Method and apparatus for establishing a key agreement protocol   

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to cryptography and, more particularly, to a system and method for facilitating cryptographic applications.

2. Description of the Prior Art

Key Agreement Protocols

It is sometimes desirable for individuals to be able to communicate with each other in a way in which third parties are unable to listen to the communication. A simple way for these individuals to communicate is to have the communications themselves proceed in private. For example if party A and party B desire to communicate in a way which will not be heard by party C, A and B can simply meet at a designated location unknown to C. Similarly, A and B can set up a designated communication line between them which excludes C. Such communication lines are expensive and inconvenient especially if A and B are geographically far apart from one another.

A first approach to facilitating private communications between A and B is to give A and B a secret key that may be used to encrypt and/or decrypt messages sent between A and B. If C does not know what the key is, it may be very difficult for C to both get a hold of a message sent between A and B and try to understand it. However, giving A and B such a key is also cumbersome, expensive and time consuming. Issues to be addressed include secretly transmitting such a key to A and B and generating a new key each time two individuals need to communicate. Also, if C does ascertain the secret key, then all communications between A and B can be decrypted and read by C.

Another approach for facilitating private communications between A and B is to assign A and B secret mathematical functions fa, fb respectively. The functions fa and fb are chosen from a set of functions, S, all of whose elements are designed so as to be commutative: applying fa followed by fb yields the same result as applying fb followed by fa (i.e., given an element x, fa(fb(x))=fb(fa(x))). Assuming the element x is known by both A and B, A can then send fa(x) to B, and B can send fb(x) to A over public channels. The secret key that can be evaluated and shared by both A and B is then, fa(fb(x))=fb(fa(x)). To insure that the system is secure (from an adversary C who knows x and can listen to all communication between A and B) it is necessary that the functions fa and fb satisfy the following property: given the value fa(x) (respectively fb(x)) it is computationally difficult to determine the function fa (respectively fb). This is called the general Diffie-Hellman key agreement protocol.

Many specific instances of the general Diffie-Hellman protocol for sending secure communications between A and B are known in the prior art (see Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, “Handbook of Applied Cryptography,” CRC Press (1997)). They all differ by their choice of the set of functions. The original Diffie-Hellman key agreement protocol is an example of the above described techniques (see W. Diffie and M. E. Hellman, “New directions in cryptography,” IEEE Transaction on Information Theory, vol. IT 22 (November 1976), pp. 644-654). Using an algorithm like the one first introduced by Diffie-Hellman, parties A and B can obtain a common shared secret by communicating over a public channel. The security of the system, in this instance, rests on the computational difficulty of computing discrete logarithms in the multiplicative group of the finite field. In more general cases the security is based on the notion of a one-way function. A function f from a set X to a set Y is termed one-way if f(x) is easy to compute for all x ∈ X but for essentially all elements y it is computationally difficult to find x ∈ X such that f(x)=y. To date a diverse array of mathematical techniques (including geometric and algebraic ones), have been used to create systems for secure communication whose security is based on one-way functions.

A problem with some of the prior art algorithms, is that most of them rely on a cost-risk analysis when generating the one-way function. That is, in order to produce a more complex and more difficult to determine secret key, each party would need to spend more time in generating such a key and may need to invest in more expensive devices. With rapidly evolving technologies, implementing the current algorithms in a cryptographically secure manner is becoming difficult. Furthermore, there are instances of resource limited devices where current algorithms are difficult to implement. Thus, there is a need in the art for a system and method which can produce a secure key relatively quickly and without employing expensive devices.

SUMMARY

An aspect of the invention is a method for securing communications from a user. The method comprises selecting a first monoid, selecting a second monoid and selecting a function, the function being a monoid homomorphism that maps the first monoid to the second monoid. The method further comprises selecting a group, selecting an action of the group on the first monoid, and determining a semi-direct product of the first monoid and the group to produce a third monoid. The method further comprises selecting a first and second submonoid of the third monoid, a pair of the first and second submonoids satisfying a criterion, the first submonoid being defined by a first set of generators, wherein the criterion satisfies a property determined by the function, a structure of the first and second monoids, and the action. The method still further comprises selecting a plurality of generators of the first set of generators to produce a private key.

Another aspect of the invention is a method for securing communications from a user. The method comprises receiving a first submonoid, the first submonoid being produced by selecting a first monoid, selecting a second monoid, selecting a function, the function being a monoid homomorphism that maps the first monoid to the second monoid, selecting a group, selecting an action of the group on the first monoid, determining a semi-direct product of the first monoid and the group to produce a third monoid, selecting a first and second submonoid of the third monoid, the pair of the first and second submonoids satisfying a criterion, the first submonoid being defined by a first set of generators, the criterion satisfying a property determined by the function, a structure of the first and second monoids, and the action. The method further comprising selecting a plurality of generators of the first set of generators to produce a private key. The method still further comprising applying the second component of an identity on a non-group component of a first generator of the private key to produce a result, wherein the identity comprises a first component, the first component being an identity of the second monoid, and the identity comprises a second component, the second component being an identity of the group. The method still further comprising applying the function to the result to produce a first modified result, multiplying the first component of the identity by the modified result to produce a first further modified result, multiplying the second component of the identity with a group component of the first generator to produce a first still further modified result, and combining the first further modified result with the first still further modified result to produce a public key.

Still another aspect of the invention is a method for securing communications among two users. The method comprises selecting a first monoid, selecting a second monoid, and selecting a function, the function being a monoid homodiorphism that maps the first monoid to the second monoid. The method further comprising selecting a group, selecting an action of the group on the first monoid, and determining a first semi-direct product of the first monoid and the group to produce a third monoid. The method still further comprising selecting a first and second submonoid of the third monoid, a pair of the first and second submonoids satisfying a criterion, the first submonoid being defined by a first set of generators, the second submonoid being defined by a second set of generators, the criterion satisfying a property determined by the function, a structure of the first and second monoids, and the action. The method further comprising at a first user, receiving the first submonoid, selecting a plurality of generators of the first set of generators to produce a first private key, and applying the second component of an identity on a non-group component of a first generator of the first private key to produce a first result, wherein the identity comprises a first component, the first component being an identity of the second monoid, and the identity comprises a second component, the second component being an identity of the group. The method further comprising at the first user applying the function to the first result to produce a first modified result, multiplying the first component of the identity by the modified result to produce a first further modified result, multiplying the second component of the identity with a group component of the first generator of the first private key to produce a first still further modified result, and combining the first further modified result with the first still further modified result to produce a first public key. The method still further comprising at the first user a. applying a group component of the first public key on a non-group component of a second generator of the first private key to produce a second result, b. applying the function to the second result to produce a second modified result, c. multiplying a non-group component of the first public key by the second modified result to produce a second further modified result, d. multiplying the group component of the first public key with a group component of the second generator of the private key to produce second still further modified result; and e. combining the first further modified result with the second still further modified result to produce a second public key. The method further. comprising at a second user receiving the second submonoid, selecting a plurality of generators of the second set of generators to produce a second private key, applying the second component of the identity on a non-group component of a first generator of the second private key to produce a third result, applying the function to the third result to produce a third modified result, multiplying the first component of the identity by the third modified result to produce a third further modified result, multiplying the second component of the identity with a group component of the first generator of the second private key to produce a third still further modified result. and combining the third further modified result with the third still further modified result to produce a third public key. The method still further comprising at the second user f. applying &group component of the third public key on a non-group component of a second generator of the second private key to produce a fourth result, g. applying the function to the fourth result to produce a fourth modified result, h. multiplying a non-group component of the third public key by the fourth modified result to produce a fourth further modified result, i. multiplying the group component of the third public key with a group component of the second generator of the second private key to produce a fourth still further modified result; and j. combining the fourth further modified result with the fourth still further modified result to produce a fourth public key.

Yet still another aspect of the invention is a transmitter comprising a memory including a first submonoid, the first submonoid being produced by selecting a first monoid, selecting a second monoid, selecting a function, the function being a monoid homomorphism that maps the first monoid to the second monoid, selecting a group, selecting an action of the group on the first monoid; determining a semi-direct product of the first monoid and the group to produce a third monoid, selecting a first and second submonoid of the third monoid, the pair of the first and second submonoids satisfying a criterion, the first submonoid being defined by a first set of generators; the criterion satisfying a property determined by the function, a structure of the first and second monoids, and the action. The transmitter further comprising a processor wherein the processor is effective to select a plurality of generators of the first set of generators to produce a private key. The processor is further effective to apply the second component of an identity on a non-group component of a first generator of the private key to produce a result, wherein the identity comprises a first component, the first component being an identity of the second monoid, and the identity comprises a second component, the second component being an identity of the group. The processor is further effective to apply the function to the result to produce a first modified result. The processor is effective to multiply the first component of the identity by the modified result to produce a first further modified result. The processor is effective to multiply the second component of the identity with a group component of the first generator to produce a first still further modified result; and the processor is effective to combine the first further modified result with the first still further modified result to produce a first public key. The processor is effective to a. apply a group component of the first public key on a non-group component of a second generator of the private key to produce a second result, b. apply the function to the second result to produce a second modified result, c. multiply a non-group component of the first public key by the second modified result to produce a second further modified result, d. multiply the group component of the first public key with a group component of the second generator of the private key to produce second still further modified result, and e. combine the first further modified result with the second still further modified result to produce a second public key.

Still another aspect of the invention is a system for securing communications between users. The system comprises a communications center, the communications center effective to select a first monoid, select a second monoid, select a function, the function being a monoid homomorphism that maps the first monoid to the second monoid, select a group, and select an action of the group on the first monoid. The communications center further effective to determine a first semi-direct product of the first monoid and the group to produce a third monoid; and select a first and second submonoid of the third monoid, a pair of the first and second submonoids satisfying a criterion, the first submonoid being defined by a first set of generators, the second submonoid being defined by a second set of generators, the criterion satisfying a property determined by the function, a structure of the first and second monoids, and the action. The system further comprising a first transmitter comprising a memory including the first submonoid and a first processor. The first processor effective to select a plurality of generators of the first set of generators to produce a first private key and apply the second component of an identity on a non-group component of a first generator of the first private key to produce a first result, wherein the identity comprises a first component, the first component being an identity of the second monoid, and the identity comprises a second component, the second component being an identity of the group. The first processor further effective to apply the function to the first result to produce a first modified result, multiply the first component of the identity by the modified result to produce a first further modified result, multiply the second component of the identity with a group component of the first generator to produce a first still further modified result and combine the first further modified result with the first still further modified result to produce a first public key. The first processor is further effective to a. apply a group component of the first public key on a non-group component of a second generator of the private key to produce a second result, b. apply the function to the second result to produce a second modified result, c. multiply a non-group component of the first public key by the second modified result to produce a second further modified result, d. multiply the group component of the first public key with a group component of the second generator of the first private key to produce second still further modified result; and e. combine the first further modified result with the second still further modified result to produce a second public key. The system further comprises a second transmitter comprising a memory including the second submonoid and a second processor. The second processor effective to select a plurality of generators of the second set of generators to produce a second private key, apply the second component of the identity on a non-group component of a first generator of the second private key to produce a third result, apply the function to the third result to produce a third modified result, and multiply the first component of the identity by the third modified result to produce a third further modified result. The second processor further effective to multiply the second component of the identity with a group component of the second generator to produce a third still further modified result and combine the third further modified result with the third still further modified result to produce a third public key. The second processor is further effective to f. apply a group component of the third public key on a non-group component of a second generator of the second private key to produce a fourth result, g. apply the function to the fourth result to produce a fourth modified result, h. multiply a non-group component of the first public key by the fourth modified result to produce a fourth further modified result, i. multiply the group component of the third public key with a group component of the second generator of the second private key to produce fourth still further modified result and j. combine the fourth further modified result with the fourth still further modified result to produce a fourth public key.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a system diagram illustrating a Π-Function module in accordance with an embodiment of the invention.

FIG. 2 is a system diagram illustrating a S-Action module in accordance with an embodiment of the invention.

FIG. 3 is a system diagram illustrating an E-Function module in accordance with an embodiment of the invention.

FIG. 4 is a system diagram illustrating the operation of an E-Function iterator module in accordance with an embodiment of the invention.

FIG. 5 is another system diagram illustrating the operation of an E-Function iterator module in accordance with an embodiment of the invention.

FIG. 6 is a system diagram illustrating a system for determining a pair of E-commuting monoids in accordance with an embodiment of the invention.

FIG. 7 is a system diagram illustrating a system for determining a private key in accordance with an embodiment of the invention.

FIG. 8 is a system diagram illustrating a system for determining a public key in accordance with an embodiment of the invention.

FIG. 9 is a system diagram illustrating a system for determining a common agreed upon secret key in accordance with an embodiment of the invention.

FIG. 10 is a flow diagram illustrating a method for determining a common agreed upon secret key and transmitting a message using that secret key in accordance with an embodiment of the invention.

FIG. 11 is a system diagram illustrating a system for determining a secret key in accordance with an embodiment of the invention.

DETAILED DESCRIPTION

The present invention introduces an algorithmically efficient one-way function. The algorithm is both rapidly computable and computationally hard to reverse. An overview in accordance with the invention is provided in FIG. 10. Parties Alice and Bob are each in possesion of a database from which they form their respective private keys (Boxes 101 and 102). They then proceed to produce their respective public keys based on their respective private keys by applying an algorithm in accordance with the invention (Boxes 103 and 104). Alice and Bob each have access to a respective transmitter and receiver. Alice and Bob use their respective transmitter and receiver to exchange their public keys. By exchanging these public keys they are each in a position to obtain a common agreed upon secret key by letting the received public key act on the respective user\'s private keys (Boxes 105 and 106). Once the shared secret key has been obtained, Alice can then encrypt a plaintext message, produce an encrypted message (Box 107), send the encrypted message (Box 108) to Bob, who can then decrypt the encrypted message (Box 109) to obtain Alice\'s plaintext message (Box 107).

Let M, N denote monoids and let S denote a group which acts on M on the left. Given an element s ∈ S, and an element m ∈ M, we denote the result of s acting on m by sm. The semidirect product of M and S, M×S is defined to be the monoid whose underlying set is M×S and whose internal binary operation

θM×S:(M×S)×(M×S)→M×S

is given by

θM×S:((m1,s1), (m2,s2))→(m1·s1m2,s1s2).

Furthermore, we let N×S denote the direct product.

An algebraic eraser is specified by a 6-tuple (M×S, N, Π, E, A, B) where M×S and N are as above, A, B are user submonoids of M, Π is an easily computable monoid homomorphism

Π: M→N,

E is a function

E: (N×S)×(M×S)→N×S

given by

E((n,s),(m1,s1))=(nΠ(sm1),ss1),

and A, B are submonoids of M×S such that for all (a, sa) ∈ A, (b, sb) ∈ B

E((Π(a),sa), (b, sb))=E((Π(b), sb), (a, sa)).

Two submonoids satisfying the above identity are termed E-Commuting.

An action of S on M does not induce an action of S on N, and given knowledge of the elements

(n,s), E((n,s), (m1,s1)) ∈ N×S

it is very difficult to obtain the element (m1, s1) ∈ M×S. The action of the element s ∈ S has been effectively erased by the algebraic eraser. A benefit lies in the efficiency of the computation of the function Π and the iterative nature of the method and apparatus for the computation of the function E.

A preferred embodiment of an apparatus to perform an algebraic key agreement protocol based on the algebraic eraser, is depicted in FIGS. 1 through 11, and begins with an apparatus to compute the function Π. The Π-Function module 13 is responsive to the data from the Π-Function module library 11, and the input element m ∈ M from 12. The Π-Function module 13 computes the element Π(m) ∈ N.

In general a group S is said to act (on the left) on a monoid M provided there is a homomorphism from S to the endomorphisms of M which satisfies certain properties. Given s ∈ S and m ∈ M, the element s maps m to a new element in M, denoted sm. The required properties are

s(m1m2)=sm1sm2, 1m=m, s1s2m=s1(s2m)

Referring to FIG. 2, S-Action module 23 is responsive to the inputs s ∈ S 21 and m ∈ M 22, and computes the image of m under the action of s yielding sm as output.

An apparatus to compute the function E is depicted in FIG. 3. The E-Function module 36 is responsive to the inputs (n, s) 31 and (m, s) 34. Given an ordered list (x, y) of two elements x, y, the first component projection of (x, y) outputs the first component x on the list. Similarly, the second component projection outputs the second component y. The input (n, s), 31, is sent to the second component projection module, 32 and the input (m1, s1) is likewise sent to a first component projection module, 33. The resulting elements of S and M of the first and second component modules 32, 33 are then forwarded to the S-Action module 23, yielding the element sm1 ∈ M. This resulting element sm1 is forwarded to the Π-Function module, 13, which outputs the element Π(sm1). The E-Function multiplier, 35, is responsive to the input (n, s), 31, the element Π(sm1) ∈ N, and the result of the input (m1, s1), 34, being entered into the second component projection module, 32. The E-Function multiplier outputs the element (n Π(sm1), s s1) ∈ N×S which is also the output of the E-Function module 36.

The semi-direct product of M and S, denoted M×S, is defined to be the monoid whose underlying set is the direct product M×S and whose binary operation is given by

(m1,s1)·(m2,s2)=(m1·s1m2, s1s2).

It is noted that given an element (n, s) ∈ N×S and two elements (m1, s1), (m2, s2) ∈ M×S, that

E((n,s), ((m1,s1)·(m2,s2)))=E(E((n,s), (m1,s1)), (m2,s2)).

Hence computing the E-Function iteratively increases the system\'s efficiency and speed.

FIG. 4 depicts an apparatus which may be used in performing the above computation. An E-Function Iterator module 42 is responsive to the input (n, s), 31, and to the input ((m1, s1), (m2, s2), . . . , (mk, sk)), 41, and outputs

(n Π(sm1)Π(ss1m2) . . . Π(ss1. . . skmk), s s1 . . . sk).

A more detailed apparatus of the E-Function Iterator module 42, is depicted in FIG. 5, begins with the input (n, s) 31 being sent to the E-Function module 36. In addition, an input <(m1, s1), (m2, s2), . . . , (mk, sk)>, 41, is sent to the choose tth component module, 53, which is a module initialized at the value t=1 and repeatedly incremented by the increment t module, 54. The tth component of the input ((m1, s1), (m2, s2), . . . , (mk, sk)) is precisely (mt, st) which is the output of 53 and sent to the E-Function module 36. Furthermore the value of t is sent to the decision box 55 which also receives the value of the E-Function (iterated t−1 times up to that point). The decision box 55 determines if t=k, at which point the computation stops, otherwise, the output of decision box 55 becomes input 31 to the E-Function module 36 to be used as the new first component of E together with the incoming entry from choose tth component module 53. The final value arrived at is given by

(n·Π(sm1)Π(ss1m2) . . . Π(ss1. . . sk−1mk), s s1 . . . sk)=(n·Π((sm1)(ss1m2) . . . (ss1. . . sk−1mk)), ss1 . . . sk).

Recall that two submonoids A, B are said to be E-Commuting provided

E((Π(a), sa),(b,sb))=E((Π(b), sb), (a, sa))

holds for all (a, sa) ∈ A, (b, sb) ∈ B. FIG. 6 illustrates an apparatus which may be used in choosing a pair of E-Commuting monoids, A, B which may be utilized in the invention. A monoid is specified by a generating set, i.e., a subset of elements of the monoid which have the property that every element of the monoid can be expressed as a product of some of these generators (in some order, with repetitions allowed). The Semidirect Product Producer 60 is responsive to the monoid M and the group S and produces the monoid M×S. The monoid M×S, together with the monoid N and the function Π are sent to the E-Commuting Monoid Producer 63, whose output is sent to the Pairs of E-Commuting Monoid Library 64. A Pseudorandom Number Generator 61 produces a random number α, a Chooser 62 then accesses the αth element of the Pairs of E-Commuting Monoid Library 63 and outputs the pair of E-Commuting monoids A1, B1 which are forwarded to Alice and Bob, respectively. Additionally the pair A1, B1 is forwarded to the User Submonoid Generator Database 65.

With the apparatuses for computing the S-Action, the functions Π and E specified, and each users submonoid in place, the algebraic eraser key agreement protocol can now be detailed. If the E-commuting monoids A1, B1, are privately assigned to Alice and Bob, then the invention functions, for example, as a symmetric cryptosystem. If the monoid M×S possesses a large library of pairs of E-Commuting submonoids which are recursively enumerable and whose internal algebraic structure is hidden then the invention can function, for example, as an asymmetric cryptosystem.

FIG. 7 illustrates a mechanism which may be used in enabling a user to generate a private key. Focusing on Alice (Bob case is analogous) a second Pseudorandom Number Generator 72 responsive to the input α*, 71, creates a list of integers e1, e2, . . . , eα* where each ei is generated in such a way that ei≦number of generators of (A1). The Sequence Encoder 73 is responsive to the list e1, e2, . . . , eα* and the User Submonoid Generator database 65, is responsive to the submonoid A1. The Sequence Encoder 73 produces the list of the user generators (me1, se1), (me2, se2), . . . , (meα*,seα*) out of the generating set of A1. The Private Key Generator 74 is responsive to Encoder 73 and produces the user private key

which is sent to a memory 75. It should be observed that the product of the elements, denoted (MA, sA),

( M A , s A ) =  ( m e 1 , s e 1 ) · ( m e 2 , s e 2 )   …   ( m e α * , s e α * ) =  ( ( m e 1

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