abstract
A key establishment protocol between a pair of correspondents
includes the generation by each correspondent of respective
signatures. The signatures are derived from information that is
private to the correspondent and information that is public. After
exchange of signatures, the integrity of exchange messages can be
verified by extracting the public information contained in the
signature and comparing it with information used to generate the
signature. A common session key may then be generated from the
pubilc and private information of respective ones of the
correspondents.
Inventors: |
Vanstone, Scott; (Waterloo,
CA) ; Menezes, Alfred John; (Auburn, AL) ; Qu,
Minghua; (Waterloo, CA) |
Correspondence
Address: |
BLAKE, CASSELS & GRAYDON LLP
BOX 25, COMMERCE COURT WEST
199 BAY STREET, SUITE 2800
TORONTO
ON
M5L 1A9
CA
|
Assignee: |
Cryptech Systems Inc.
|
Family ID: |
32909096 |
Appl. No.: |
10/899303 |
Filed: |
July 27, 2004 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
10899303 |
Jul 27, 2004 |
|
|
|
09558256 |
Apr 25, 2000 |
|
|
|
6785813 |
|
|
|
|
09558256 |
Apr 25, 2000 |
|
|
|
08966766 |
Nov 7, 1997 |
|
|
|
6122736 |
|
|
|
|
08966766 |
Nov 7, 1997 |
|
|
|
08426712 |
Apr 21, 1995 |
|
|
|
Current U.S.
Class: |
713/171 |
Current CPC
Class: |
G06F 7/725 20130101;
H04L 9/0841 20130101; H04L 9/3247 20130101 |
Class at
Publication: |
713/171 |
International
Class: |
H04L 009/00 |
Claims
We claim:
1. 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.sub.A,p.sub.B derived from a generator .alpha. and respective
ones of said private keys a,b, said method including the steps of
i) a first of said correspondents A selecting a first random
integer x and exponentiating a function f(.alpha.) including said
generator to a power g.sup.(x) to provide a first exponentiated
function f(.alpha.).sup.2(x); ii) said first correspondent A
generating a first signature s.sub.A from said random integer x and
said exponentiated function f(.alpha.).sup.2(x); iii) said first
correspondent A forwarding to a second correspondent B a message
including said first exponentiated function f(.alpha.).sup.g(x) and
said signature s.sub.A; iv) said correspondent B selecting a second
random integer y and exponentiating a function f'(.alpha.)
including said generator to a power g.sup.(y) to provide a second
exponentiated function f'(.alpha.).sup.g(y) and generating a
signature s.sub.B obtained from said second integer y and said
second exponentiated function f'(.alpha.).sup.g(y); v) said second
correspondent B forwarding a message to said first correspondent A
including said second exponential function f'(.alpha.).sup.g(y) and
said signature s.sub.g; 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; vii) each
of said correspondents constructing a session key K by
exponentiating information made public by another correspondent
with said random integer that is private to themselves.
2. A method of claim 1 wherein said message forwarded by said first
correspondent includes an identification of the first
correspondent.
3. A method according to claim 1 wherein said message forwarded by
said second correspondent includes an identification of said second
correspondent.
4. A method according to claim 3 wherein said message forwarded by
said first correspondent includes an identification of the first
correspondent.
5. A method according to claim 1 wherein said first function
including said generator f(.alpha.) is said generator itself.
6. A method according to claim 1 wherein said second function
f(.alpha.) including said generator is said generator itself.
7. A method according to claim 6 wherein said first function
f(.alpha.) including said generator is said generator itself.
8. A method according to claim 1 wherein said first function
including said generator f(.alpha.) includes the public key p.sub.B
of said second correspondent.
9. A method according to claim 1 wherein said second function
including said generator f'.alpha. includes the public key p.sub.A
of said first correspondent.
10. A method according to claim 1 wherein said signature generated
by a respective one of the correspondents combine the random
integer, exponentiated function and private key of that one
correspondent.
11. A method according to claim 10 wherein said signature of
correspondent A is of the form x-r.sub.A a mod (p-1).
12. A method according to claim 10 wherein said signature of
correspondent A is of the form x+a(p.sub.B).sup.x mod (p-1).
13. A method according to claim 10 wherein said signature of
correspondent A is of the form
x.GAMMA..sub.x.sup..sup.1-(.GAMMA..sub.A).sup..GAMMA..su-
p..sub.x1a mod (p-1) where x.sub.1 is a second random integer
selected by A and r.sub.x.sup..sup.1=.alpha..sup.x.sup..sub.1.
14. A method according to claim 10 wherein said signature of
correspondent B is of the form y.sub.B-r.sub.Bb mod (p-1).
15. A method according to claim 10 wherein said signature of
correspondent B is of the form y+b(p.sub.A).sup.y mod (p-1).
16. A method according to claim 10 wherein said signature of
correspondent B is of the form
y.GAMMA..sub.y.sup..sup.1-(.GAMMA..sub.B).sup..GAMMA..su-
p..sub.y-b mod (p-1) where y.sub.1 is a second integer selected by
correspondent B and
.GAMMA..sub.y.sup..sup.1=.alpha..sup.y.sup..sub.1
17. A method according to claim 11 wherein said correspondent A
selects a second integer x.sub.1 and forwards
.GAMMA..sub.A.sub..sub.1 to correspondent B where
.GAMMA..sub.A.sub..sub.1=.alpha..sup.x.sup..sub.1 and said
correspondent B selects a second random integer y.sub.1 and sends
.GAMMA..sub.B.sub..sub.1 to correspondent A, where
.GAMMA..sub.B.sub..sub.1=.alpha..sup.y.sup..sub.1 each of said
correspondents computing a pair of keys k.sub.1,k.sub.2. equivalent
to .alpha..sup.a and .alpha..sup.x.sup..sub.1.sup.y.sup..sub.s
respectively, said session key K being generated by XORing k.sub.1
and k.sub.2.
Description
[0001] This application is a continuation of U.S. divisional patent
application Ser. No. 09/558,256 filed on Apr. 25, 2000, which is a
divisional of U.S. patent application Ser. No. 08/966,766 filed on
Nov. 7, 1997 which is a file wrapper continuation of U.S. patent
application Ser. No. 08/426,712 filed on Apr. 12, 1995.
[0002] The present invention relates to key agreement protocols for
transfer and authentication of encryption keys.
[0003] To 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.
[0004] In 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.
[0005] Public 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.
[0006] Key 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.
[0007] There are various prior proposals which claim to provide
implicit key authentication.
[0008] Examples include the Nyberg-Rueppel one-pass protocol and
the Matsumoto-Takashima-Imai (MTI) and the Goss and Yacobi two-pass
protocols for key agreement.
[0009] The 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.
[0010] For example, the MTI/AO key agreement protocol establishes a
shared secret K, known to the two correspondents, in the following
manner:
[0011] 1. During initial, one-time setup, key generation and
publication is undertaken by selecting and publishing an
appropriate system prime p and generator .alpha. of the
multiplicative group Z.sup.-.sub.p, that is, .alpha.s
Z.sup.-.sub.p; in a manner guaranteeing authenticity. Correspondent
A selects as a long-term private key a random integer
"a",1<a<p-1, and computes a long-term public key
Z.sub.A=.alpha..sup.z mod p. Correspondent B generates analogous
keys b, z.sub.B. Correspondents A and B have access to
authenticated copies of each other's long-term public key.
[0012] 2. The protocol requires the exchange of the following
messages.
A.fwdarw.B: .alpha..sup.x mod p (1)
A.fwdarw.B: .alpha..sup.y mod p (2)
[0013] where x and y are integers selected by correspondents A and
B respectively.
[0014] The values of x and y remain secure during such
transmissions as it is impractical to determine the exponent even
when the value of .alpha. and the exponentiation is known provided
of course that p is chosen sufficiently large.
[0015] 3. To implement the protocol the following steps are
performed each time a shared key is required.
[0016] (a) A chooses a random integer x, 1.ltoreq.x.ltoreq.p-2, and
sends B message (1) i.e. .alpha..sup.x mod p.
[0017] (b) B chooses a random integer y, 1.ltoreq.y.ltoreq.p-2, and
sends A message (2) i.e. .alpha..sup.y mod p.
[0018] (c) A computes the key K=(.alpha..sup.y).sup.az.sub.B.sup.x
mod p.
[0019] (d) B computes the key K=(.alpha..sup.x).sup.bz.sub.A.sup.y
mod p.
[0020] (e) Both share the key K=.alpha..sup.bx+ay.
[0021] In 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 a 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.
[0022] As such this and related protocols have been considered
satisfactory for key establishment and resistant to conventional
eavesdropping or man-in-the middle attacks.
[0023] In some circumstances it may be advantageous for an
adversary to mislead one correspondent as to the true identity of
the other correspondent.
[0024] In 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 5 may be useful.
[0025] A 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.
[0026] It is therefore an object of the present invention to
provide a protocol in which the above disadvantages are obviated or
mitigated.
[0027] According therefore to the present invention there is
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.sub.A,p.sub.B derived from a generator .alpha. and respective
ones of said private keys a,b, said method including the steps
of
[0028] i) a first of said correspondents A selecting a first random
integer x and exponentiating a function f(.alpha.) including said
generator to a power g(x) to provide a first exponentiated function
f(.alpha.).sup.g(x);
[0029] ii) said first correspondent A generating a first signature
s.sub.A from said random integer x and said first exponentiated
function f(.alpha.).sup.g(x);
[0030] iii) said first correspondent A forwarding to a second
correspondent B a message including said first exponentiated
function f(.alpha.).sup.g(x) and the signature s.sub.A;
[0031] iv) said correspondent B selecting a second random integer y
and exponentiating a function f'(.alpha.) including said generator
to a power g(y) to provide a second exponentiated function
f'(.alpha.).sup.g(y) and a signature s.sub.B obtained from said
second integer y and said second exponentiated function
f'(.alpha.).sup.g(y);
[0032] v) said second correspondent B forwarding a message to said
first correspondent A including said second exponentiated function
f'(.alpha.).sup.g(y) and said signature s.sub.B.
[0033] 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;
[0034] 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.
[0035] Thus although the interloper E can substitute her public key
p.sub.E=.alpha..sup.ac in the transmission as part of the message,
B will use p.sub.E rather than p.sub.A when authenticating the
message. Accordingly the computed and transmitted values of the
exponential functions will not correspond.
[0036] Embodiments of the invention will now be described by way of
example only with reference to the accompanying drawings in
which:
[0037] FIG. 1 is a schematic representation of a data communication
system.
[0038] FIG. 2 is a flow chart illustrating the steps of
authenticating the correspondents shown in FIG. 1 according to a
first protocol.
[0039] Referring therefore to FIG. 1, 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.
[0040] In operation, a message generated by correspondent A, 10, is
encrypted by the unit 16 with the key 20 and transmitted as
ciphertext over channel 14 to the unit 18.
[0041] The 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 the correspondent
12 will be that sent by the correspondent 10.
[0042] In order for the system shown in FIG. 1 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.
[0043] The system parameters for these protocols are a prime number
p and a generator .alpha..sup.a of the multiplicative group
Z.sup.-.sub.p. Correspondent A has private key a and public key
p.sub.A=.alpha..sup.a. Correspondent B has private key b and b
public key p.sub.B=.alpha..sup.b. In the protocol exemplified
below, text.sub.A refers to a string of information that identifies
party A. If the other correspondent B possesses an authentic copy
of correspondent A's public key, then text.sub.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.
[0044] In 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.ltoreq.e.ltoreq.p-2, computes
p.sub.E=(p.sub.A).sup.e=.alpha..sup.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.sub.E for text.sub.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 her 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.
[0045] To avoid interloper E convincing B that he is communicating
with E, the following protocol is adapted, as exemplified in FIG.
2.
[0046] The 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.
[0047] The system parameters for this protocol are a prime number p
and a generator .alpha. of the multiplicative group
Z.sup..alpha..sub.p. User A has private key a and public key
p.sub.A=.alpha..sup.a. User B has private key b and public key
p.sub.B=.alpha..sup.b.
[0048] First Protocol
[0049] 1. A picks a random integer x,1.ltoreq.x.ltoreq.p-2, and
computes a value r.sub.A=.alpha..sup.x and a signature
s.sub.A=x-r.sub.Aa mod (p-1). A sends {r.sub.A,s.sub.A,text.sub.A}
to B.
[0050] 2. B picks a random integer y,1.ltoreq.y.ltoreq.p-2, and
computes a value r.sub.B=.alpha..sup.y and a signature
s.sub.B=y-r.sub.Bb mod (p-1). B sends {r.sub.B, s.sub.B,
text.sub.B} to A.
[0051] 3. A computes .alpha..sup.s.sup..sub.B
(p.sub.B).sup.r.sup..sub.B and verifies that this is equal to
r.sub.B. A computes the session key
K=(r.sub.B).sup.x=.alpha..sup.xy.
[0052] 4. B computes
.alpha..sup.s.sup..sub.A(p.sub.A).sup.r.sup..sub.A and verifies
that this is equal to r.sub.A. B computes the session key
K=(r.sub.A).sup.y=.alpha..sup.xy.
[0053] Should E replace text.sub.A with text.sub.E, B will compute
.alpha..sup.s.sup..sub.B(p.sub.E).sup.r.sup..sub.A which will not
correspond with the transmitted value of r.sub.A. B will thus be
alerted to the interloper E and will proceed to initiate another
session key.
[0054] One draw back 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.
[0055] Modified First Protocol.
[0056] In step 1, A also sends .alpha..sup.x.sup..sub.1 to B, where
x.sub.1 is a second random integer generated by A. Similarly, in
step 2 above, B also sends .alpha..sup.y.sup..sub.1 to A, where
y.sub.1 is a random integer. A and B now compute the key
K=.alpha..sup.xy{circle over
(+)}.alpha..sup.x.sup..sub.1.sup.y.sup..sub.1.
[0057] Another 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.sub.A=x-r.sub.Aa {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.
[0058] Second Protocol
[0059] A second protocol set out below addresses these two
drawbacks.
[0060] 1. A picks a random integer X,1.ltoreq.x.ltoreq.p-2, and
computes (p.sub.B).sup.x, .alpha..sup.x and a signature
s.sub.A=x.div.a(p.sub.B).s- up.x {mod (p-1)}. A sends
{.alpha..sup.x,s.sub.A,text.sub.A} to B.
[0061] 2. B picks a random integer y,1.ltoreq.y.ltoreq.p-2, and
computes (p.sub.A).sup.y,.alpha..sup.y and a signature
s.sub.B=Y.div.b(p.sub.A).su- p.y {mod (p-1)}. B sends
{.alpha..sup.Y,s.sub.B,text.sub.B} to A.
[0062] 3. A computes (.alpha..sup.y).sup.a and verifies that
.alpha..sup.s.sup..sub.B(p.sub.B).sup.-.alpha..sup..sup.ay=.alpha..sup.y.
A then computes session key=.alpha..sup.ay(p.sub.B).sup.x.
[0063] 4. B computes (.alpha..sup.x).sup.b and verifies that
.alpha..sup.s.sup..sub.A
(p.sub.A).sup.-.alpha..sup..sup.bx=.alpha..sup.x- . A then computes
session key K=.alpha..sup.bx(p.sub.A).sup.y.
[0064] The second protocol improves upon the first protocol in the
sense that if 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 he uses it in step 1 of
the protocol.
[0065] If 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.
[0066] The quantity s.sub.A serves as A's signature on the value
.alpha..sup.x. This signature has the novel property that it can
only be verified by part B. This idea can be generalized to all
ElGamal-like signatures schemes.
[0067] A further protocol is available for parties A and B to
establish a session key K.
[0068] Third Protocol
[0069] The system parameters for this protocol are a prime number p
and a generator .alpha. for the multiplicative group
Z.sup..alpha..sub.p. User A has private key a and public key
p.sub.A=.alpha..sup.a. User B has private key b and public key
p.sub.B=.alpha..sup.b.
[0070] 1. A picks two random integers
x,x.sub.1,1.ltoreq.x,x.sub.1.ltoreq.- p-2, and computes
.gamma.x.sub.1,=.alpha..sup.x1,.GAMMA..sub.A=.alpha..sup- .x and
(.GAMMA..sub.A).sup..GAMMA.x1, then computes a signature
s.sub.A=x.GAMMA..sub.x.sub..sub.1.multidot.(.gamma..sub.A).sup..GAMMA..su-
p..sub.x1 a mod (p-1). A sends
{.GAMMA..sub.A,s.sub.A,.alpha..sup.x.sup..s- ub.1, text.sub.A} to
B.
[0071] 2. B picks two random integers
y,y.sub.1,1.ltoreq.y,y.sub.1.ltoreq.- p-2, and computes
.GAMMA..sub.y1=.alpha..sup.y1,.DELTA..sub.B=.alpha..sup.- y and
(.GAMMA..sub.B).sup..GAMMA..sup..sup.y1, then computes a signature
s.sub.B=yr.sub.y1.multidot.(.GAMMA..sub.B).sup..GAMMA..sup..sup.y1
b {mod (p-1)}. B sends {.gamma..sub.B,s.sub.B,
.alpha..sup.y1,text.sub.B} to A.
[0072] 3. A computes
.alpha..sup.s.sup..sub.B(p.sub.B).sup..GAMMA..sup..su-
b.B).sup..sup..GAMMA..sup..sub.y1 and verifies that is equal to
(.GAMMA..sub.B).sup..GAMMA..sup..sub.y1.
[0073] A computes session key
K=(.alpha..sup.y1).sup.x.sup..sub.1=.alpha..-
sup.x.sup..sub.1.sup.y1.
[0074] 4. B computes
.alpha..sup.s.sup..sub.A(p.sub.A).sup.(.GAMMA..sup..s-
ub.A.sup.).sup..sup..GAMMA..sup..sub.x1 and verifies that this is
equal to (.GAMMA..sub.A).sup..GAMMA..sup..sub.x1.
[0075] B computes session key
K=(.alpha..sup.x.sup..sub.1).sup.y1=.alpha..-
sup.x.sup..sub.1.sup.y.sup..sub.1
[0076] In these protocols, (.GAMMA..sub.A,s.sub.A) can be thought
of as the signature of .GAMMA..sub.x.sup..sup.1, with the property
that only A can sign the message .GAMMA..sub.x.sup..sup.1.
[0077] Key Transport Protocol
[0078] The 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.
[0079] 1. A picks a random integer x, 1.ltoreq.x.ltoreq.p-2, and
computes r.sub.A=.alpha..sup.x and a signature s.sub.A=x-r.sub.Aa
{mod (p-1)}. A computes session key K=(p.sub.B).sup.x and sends
{r.sub.A,s.sub.A,text.su- b.A} to B.
[0080] 2. B computes
.alpha..sup.s.sup..sub.A(p.sub.A).sup.r.sup..sub.A and verifies
that this quantity is equal to r.sub.A. B computes session key
K=(r.sub.A).sup.b.
[0081] All 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.).
[0082] This 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{circle over (-)}K, i.e. k XOR'd with K.
[0083] All the protocols discussed above have been described in the
setting of the multiplicative group Z.sup..alpha..sub.p. 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 field (in
particular the elliptic curve defined over a finite field. In each
case an appropriate generator .alpha. will be used to define the
public keys.
[0084] The 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 a (or analogous parameters if a
group other than Z.sup..alpha..sub.p is used).
* * * * *