PATN Patent Bibliographic Information
WKU Patent Number: 05276737
SRC Series Code: 7
APN Application Number: 8709351
APT Application Type: 1
ART Art Unit: 222
APD Application Filing Date: 19920420
TTL Title of Invention: Fair cryptosystems and methods
of use
ISD Issue Date: 19940104
NCL Number of Claims: 18
ECL Exemplary Claim Number: 1
EXP Primary Examiner: Swann; Tod R.
NDR Number of Drawings Sheets: 1
NFG Number of Figures: 2
INVT Inventor Information
NAM Inventor Name: Micali; Silvio
STR Inventor Street: 224 Upland Rd.
CTY Inventor City: Cambridge
STA Inventor State: MA
ZIP Inventor Zip Code: 02140
CLAS Classification
OCL Original U.S. Classification: 380 30
XCL Cross Reference Classification: 380 23
XCL Cross Reference Classification: 380 25
EDF International Classification Edition Field: 5
ICL International Classification: H04K 100
FSC Field of Search Class: 380
FSS Field of Search Subclass: 23;24;25;28;29;30
UREF U.S. Patent Reference
PNO Patent Number: 4375579
ISD Issue Date: 19830300
NAM Patentee Name: Davida et al.
OCL Original U.S. Classification: 380 28
UREF U.S. Patent Reference
PNO Patent Number: 4933970
ISD Issue Date: 19900600
NAM Patentee Name: Shamir
OCL Original U.S. Classification: 380 30
UREF U.S. Patent Reference
PNO Patent Number: 5005200
ISD Issue Date: 19910400
NAM Patentee Name: Fischer
OCL Original U.S. Classification: 380 30
UREF U.S. Patent Reference
PNO Patent Number: 5018196
ISD Issue Date: 19910500
NAM Patentee Name: Takaragi et al.
OCL Original U.S. Classification: 380 30
UREF U.S. Patent Reference
PNO Patent Number: 5136643
ISD Issue Date: 19920800
NAM Patentee Name: Fischer
OCL Original U.S. Classification: 380 23
UREF U.S. Patent Reference
PNO Patent Number: 5150411
ISD Issue Date: 19920900
NAM Patentee Name: Maurer
OCL Original U.S. Classification: 380 30
LREP Legal Information
FR2 Combined Principal Attorney(s): Judson; David H.
ABST Abstract
A method, using a public-key cryptosystem, for enabling a predetermined
entity to monitor communications of users suspected of unlawful activities
while protecting the privacy of law-abiding users, wherein each user is
assigned a pair of matching secret and public keys. According to the
method, each user's secret key is broken into shares. Then, each user
provides a plurality of "trustees" pieces of information. The pieces of
information provided to each trustee enable that trustee to verify that
such information includes a "share" of a secret key of some given public
key. Each trustee can verify that the pieces of information provided
include a share of the secret key without interaction with any other
trustee or by sending messages to the user. Upon a predetermined request
or condition, e.g., a court order authorizing the entity to monitor the
communications of a user suspected of unlawful activity, the trustees
reveal to the entity the shares of the secret key of such user. This
enables the entity to reconstruct the secret key and monitor the suspect
user's communications.
BSUM Brief Summary
TECHNICAL FIELD
The present invention relates generally to cryptosystems and more
particularly to methods for enabling a given entity to monitor
communications of users suspected of unlawful activities while protecting
the privacy of law-abiding users.
BACKGROUND OF THE INVENTION
In a single-key cryptosystem a common secret key is used both to encrypt
and decrypt messages. Thus only two parties who have safely exchanged such
a key beforehand can use these systems for private communication. This
severely limits the applicability of single-key systems.
In a double-key cryptosystem, the process of encrypting and decrypting is
instead governed by different keys. In essence, one comes up with a pair
of matching encryption and decryption keys. What is encrypted using a
given encryption key can only be decrypted using the corresponding
decryption key. Moreover, the encryption key does not "betray" its
matching decryption key. That is, knowledge of the encryption key does not
help to find out the value of the decryption key. The advantage of
double-key systems is that they can allow two parties who have never
safely exchanged any key to privately communicate over an insecure
communication line (i.e., one that may be tapped by an adversary). They do
this by executing an on-line, private communication protocol.
In particular, Party A alerts Party B that he wants to talk to him
privately. Party B then computes a pair of matching encryption and
decryption keys (E.sub.B, D.sub.B). B then sends A key E.sub.B. Party A
now encrypts his message m, obtaining the ciphertext c=E.sub.B (m), and
sends c to B over the insecure channel. B decrypts the ciphertext by
computing m=D.sub.B (c). If an adversary eavesdrops all communication
between A and B, he will then hear both B's encryption key, E.sub.B, and
A's ciphertext, c. However, since the adversary does not know B's
decryption key, D.sub.B, he cannot compute m from c.
The utility of the above protocol is still quite limited since it suffers
from two drawbacks. First, for A to send a private message to B it is
necessary also that B send a message to A, at least the first time. In
some situations this is a real disadvantage. Moreover, A has no guarantee
(since the line is insecure anyway) that the received string D.sub.B
really is B's encryption key. Indeed, it may be a key sent by an
adversary, who will then understand the subsequent, encrypted
transmission.
An ordinary public-key cryptosystem ("PKC") solves both difficulties and
greatly facilitates communication. Such a system essentially consists of
using a double-key system in conjunction with a proper key management
center. Each user X comes up with a pair of matching encryption and
decryption keys (E.sub.X, D.sub.X) of a double-key system. He keeps
D.sub.X for himself and gives E.sub.X to the key management center. The
center is responsible for updating and publicizing a directory of correct
public keys for each user, that is, a correct list of entries of the type
(X, E.sub.X). For instance, upon receiving the request from X to have
E.sub.X as his public key, the center properly checks X's identity, and
(digitally) signs the pair (X, E.sub.X), together with the current date if
every encryption key has a limited validity. The center publicizes E.sub.X
by distributing the signed information to all users in the system. This
way, without any interaction, users can send each other private messages
via their public, encryption key that they can look up in the directory
published by the center. The identity problem is also solved, since the
center's signature of the pair (X, E.sub.X) guarantees that the pair has
been distributed by the center, which has already checked X's identity.
The convenience of a PKC depends on the key management center. Because
setting up such a center on a grand scale requires a great deal of effort,
the precise protocols to be followed must be properly chosen. Moreover,
public-key cryptography has certain disadvantages. A main disadvantage is
that any such system can be abused, for example, by terrorists and
criminal organizations who can use their own PKC (without knowledge of the
authorities) and thus conduct their illegal business with great secrecy
and yet with extreme convenience.
It would therefore be desirable to prevent any abuse of a public key
cryptosystem while maintaining all of its lawful advantages.
BRIEF SUMMARY OF THE INVENTION
It is an object of the present invention to provide methods for enabling a
given entity, such as the government, to monitor communications of users
suspected of unlawful activities while at the same time protecting the
privacy of law-abiding users.
It is a further object of the invention to provide such methods using
either public or private key cryptosystems.
It is a still further object of the invention to provide so-called "fair"
cryptosystems wherein an entity can monitor communications of suspect
users only upon predetermined occurrences, e.g., the obtaining of a court
order.
It is another object to describe methods of constructing fair cryptosystems
for use in such communications techniques.
In one embodiment, these and other objects of the invention are provided in
a method, using a public-key cryptosystem, for enabling a predetermined
entity to monitor communications of users suspected of unlawful activities
while protecting the privacy of law-abiding users, wherein each user is
assigned a pair of matching secret and public keys. According to the
method, each user's secret key is broken into shares. Then, each user
provides a plurality of "trustees" pieces of information. The pieces of
information provided to each trustee enable that trustee to verify that
such information includes a "share" of a secret key of some given public
key. Further, each trustee can verify that the pieces of information
provided include a share of the secret key without interaction with any
other trustee or by sending messages to the user. Upon a predetermined
request or condition, e.g., a court order authorizing the entity to
monitor the communications of a user suspected of unlawful activity, the
trustees reveal to the entity the shares of the secret key of such user to
enable the entity to reconstruct the secret key and monitor the suspect
user's communications.
The method can be carried out whether or not the identity of the suspect
user is known to the trustees, and even if less than all of the shares of
the suspect user's secret key are required to be revealed in order to
reconstruct the secret key. The method is robust enough to be effective if
a given minority of trustees have been compromised and cannot be trusted
to cooperate with the entity. In addition, the suspect user's activities
are characterized as unlawful if the entity, after reconstructing or
having tried to reconstruct the secret key, is still unable to monitor the
suspect user's communications.
According to another more generalized aspect of the invention, a method is
described for using a public-key cryptosystem for enabling a predetermined
entity to monitor communications of users suspected of unlawful activities
while protecting the privacy of law-abiding users. The method comprises
the step of "verifiably secret sharing" each user's secret key with a
plurality of trustees so that each trustee can verify that the share
received is part of a secret key of some public key.
The foregoing has outlined some of the more pertinent objects of the
present invention. These objects should be construed to be merely
illustrative of some of the more prominent features and applications of
the invention. Many other beneficial results can be attained by applying
the disclosed invention in a different manner or modifying the invention
as will be described. Accordingly, other objects and a fuller
understanding of the invention may be had by referring to the following
Detailed Description of the preferred embodiment.
DRWD Drawing Description
BRIEF DESCRIPTION OF THE DRAWINGS
For a more complete understanding of the present invention and the
advantages thereof, reference should be made to the following Detailed
Description taken in connection with the accompanying drawings in which:
FIG. 1 is a simplified diagram of a communications system over which a
government entity desires to monitor communications of users suspected of
unlawful activities;
FIG. 2 is a block diagram of a preferred hierarchy of entities that may use
the methods of the present invention to monitor communications of users
suspected of unlawful activities.
DETD Detail Description
DETAILED DESCRIPTION
FIG. 1 represents a simple communications system 10 comprising a telephone
network connected between a calling station 12 and a called station 14.
One or more local central offices or telephone switches 16 connect
telephone signals over the network in a well-known fashion. Referring now
also to FIG. 2, assume that a government entity, such as local law
enforcement agency 18, desires to monitor communications to and/or from
calling station 12 because the user of such calling station is suspected
of unlawful activity. Assume further that the user of the calling station
12 communicates using a PKC. Following accepted legal practices, the
agency 18 obtains a court order from court 20 to privately monitor the
line 15. According to the present invention, the agency's is able to
monitor the line 15 while at the same time the privacy rights of other
law-abiding users of the network are maintained. This is accomplished as
will be described by requiring that each user "secret share" the user's
secret key (of the PKC) with a plurality of trustees 22a . . . 22n.
According to the invention, a "fair" PKC is a special type of public-key
cryptosystem. Every user can still choose his own keys and keep secret his
private one; nonetheless, a special agreed-upon party (e.g., the
government), and solely this party, under the proper circumstances
envisaged by the law (e.g., a court order), and solely under these
circumstances, is authorized to monitor all messages sent to a specific
user. A fair PKC improves the security of the existing communication
systems (e.g., the telephone service 10) while remaining within the
constraints of accepted legal procedures.
In one embodiment, fair PKC's are constructed in the following general way.
Referring now to FIGS. 1-2, it is assumed that there are five (5) trustees
22a . . . 22e and that the government desires, upon receiving a court
order, to monitor the telephone communications to or from the calling
station 12. Although the above-description is specific, it should be
appreciated that users of the communications system and trustees may be
people or computing devices. It is preferable that the trustees are chosen
to be trustworthy. For instance, they may be judges (or computers
controlled by them), or computers specially set up for this purpose. The
trustees, together with the individual users, play a crucial role in
deciding which encryption keys will be published in the system.
Each user independently chooses his own public and secret keys according to
a given double-key system (for instance, the public key consists of the
product of two primes, and the secret key one of these two primes). Since
the user has chosen both of his keys, he can be sure of their "quality"
and of the privacy of his decryption key. He then breaks his secret
decryption key into five special "pieces" (i.e., he computes from his
decryption key 5 special strings/numbers) possessing the following
properties:
(1) The private key can be reconstructed given knowledge of all five,
special pieces;
(2) The private key cannot be guessed at all if one only knows (any) 4, or
less, of the special pieces;
(3) For i-1, . . . 5, the i-th special piece can be individually verified
to be correct.
Given all 5 special pieces or "shares", one can verify that they are
correct by checking that they indeed yield the private decryption key.
According to one feature of the invention, property (3) insures that each
special piece can be verified to be correct (i.e., that together with the
other 4 special pieces it yields the private key) individually, i.e.,
without knowing the secret key at all and without knowing the value of any
of the other special pieces.
The user then privately (e.g., in encrypted form) gives trustee 22i his own
public key and the i-th piece of its associated secret key. Each trustee
22 individually inspects his received piece, and, if it is correct,
approves the public key (e.g. signs it) and safely stores the piece
relative to it. These approvals are given to a key management center 24,
either directly by the trustees, or (possibly in a single message) by the
individual user who collects them from the trustees. The center 24, which
may or may not coincide with the government, itself approves (e.g., signs)
any public key that is approved by all trustees. These center-approved
keys are the public keys of the fair PKC and they are distributed and used
for private communication as in an ordinary PKC.
Because the special pieces of each decryption key are privately given to
the trustees, an adversary who taps the communication line of two users
possesses the same information as in the underlying, ordinary PKC. Thus if
the underlying PKC is secure, so is the fair PKC. Moreover, even if the
adversary were one of the trustees himself, or even a cooperating
collection of any four out of five of the trustees, property (2) insures
that the adversary would still have the same information as in the
ordinary PKC. Because the possibility that an adversary corrupts five out
of five judges is absolutely remote, the security of the resulting fair
PKC is the same as in the underlying PKC.
When presented with a court order, for example, the trustees 22 reveal to
the government 20 the pieces of a given decryption key in their
possession. According to the invention, the trustees may or may not be
aware of the identity of the user who possesses the given decryption key.
This provides additional security against "compromised" trustees who might
otherwise tip off the suspect user once a request for that user's
decryption key share is received by the trustee.
Upon receiving the shares, the government reconstructs the given decryption
key. By property (3), each trustee previously verified whether he was
given a correct special piece of a given decryption key. Moreover, every
public key was authorized by the key management center 24 only if it was
approved by all trustees 22. Thus, the government is guaranteed that, in
case of a court order, it will be given all special pieces of any
decryption key. By property (1), this is a guarantee that the government
will be able to reconstruct any given decryption key if necessary to
monitor communications over the network.
Several types of fair PKC's are now described in more detail.
Diffie and Hellman's PKC
The Diffie and Hellman public-key cryptosystem is known and is readily
transformed into a fair PKC by the present invention. In the Diffie and
Hellman scheme, each pair of users X and Y succeeds, without any
interaction, in agreeing upon a common, secret key S.sub.xy to be used as
a conventional single-key cryptosystem. In the ordinary Diffie-Hellman
PKC, there are a prime p and a generator (or high-order element) g common
to all users. User X secretly selects a random integer Sx in the interval
[1, p-1] as his private key and publicly announces the integer Px=g.sup.Sx
mod p as his public key. Another user, Y, will similarly select Sy as his
private key and announce Py=g.sup.Sy mod p as his public key. The value of
this key is determined as S.sub.xy =g.sup.SxSy mod p. User X computes Sxy
by raising Y's public key to his private key mod pX, and user Y by raising
X's public key to his secret key mod p. In fact:
EQU (g.sup.Sx).sup.Sy =g.sup.SxSy =Sxy=g.sup.SySx =(g.sup.Sy).sup.Sx mod p.
While it is easy, given g, p and x, to compute y=g.sup.x mod p, no
efficient algorithm is known for computing, given y and p, x such that
g.sup.x =y mod p when g has high enough order. This is the discrete
logarithm problem. This problem has been used as the basis of security in
many cryptosystems. The Diffie and Hellman's PKC is transformed into a
fair one in the following manner.
Each user X randomly chooses 5 integers Sx1, . . . Sx5 in the interval [1,
p-1] and lets Sx be their sum mod p. It should be understood that all
following operations are modulo p. User X then computes the numbers:
EQU t1=g.sup.Sx 1. . . , t5=g.sup.Sx5 and Px=g.sup.Sx.
Px will be User X's public key and Sx his private key. The ti's will be
referred to as the public pieces of Px, and the Sxi's as the private
pieces. It should be noted that the product of the public pieces equals
the public key Px. In fact:
EQU t1. . . t5=g.sup.Sx1. . . g.sup.Sx5 =g.sup.(Sx1+. . .+Sx5) =g.sup.Sx.
Let T1, . . . T5 be the five trustees. User X now gives Px, the public
pieces and Sx1 to trustee T1, Px, the public pieces and Sx2 to trustee T2,
and so on. Piece Sxi is privately given to trustee Ti. Upon receiving
public and private pieces ti and Sxi, trustee Ti verifies whether
g.sup.Sxi =Ti. If so, the trustee stores the pair (Px, Sxi), signs the
sequence (Px,t1,t2,t3,t4,t5) and gives the signed sequence to the key
management center 24 (or to user X, who will then give all of the signed
public pieces at once to the key management center). Upon receiving all
the signed sequences relative to a given public key Px, the key management
center verifies that these sequences contain the same subsequence of
public pieces t1 . . . t5 and that the product of the public pieces indeed
equals Px. If so, center 24 approves Px as a public key and distributes it
as in the original scheme (e.g., signs it and gives it to user X). The
encryption and decryption instructions for any pair of users X and Y are
exactly as in the Diffie and Hellman scheme (i.e., with common, secret key
Sxy).
This way of proceeding matches the previously-described way of constructing
a fair PKC. A still fair version of the Diffie-Hellman scheme can be
obtained in a simpler manner by having the user give to each trustee Ti
just the public piece ti and its corresponding private piece Sxi, and have
the user give the key management center the public key Px. The center will
approve Px only if it receives all public pieces, signed by the proper
trustee, and the product of these public pieces equals Px. In this way,
trustee Ti can verify that Sxi is the discrete logarithm of public piece
ti. Such trustee cannot quite verify that Sxi is a legitimate share of Px
since the trustee has not seen Px or the other public pieces. Nonetheless,
the result is a fair PKC based on the Diffie-Hellman scheme because
properties (1)-(3) described above are still satisfied.
Either one of the above-described fair PKC has the same degree of privacy
of communication offered by the underlying Diffie-Hellman scheme. In fact,
the validation of a public key does not compromise the corresponding
private key. Each trustee Ti receives, as a special piece, the discrete
logarithm, Sxi, of a random number, ti. This information is clearly
irrelevant for computing the discrete logarithm of Px. The same is
actually true for any 4 of the trustees taken together, since any four
special pieces are independent of the private decryption key Sx. Also the
key management center does not possess any information relevant to the
private key; i.e., the discrete logarithm of Px. All the center has are
the public pieces respectively signed by the trustees. The public pieces
simply are 5 random numbers whose product is Px. This type of information
is irrelevant for computing the discrete logarithm of Px; in fact, any one
could choose four integers at random and setting the fifth to be Px
divided by the product of the first four. The result would be integral
because division is modulo p. As for a trustee's signature, this just
represents the promise that someone else has a secret piece.
Even the information in the hands of the center together with any four of
the trustees is irrelevant for computing the private key Sx. Thus, not
only is the user guaranteed that the validation procedure will not betray
his private key, but he also knows that this procedure has been properly
followed because it is he himself that computes his own keys and the
pieces of his private one.
Second, if the key management center validates the public key Px, then its
private key is guaranteed to be reconstructable by the government in case
of a court order. In fact, the center receives all 5 public pieces of Px,
each signed by the proper trustee. These signatures testify that trustee
Ti possesses the discrete logarithm of public piece ti. Since the center
verifies that the product of the public pieces equals Px, it also knows
that the sum of the secret pieces in storage with the trustees equals the
discrete logarithm of Px; i.e, user X's private key. Thus the center knows
that, if a court order were issued requesting the private key of X, the
government is guaranteed to obtain the needed private key by summing the
values received by the trustees.
RSA Fair PKC
The following describes a fair PKC based on the known RSA function. In the
ordinary RSA PKC, the public key consists of an integer N product of two
primes and one exponent e (relatively prime with f(N), where F is Euler's
quotient function). No matter what the exponent, the private key may
always be chosen to be N's factorization. By way of brief background, the
RSA scheme has certain characteristics that derive from aspects of number
theory:
Fact 1. Let Z.sub.N.sup.* denote the multiplicative group of the integers
between 1 and N and relatively prime with N. If N is the product of two
primes N=pq (or two prime powers: N=p.sup.a p.sup.b), then
(1) a number s in Z.sub.N.sup.* is a square mod N if and only if it has
four distinct square-roots mod N: x, -x mod N, y, and -y mod N (i.e.,
x.sup.2 =y.sup.2 =s mod N). Moreover, from the greatest common divisor of
+-x+-y and N, one easily computes the factorization of N. Also;
(2) one in four of the numbers in Z.sub.N.sup.* is a square mod N.
Fact 2. Among the integers in Z.sub.N.sup.* is defined a function, the
Jacobi symbol, that evaluates easily to either 1 or -1. The Jacobi symbol
of x is denoted by (s/N). The Jacobi symbol is multiplicative; i.e.,
(x/N)(Y/N)=(xy/N). If N is the product of two primes N=pq (or two prime
powers: N=p.sup.a p.sup.b), the p and 1 are congruent to 3 mod 4. Then, if
+-x and +-y are the four square roots of a square mod N (s/N)=(-x/N)=+1
and (y/N)=(-y/N)=-1. Thus, because of Fact 1, if one is given a Jacobi
symbol 1 root and a Jacobi symbol -1 root of any square, he can easily
factor N.
With this background, the following describes how the RSA cryptosystem can
be made fair in a simple way. For simplicity again assume there are five
trustees and that all of them must collaborate to reconstruct a secret
key, while no four of them can even predict it. The RSA cryptosystem is
easily converted into a fair PKC by efficiently sharing with the trustee's
N's factorization. In particular, the trustees are privately provided
information that, perhaps together with other given common information,
enables one to reconstruct two (or more) square roots x and y (x different
from .+-.y mod N) of a common square mod N. The given common information
may be the -1 Jacobi symbol root of X.sup.2, which is equal to y.
A user chooses P and Q primes congruent to 3 mod 4, as his private key and
N=PQ as his public key. Then he chooses 5 Jacobi 1 integers X.sub.1,
X.sub.2, X.sub.3, X.sub.4 and X.sub.5 (preferably at random) in
Z.sub.N.sup.* and computes their product, X, and X.sub.i.sup.2 mod N for
all i=1, . . . , 5. The product of the last 5 squares, Z, is itself a
square. One square root of Z mod N is X, which has Jacobi symbol equal to
1 (since the Jacobi symbol is multiplicative). The user computes Y, one of
the Jacobi -1 roots mod N. X.sub.1, . . . X.sub.5 will be the public
pieces of public key N and the X.sub.i 's the private pieces. The user
gives trustee Ti private piece X.sub.i (and possibly the corresponding
public piece, all other public pieces and Px, depending on whether it is
desired that the verification of the shares so as to satisfy properties
(1)-(3) is performed by both trustees and the center, or the trustees
alone). Trustee Ti squares Xi mod N, gives the key management center his
signature of X.sub.i.sup.2, and stores X.sub.i.
The center first checks that (-1/N)=1, i.e., for all x: (x/N)=(-x/N). This
is partial evidence that N is of the right form. Upon receiving the valid
signature of the public pieces of N and the Jacobi -1 value Y from the
user, the center checks whether mod N the square of Y equals the product
of the five public pieces. If so, it checks, possibly with the help of the
user, that N is the product of two prime powers. If so, the center
approves N.
The reasoning behind the scheme is as follows. The trustees' signatures of
the X.sub.i.sup.2 's (mod N) guarantee the center that every trustee Ti
has stored a Jacobi symbol 1 root of X.sub.i.sup.2 mod N. Thus, in case of
a court order, all these Jacobi symbol 1 roots can be retrieved. Their
product, mod N, will also have Jacobi symbol 1, since this function is
multiplicative, and will be a root of X.sup.2 mod N. But since the center
has verified that Y.sup.2 =X.sup.2 mod N, one would have two roots X and Y
of a common square mod N. Moreover, Y is different from X since it has
different Jacobi symbol, and Y is also different from -x, since
(-x/N)=(s/N) because (a) (-1/N) has been checked to be 1 and (b) the
Jacobi symbol is multiplicative. Possession of such square roots, by Facts
1 and 2, is equivalent to having the factorization of N, provided that N
is product of at most two prime powers. This last property has also been
checked by the center before it has approved N.
Verification that N is the product of at most two prime powers can be
performed in various ways. For instance, the center and user can engage in
a zero-knowledge proof of this fact. Alternatively, the user may provide
the center with the square root mod N for roughly 1/4 of the integers in a
prescribed and random enough sequence of integers. For instance, such a
sequence could be determined by one-way hashing N to a short seed and then
expanding it into a longer sequence using a psuedo-random generator. If a
dishonest user has chosen his N to be the product of three or more prime
powers, then it would be foolish for him to hope that roughly 1/4of the
integers in the sequence are squares mod N. In fact, for his choice of N,
at most 1/8 of the integers have square roots mod N.
Variations
The above schemes can be modified in many ways. For instance, the proof
that N is product of two prime powers can be done by the trustees (in
collaboration with the user), who then inform the center of their
findings. Also, the scheme can be modified so that the cooperation of the
majority of the trustees is sufficient for reconstructing the secret key,
while any minority cannot gain any information about the secret key. Also,
as with all fair cryptosystems, one can arrange that when the government
asks a trustee for his piece of the secret key of a user, the trustee does
not learn about the identity of the user. The variations are discussed in
more detail below.
In particular, the schemes described above are robust in the sense that
some trustees, accidentally or maliciously, may reveal the shares in their
possession without compromising the security of the system. However, these
schemes rely on the fact that the trustees will collaborate during the
reconstruction stage. In fact, it was insisted that all of the shares
should be needed for recovering a secret key. This requirement may be
disadvantageous, either because some trustees may reveal to be
untrustworthy and refuse to give the government the key in their
possession, or because, despite all file backups, the trustee may have
genuinely lost the information in its possession. Whatever the reason, in
this circumstance the reconstruction of a secret key will be prevented.
This problem is also solved by the present invention.
By way of background, "secret sharing" (with parameters n,T,t) is a prior
cryptographic scheme consisting of two phases: in phase one a secret value
chosen by a distinguished person, the dealer, is put in safe storage with
n people or computers, the trustees, by giving each one of them a piece of
information. In phase two, when the trustees pool together the information
in their possession, the secret is recovered. Secret sharing has a major
disadvantage--it presupposes that the dealer gives the trustees correct
shares (pieces of information) about his secret value. "Verifiable Secret
Sharing" (VSS) solves this "honesty" problem. In a VSS scheme, each
trustee can verify that the share given to him is genuine without knowing
at all the shares of other trustees of the secret itself. Specifically,
the trustee can verify that, if T verified shares are revealed, the
original secret will be reconstructed, no matter what the dealer or
dishonest trustees might do.
The above-described fair PKC schemes are based on a properly structured,
non-interactive verifiable secret sharing scheme with parameters n=5, T=5
and t=4. According to the present invention, it may be desirable to have
different values of these parameters, e.g., n=5, T=3 and t=2. In such
case, any majority of the trustees can recover a secret key, while no
minority of trustees can predict it all. This is achieved as follows (and
be simply generalized to any desired values of n, T and t in which T>t).
Subset Method for the Diffie-Hellman Scheme
After choosing a secret key Sx in [1, p-1], user X computes his public key
Px=g.sup.Sx mod p (with all computations below being mod p). User X now
considers all triplets of numbers between 1 and 5: (1,2,3), (2,3,4) etc.
For each triplet (a,b,c), user X randomly chooses three integers S1abc, .
. . S3abc in the interval [1, p-1] so that their sum mod p equals Sx. Then
he computes the numbers:
EQU t1abc=g.sup.S1abc, t2abc=g.sup.S2abc, t3abc=g.sup.S3abc
The tiabc's will be referred to as public pieces of Px, and the Siabc's as
private pieces. Again, the product of the public pieces equals the public
key Px. In fact,
EQU t1abc.multidot..multidot.t2abc.multidot..multidot.t3abc=g.sup.S1abc
.multidot.g.sup.S 2abc.multidot.gS3abc=g (.sup.S1abc
+.multidot..multidot..multidot.+.sup.S3abc)=g.sup.Sx =Px
User X then gives trustee Ta t1abc and S1abc, trustee Tb t2abc and S2abc,
and trustee Tc t3abc and S3abc, always specifying the triplet in question.
Upon receiving these quantities, trustee Ta (all other trustees do
something similar) verifies that t1abc=g.sup.S1abc, signs the value (Px,
t1abc, (a,b,c)) and gives the signature to the management center.
The key management center, for each triple (a,b,c), retrieves the values
t1abc, t2abc and t3abc from the signed information received from trustees,
Ta, Tb and Tc. If the product of these three values equals Px and the
signatures are valid, the center approves Px as a public key.
The reason the scheme works, assuming that at most 2 trustees are
untrustworthy, is that all secret pieces of a triple are needed for
computing (or predicting) a secret key. Thus no secret key in the system
can be retrieved by any 2 trustees. On the other hand, after a court order
at least three trustees reveal all the secret pieces in their possession
about a given public key. The government then has all the necessary secret
pieces for at least one triple, and thus can compute easily the desired
secret key.
Alternatively, each trustee is replaced by a group of new trustees. For
instance, instead of a single trustee Ta, there may be three trustees:
Ta1, Ta2 and Ta3. Each of these trustees will receive and check the same
share of trustee Ta. In this way it is very unlikely that all three
trustees will refuse to surrender their copy of the first share.
After having insured that a few potentially malicious trustees cannot
prevent reconstruction of the key, there are still further security issues
to address, namely, a trustee--requested by a court order to surrender his
share of a given secret key--may alert the owner of that key that his
communications are about to be monitored. This problem is also solved by
the invention. A simple solution arises if the cryptosystem used by the
trustees possess certain algebraic properties. This is illustrated for the
Diffie-Hellman case, though the same result occurs for the RSA scheme. In
the following discussion, for simplicity it is assumed that all trustees
collaborate in the reconstruction of the secret key.
Oblivious and Fair Diffie-Hellman Scheme
Assume that all trustees use deterministic RSA for receiving private
messages. Thus, let Ni be the public RSA modulus of trustee Ti and ei his
encryption exponent (i.e., to send Ti a message m in encrypted form, one
would send m.sup.ei mod Ni).
User U prepares his public and secret key, respectively Px and Sx (thus
Px=g.sup.Sx mod p), as well as his public and secret pieces of the secret
key, respectively ti and Sxi's (thus Px=t1, t2 . . . t5 mod p and
ti=g.sup.Sxi mod p for all i). Then, the user gives to the key management
center Px, all of the ti's and the n values Ui=(Sxi).sup.3 mod Ni; i.e.,
he encrypts the i-th share with the public key of trustee Ti. Since the
center does not know the factorization of the Ni's, this is not useful
information to predict Sx, nor can the center verify that the decryption
of the n ciphertexts are proper shares of Sx. For this, the center will
seek the cooperation of the n trustees, but without informing them of the
identity of the user as will be described.
The center stores the values tj's and Uj's relative to user U and then
forwards Ui and ti to trustee Ti. If every trustee Ti verified that the
decryption of Ui is a proper private piece relative to ti, the center
approves Px.
Assume now that the judicial authority decides to monitor user U's
communications. To lawfully reconstruct secret key Sx without leaking to a
trustee the identity of the suspected user U, a judge (or another
authorized representative) randomly selects a number Ri mod Ni and
computes yi=Ri .sup.ei mod Ni. Then, he sends trustee Ti the value
zi=Ui-yi mod Ni, asking with a court order to compute and send back wi,
the ei-th root of zi mod Ni. Since zi is a random number mod Ni, no matter
what the value of Ui is, trustee Ti cannot guess the identity of the user
U in question. Moreover, since zi is the product of Ui and yi mod Ni, the
ei-th root of zi is the product mod Ni of the ei-th root of Ui (i.e., Sxi)
and the ei-th root of yi (i.e., Ri). Thus, upon receiving wi, the judge
divides it by yi mod Ni, thereby computing the desired Sxi. The product of
these Sxi's equals the desired Sx.
Further variation
In other variations of the invention, in case of a court order, the
government is only authorized to understand the messages concerning a
given user for a limited amount of time. The collective approval of all
trustees may stand for the government approval. Also, trustees need not
store their piece of the private key. The encryption of this piece--in the
trustee's public key and signed by the trustee--can be made part of the
user's public key. In this way, the public key carries the proof of its
own authenticity and verification. In the latter case it may be
advantageous to break the trustee's private keys into pieces.
If the user is an electronic device, such as an integrated circuit chip,
the basic process of key selection and public-key validation can be done
before the device leaves the factory. In this case, it may be advantageous
that a "copy" of the trustee can be maintained within the factory. A copy
of a trustee is a physically secure chip--one whose data cannot be
read--containing a copy of the trustee's decryption key. The trustee
(i.e., the party capable of giving the piece of a private key under a
court order) need not necessarily coincide with this device.
In another variation, it may be arranged that the trustees each a have
piece of the government private key, and that each user's private key is
encrypted with the public key of the government.
While the use of a fair PKC in a telecommunications network (and under the
authority of the government) has been described, such description is not
meant to be taken by way of limitation. A fair PKC can be used in private
organizations as well. For example, in a large organization where there is
a need for privacy, assume there is an established "superior" but not all
employees can be trusted since there are too many of them. The need for
privacy requires the use of encryption. Because not all employees can be
trusted, using a single encryption key for the whole company is
unacceptable, as is using a number of single-key cryptosystems (since this
would generate enormous key-distribution problems). Having each employee
use his own double-key system is also dangerous, since he or she might
conspire against the company with great secrecy, impunity and convenience.
In such application of a fair PKC, numerous advantages are obtained. First,
each employee is in charge of choosing his own keys. While enjoying the
advantages of a more distributed procedure, the organization retains
absolute control because the superior is guaranteed to be able to decrypt
every employee's communications when necessary. There is no need to change
keys when the superior changes because the trustees need not be changed.
The trustees' storage places need less surveillance, since only
compromising all of them will give an adversary any advantage.
For making fair a private key cryptosystem, but also for a PKC, it is
desirable that each trustee first deposits an encrypted version or
otherwise committed version of his share, so that, when he is asked to
reveal what his share was, he cannot change his mind about its value.
Also, it is desirable that the user gives his shares to the trustees
signed; such signatures can be relative to a different public key (if they
are digital signatures) or to the same new public key if the new key can
be used for signing as well. In this way, the share revealed by the
trustee clearly proves that it way originated. Better still, the user may
sign (with the trustee's key) the encryption of the share given to a
trustee, and the signature can be revealed together with the share. This
approach insures that one can both be certain that what was revealed was a
share approved by the user and also that the trustees and the user cannot
collaborate later on in changing its value.
It should be appreciated by those skilled in the art that the specific
embodiments disclosed above may be readily utilized as a basis for
modifying or designing other techniques and processes for carrying out the
same purposes of the present invention. It should also be realized by
those skilled in the art that such equivalent constructions do not depart
from the spirit and scope of the invention as set forth in the appended
claims.
CLMS Claims
STM Claim Statement: What is claimed is:
NUM Claim Number: 1.
1. A method, using a public-key cryptosystem, for enabling a predetermined
entity to monitor communications of users suspected of unlawful activities
while protecting the privacy of law-abiding users, wherein each user is
assigned a pair of matching secret and public keys, comprising the steps
of:
breaking each user's secret key into shares;
providing trustees pieces of information enabling the trustees to verify
that the pieces of information include shares of a secret key of some
given public key; and
upon a predetermined request, having the trustees reveal the shares of the
secret key of a user suspected of unlawful activity to enable the entity
to attempt reconstruction of the secret key for monitoring communications
to the suspect user.
NUM Claim Number: 2.
2. The method as described in claim 1 wherein the predetermined entity is a
government agency and the predetermined request is a court order.
NUM Claim Number: 3.
3. The method as described in claim 1 wherein the identity of the suspect
user is known to the trustees.
NUM Claim Number: 4.
4. The method as described in claim 1 wherein the identity of the suspect
user is unknown to the trustees.
NUM Claim Number: 5.
5. The method as described in claim 1 further including the step of:
characterizing the suspect user's activities as unlawful if the entity is
unable to monitor the suspect user's communications.
NUM Claim Number: 6.
6. The method as described in claim 1 wherein less than all of the shares
of the suspect user's secret key are required to be revealed in order to
reconstruct the secret key.
NUM Claim Number: 7.
7. The method as described in claim 1 wherein the shares are revealed to
the entity upon the predetermined request.
NUM Claim Number: 8.
8. The method as described in claim 1 wherein a given minority of trustees
are unable to reconstruct the secret key.
NUM Claim Number: 9.
9. The method as described in claim 1 wherein each trustee can verify that
the pieces of information provided include a share of the secret key
without interaction with any other trustee.
NUM Claim Number: 10.
10. A method, using a public-key cryptosystem, for enabling a predetermined
entity to monitor communications of users suspected of unlawful activities
while protecting the privacy of law-abiding users, wherein each user is
assigned a pair of matching secret and public keys, comprising the steps
of:
breaking each user's secret key into shares;
providing trustees pieces of information that include shares of a secret
key of some give public key; and
upon a predetermined request, having the trustees reveal the shares of the
secret key of a user suspected of unlawful activity to enable the entity
to reconstruct the secret key and monitor communications to the suspect
user.
NUM Claim Number: 11.
11. A method, using a public-key cryptosystem into a cryptosystem, for
enabling a predetermined entity to monitor communications of users
suspected of unlawful activities while protecting the privacy of
law-abiding users, comprising the step of:
verifying secret sharing each user's secret key with a plurality of
trustees so that each trustee can verify that the share received is part
of a secret key of some public key.
NUM Claim Number: 12.
12. The method as described in claim 11 further including the step of:
characterizing the suspect user's activities as unlawful if the entity is
unable to monitor the suspect user's communications.
NUM Claim Number: 13.
13. The method as described in claim 11 wherein a given minority of
trustees are unable to reconstruct the secret key.
NUM Claim Number: 14.
14. The method as described in claim 11 wherein each trustee can verify
that the pieces of information provided include a share of the secret key
without interaction with any other trustee.
NUM Claim Number: 15.
15. A method, using a cryptosystem, for enabling a predetermined entity to
monitor communications of users suspected of unlawful activities while
protecting the privacy of law-abiding users, wherein a group of users has
a secret key, comprising the steps of:
breaking the secret key into shares;
providing trustees pieces of information that include shares of the secret
key; and
upon a predetermined request, having the trustees reveal the shares of the
secret key of a user suspected of unlawful activity to enable the entity
to reconstruct the secret key and monitor communications to the suspect
user.
NUM Claim Number: 16.
16. The method as described in claim 15 further including the step of:
characterizing the suspect user's activities as unlawful if the entity is
unable to monitor the suspect user's communications.
NUM Claim Number: 17.
17. The method as described in claim 15 wherein a given minority of
trustees are unable to reconstruct the secret key.
NUM Claim Number: 18.
18. The method as described in claim 15 wherein each trustee can verify
that the pieces of information provided include a share of the secret key
without interaction with any other trustee.