Patent Publication Number: US-2006002562-A1

Title: Method and apparatus for geometric key establishment protocols based on topological groups

Description:
CROSS REFERENCE TO RELATED APPLICATIONS  
      U.S. Pat. No. 5,696,826, December/1997, by Gao; U.S. Pat. No. 6,493,449, December/2002, Anshel et al; U.S. patent application Ser. No. 10/605,935, November/2003, Berenstein and Chernyak. 
    
    
     BACKGROUND OF INVENTION  
      Description of the Prior Art: Key Establishment Protocols  
      The concepts, terminology and framework for understanding cryptographic key establishment protocols is given in Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, “Handbook of Applied Cryptography,” CRC Press (1997), pages 490-491.  
      A ‘protocol ’ is a multi-party algorithm, defined by a sequence of steps specifying the actions required of two or more parties in order to achieve a specified objective.  
      A ‘key establishment’ protocol is a protocol whereby a shared secret becomes available to two or more parties, for subsequent cryptographic applications.  
      A ‘key transport’ protocol is a key establishment protocol where one party creates or obtains a secret value, and securely transfers it to the other participating parties.  
      A ‘key agreement’ protocol is a key establishment protocol in which a shared secret is derived by two (or more) parties as a function of information contributed by, or associated with, each of the participating parties such that no party can predetermine the resulting value.  
      A ‘key distribution ’ protocol is a key establishment protocol whereby the established keys are completely determined a priori by initial keying material.  
      The Diffie-Hellman key establishment protocol (also called ‘exponential key exchange’) is a fundamental algebraic protocol. It is presented in W. Diffie and M. E. Hellman, “New Directions in Cryptography,” IEEE Transaction on Information Theory vol. IT 22 (November 1976), pp. 644-654. The Diffie-Hellman protocol provided the first practical solution to the key distribution problem, allowing two parties, never having met in advance or sharing keying material, to establish a shared secret by exchanging messages over an open channel.  
      The security of this protocol rests on the intractability of the Diffie-Hellman problem and the related problem of computing discrete logarithms in the multiplicative group of the finite field GF(p) where p is a large prime, cf. Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, “Handbook of Applied Cryptography,” CRC Press (1997), page 113.  
      Most of known applications of Diffie-Hellman protocol deal with finite groups. Recently there emerged versions of Diffie-Hellman protocol for infinite, but yet discrete groups (see for example, U.S. Pat. No. 6,493,449 by Anshel et al).  
      Unlike approaches existing in the prior art, the present invention is based not on finite or discrete groups, but rather on the connected compact topological groups.  
      Brief overview of connected compact topological groups  
      The basic reference for concepts, terminology and historical framework in topological group are given in the monograph by Philip J Higgins,  Introduction to topological groups , Cambridge: University Press, 1974, and in the monograph by John F. Price,  Lie groups and compact groups , Cambridge [Eng]; New York: Cambridge University Press, 1977.  
      A group (G,*) is defined as a set G together with a binary operation *: G×G→G satisfying the following axioms:  
      Associativity: For all a, b and c in G, (a*b)*C=a*(b*c).  
      Identity element: There is an element e in G such that for all a in G, e*a=a=a*e.  
      Inverse element: For all a in G, there is an element b in G such that a*b=e=b*a, where e is the identity element from the previous axiom.  
      A topological group G is a group which is also a topological space such that the group multiplication G×G→G and the operation of taking inverses G→G are continuous maps. (Here, G×G is viewed as a topological space by using the product topology).  
      A topological group G is called compact if the underlying topological space is compact, i.e., if any open cover of the space G has a finite sub-cover.  
      A first example of compact topological groups is any finite group (equipped with the discrete topology). Such groups provide examples of compact disconnected topological groups.  
      Another class of compact topological groups is connected compact topological groups. A topological group is connected if the underlying topological space is connected. This class contains such groups as SO(V), where SO(V) is the group of all special orthogonal transformations of a Euclidean vector space V (therefore, there are at least as many compact connected topological groups as there are Euclidean vector spaces).  
      The present invention implements the ideas and algorithms of Diffie-Hellman protocol for the case of connected compact topological groups. This approach allows one to bypass and, in some cases, to completely eliminate the computational complexity of the exponentiation operation. Such an approach does not exist in the prior art.  
     SUMMARY OF INVENTION  
      Geometric key establishment system of the present invention allows for easy, secure, and rapid creation and distribution of encryption/decryption keys for major cryptosystems. The procedures of creation and distribution of keys are performed extremely rapidly and have very low computer memory requirements.  
      The present invention proposes a continuous version of Diffie-Hellman protocol. Based on this continuous Diffie-Hellman protocol, a method for public distribution of keys for encryption/decryption systems is implemented. An embodiment of the method, while providing an extremely high security level, is several orders of magnitude faster than existing key distribution systems.  
      The key creation process of the system hereof uses the operation of linear combination with integer coefficients of irrational numbers and the operation of taking fractional parts of real numbers. In more advanced implementations the operation of taking fractional parts can be replaced by the exponentiation from the compact Lie algebra into the corresponding compact Lie group.  
      The system of the present invention constructs encryption/decryption keys on the fly out of a publicly chosen n-tuple g of irrational numbers and a pair of secret integer n×n matrices A and B, where the first integer matrix A is generated by the first communicating party and the second integer matrix B—by the second, and the matrices commute, i.e., A·B=B·A. Absolute values of matrix coefficients of these matrices are bounded by a publicly available constant 10 N  that may be arbitrarily big. Thus the keys created and distributed by the system hereof can be of any given in advance size. The present invention combines the idea of Diffie-Hellman protocol of key distribution with the idea of the geometric cryptosystem developed in the U.S. patent application Ser. No. 10/605,935 entitled GEOMETRY-BASED SYMMETRIC CRYPTOSYSTEM METHOD by the authors Arkady Berenstein and Leon Chernyak.  
      The security of the system of the present invention is based on the following well-known paradigm from Number Theory. Let β 1 , β 2 , . . . be a sequence of irrational numbers (or more generally, of irrational elements of a compact Lie group) and let γ be an irrational number computed with the precision of K decimal places. Then any algorithm that recognizes γ as an element of the sequence β 1 , β 2 , . . . and identifies the index n such that γ=β n  n must work at least C·10 K  units of time where C is an a priori given constant. 
    
    
     BRIEF DESCRIPTION OF DRAWINGS  
       FIG. 1  is a block diagram of the mathematical apparatus that can be used in practicing embodiments of the invention.  
       FIG. 2  is a flow diagram of the geometric key establishment system which shows generation of commuting matrices with integer coefficients; when taken with the subsidiary flow diagrams referred to therein, can be used in implementing embodiments of the invention.  
       FIG. 3  is a flow diagram of the geometric key establishment system which shows the exponentiation of n-tuples of group elements into matrix powers; when taken with the subsidiary flow diagrams referred to therein, can be used in implementing embodiments of the invention.  
       FIG. 4  is a flow diagram of the geometric key establishment system which shows the fractional multiplication of n-tuples of real numbers by integer n×n matrices; when taken with the subsidiary flow diagrams referred to therein, can be used in implementing embodiments of the invention.  
       FIG. 5  is a block diagram of the geometric key establishment system that can be used in practicing n-dimensional embodiments of the invention.  
       FIG. 6  is a block diagram of the geometric key establishment system that can be used in practicing one-dimensional embodiments of the invention.  
       FIG. 7  is a block diagram of the geometric key establishment system that can be used in practicing preferred n-dimensional embodiments of the invention in the case when the group operation consists of taking the fractional part of sum of real numbers.  
      geometric key establishment system the is a block diagram of  FIG. 8  that can be used in practicing preferred one-dimensional embodiments of the invention in the case when the group operation consists of taking the fractional part of sum of real numbers. 
    
    
     DETAILED DESCRIPTION  
      The key creation and distribution techniques of an embodiment of the geometric key establishment system hereof are based on the operation of multiplication of real numbers by integers and the operation of evaluating fractional parts of real numbers. More specifically, the n-dimensional embodiment of the system hereof is based on the operation of multiplication of real vectors by integer matrices and on the operation of evaluating fractional parts of coordinates of the vectors.  
      In more advanced implementations the operation of evaluating fractional parts can be replaced by the exponentiation from the compact Lie algebra into the corresponding compact Lie group.  
      A preferred exemplary embodiment of such an apparatus is depicted with block diagram in  FIG. 1 , and is described as follows.  
      Let G be a compact connected group whose law of composition 
 
G×G→G 
 
 is feasibly computable. There are among such groups the special orthogonal group, the unitary group and their closed connected subgroups. The block  101  generates such groups. Since each such group has uncountably many elements, the block  102  selects an element g of G essentially at random. The block  103  generates a n-tuple g=(g 1 ,g 2 , . . . , g n ) of pairwise commuting elements g 1 , g 2 , . . . , g n  of G. This generator may proceed by choosing a commutative subgroup of G and then selecting elements g 1 , g 2 , . . . ,g n  from this subgroup. Alternatively, the group generator  101  can start with choosing a commutative group G. The block  104  is designed for independent generation of commuting integer matrices, which procedure is depicted in more details in  FIG. 2 . The block  105  is designed for raising each n-tuple g=(g 1 , g 2 , . . . , g n ) into an integer matrix power, which procedure is depicted in more details in  FIG. 3 . The block  106  rounds each element g of the group G to the nearest element [g] of G. This procedure is depicted in more details in subsequent flow diagrams of  FIG. 7  and  FIG. 8 , where, as a preferred embodiment of the invention hereof, the group operation of G consists of taking the fractional part of sum of real numbers. The block  107  applies the procedure of rounding of the block  106  to each coordinate of a given n-tuple g=(g 1 , g 2 , . . . , g n ). 
 
       FIG. 2  illustrates a basic procedure of generation of commuting n×n matrices A and B independently by the first and the second communicating parties.  
      In the block  201  a public n×n matrix S is selected.  
      In the block  202  the first communicating party chooses at random secret integers a 0 , a 1 , . . . , a n−1 , and in the block  204  creates a matrix A according to the formula: 
 
 A=a   0   ·I+a   1   ·S+a   2   ·S   2   + . . . +a   n−1   ·S   n−1 . 
 
      In a similar manner and independently, in the block  203  the second communicating party chooses at random secret integers b 0 , b 1 , . . . , b n−1 , and in the block  205  creates a matrix B according to the formula: 
 
 B=b   0   ·I+b   1   ·S+b   2   ·S   2   + . . . +b   n−1   ·S   n−1 . 
 
      By the design, the matrices A and B commute: A·B=B·A.  
       FIG. 3  represents a basic procedure of raising a n-tuple g into the A-th power, where A is an n×n matrix.  
      In the block  301  an n-tuple g=(g 1 , g 2 , . . . , g n ) of elements of the group G is generated.  
      Independently, in the block  302  an n×n matrix A is generated.  
      And, in the block  303  the power g A  is computed according to the formula: 
 
g A =(y 1 , y 2 , . . . , y n ), 
 
where 
 
y j =g 1   A1,j ·g 2   A2,j · . . . ·g n   An,j  
 
 for j=1, 2, . . . , n, where A ij  is the (i,j)-th matrix coefficient of A. 
 
       FIG. 4  represents a basic procedure of implementing the routine of  FIG. 3  in the case when the group operation consists of taking the fractional part of sum of real numbers.  
      In the block  401  an n-dimensional real vector g=(g 1 , g 2 , . . . , g n ) is generated.  
      Independently, in the block  402  an n×n matrix A is generated.  
      And, in the block  403  the fractional product {g·A} is computed according to the formula: 
 
{ g·A }=( y   1   , y   2   , . . . , y   n ), 
 
where 
 
 y   j   ={g   1   A   1,j   +g   2   A   2,j   + . . . +g   n   A   n,j }
 
 for j=1, 2, . . . , n, where A ij  is the (i,j)-th matrix coefficient of A, and where {z} stands for the fractional part of the real number z (for example, {1.7}=0.7, {−1.7}=0.3). 
 
       FIG. 5  illustrates creation, establishment, and distribution of a geometric key in an n-dimensional embodiment of the system of the present invention. It refers to the routines illustrated by other referenced flow diagrams ( FIG. 1 ,  FIG. 2 ,  FIG. 3 ) which describe features in accordance with an embodiment of the invention.  
      The block  501  represents generation of the public compact connected commutative topological group G and a choosing at random a public n-tuple g=(g 1 , g 2 , . . . , g n ) of elements of G. A public n×n matrix S is also chosen in this block. Both, g and S are to be used by both communicating parties.  
      The block  502  represents the routine that can be used by the first communicating party for generating a private matrix A according to the routine of  FIG. 2 .  
      Similarly, the block  503  represents the routine that can be used by the second communicating party for generating a private matrix B according to the routine of  FIG. 2 .  
      The block  504  represents computation (by the first communicating party) of the n-tuple g A  according to the routine of  FIG. 3 , and rounding g A  to the nearest n-tuple [g]. The rounded n-tuple [g A ] is then transmitted over an open (public) channel to the second communicating party.  
      Similarly, the block  505  represents computation (by the first communicating party) of the n-tuple g B  according to the routine of  FIG. 3 , and rounding g B  to the nearest n-tuple [g B ]. The rounded n-tuple [g B ] is then transmitted over an open (public) channel to the first communicating party.  
      The block  506  represents the routine that can be used by the second communicating party for generating the n-tuple [g A ] B  (according to the routine of  FIG. 3 ) and rounding it to the nearest n-tuple [[g A ] B ].  
      Similarly, the block  507  represents the routine that can be used by the first communicating party for generating the n-tuple [g B ] A  (according to the routine of  FIG. 3 ) and rounding it to the nearest n-tuple [[g B ] A ].  
      By the design, the n-tuples [[g A ] B ] and [[g B ] A ] are equal, and thus comprise the common secret geometric key in possession of both communicating parties.  
       FIG. 6  illustrates creation, establishment, and distribution of a geometric key in a one-dimensional embodiment of the system of the present invention. It refers to the routines illustrated by other referenced flow diagrams which describe features in accordance with an embodiment of the invention.  
      The block  601  represents generation of the public compact connected topological group G and a choosing at random a public element g of G, which g is to be used by both communicating parties.  
      The block  602  represents the routine that can be used by the first communicating party for generating a private integer a, computing the a-th power g a  of g, and rounding g a  to the nearest element [g a ] in G. This rounded element [g a ] is then transmitted over an open (public) channel to the second communicating party.  
      Similarly, the block  603  represents the routine that can be used by the second communicating party for generating a private integer b, computing the b-th power g b  of g, and rounding g b  to the nearest element [g b ] in G. This rounded element [g b ] is then transmitted over an open (public) channel to the second communicating party.  
      The block  604  represents the routine that can be used by the second communicating party for generating the element [g a ] b  and rounding it to the nearest element [[g a ] b ].  
      Similarly, the block  605  represents the routine that can be used by the first communicating party for generating the element [g b ] a  and rounding it to the nearest element [[g b ] a ] 
      By the design, the elements [g a ] b  and [g b ] a  are equal in G, and thus comprise the common secret geometric key in possession of both communicating parties.  
       FIG. 7  represents creation, establishment, and distribution of a key in an embodiment of the geometric key establishment system of present invention.  
      First, public natural numbers D, N, K are generated in the block  701 . Next, a public n-dimensional decimal vector g=(g 1 , g 2 , . . . , g n ) having D+2N+K after dot is generated in the same block  701 . And an integer n×n matrix S is chosen.  
      Then in the block  702  private integers a 0 , a 1 , . . . , a n1  (each between −10 N  and 10 N ) are generated at random; next, a private matrix A is generated according to routine of  FIG. 2 .  
      In a similar manner, in the block  703  private integers b 0 , b 1 , . . . , b n1  (each between −10 N  and 10 N ) are generated at random; next, a private matrix B is generated according to routine of  FIG. 2 .  
      In the block  704  the fractional vector {g●A} is computed according to the routine of  FIG. 4 . Next, each coordinate of the resulted vector is rounded to D+N+K decimal places. The rounded fractional vector {g●A}is then transmitted to the second communicating party.  
      In a similar manner, in the block  705  the fractional vector {g●B} is computed according to the routine of  FIG. 4 . First, the vector g is multiplied by the matrix B. Next, each coordinate of the resulted vector is rounded to D+N+K decimal places. The rounded fractional vector {g●B} is then transmitted to the second communicating party.  
      The block  706  represents the routine that can be used by the second communicating party for computing the fractional vector {{g●A}●B}. The loop  708  is used in the case when the vector {{g●A}●B} is not (K,D)-consistent (that is, in the case when the sequence of the digits d K+1 , d K+2 , . . . d K+D  of at least one coordinate of the vector {{g●A}●B} is either 0, 0, . . . , 0 or 9, 9, . . . , 9.) The loop  708  is continued until the vector {{g●A}●B} becomes (K,D)-consistent. [The probability of a vector {{g●A}●B} to be not (K,D)-consistent is extremely low. Namely, this probability is measured as at most 1−(1−2·10 −D ) n . The probability of the need for the second run of the loop  708  is measured as at most (1−(1−2·10 −D ) n ) 2 ]. The block  710  is then entered, this block represents the generation of a vector γ which is the rounding of the (K, D)-consistent vector {{g●A}●B} to K decimal places.  
      In a similar manner the block  707  represents the routine that can be used by the first communicating party for computing the fractional vector {{g●B}●A}. The loop  709  is used in the case when the vector {{g●B}●A} is not (K,D)-consistent (that is, in the case when the sequence of the digits d K+1 , d K+2 , . . . d K+D  of at least one coordinate of the vector {{g●B}●A} is either 0, 0, . . . , 0 or 9, 9, . . . , 9.) The loop  709  is continued until the vector {{g●B}●A} becomes (K,D)-consistent. [The probability of a vector {{g●B}●A} to be not (K,D)-consistent is extremely low. Namely, this probability is measured as at most 1−(1−2·10 −D ) n . The probability of the need for the second run of the loop  709  is measured as at most (1−(1−2·10 −D ) n ) 2 ]. The block  711  is then entered, this block represents the generation of a vector γ′ which is the rounding of the (K, D)-consistent vector {{g●B}●A} to K decimal places.  
      By the design, the vectors γ and γ′ are equal, and thus comprise the common secret key in possession of both communicating parties.  
       FIG. 8  represents creation, establishment, and distribution of a key in an embodiment of the geometric key establishment system of present invention.  
      First, a public real number α and public natural numbers D, N, K are generated in the block  801 .  
      Then in the block  802  a private integer a (between −10 N  and 10 N ) is generated at random, and the number {αa} is computed and rounded to D+N+K decimal places by the first communicating party. The rounded number {αa} is then transmitted to the second communicating party.  
      In a similar manner, in the block  803  a private integer b (between −10 N  and 10 N ) is generated at random, and the number {αb} is computed and rounded to D+N+K decimal places by the second communicating party. The rounded number {αb} is then transmitted to the first communicating party.  
      The block  804  represents the routine that can be used by the second communicating party for computing the number {{αa}b}. The loop  806  is used in the case when the number {{αa}b} is not (K, D)-consistent (that is, in the case when the sequence of the digits d K+1 , d K+2 , . . . , d K+D  of the number {{αa}b} is either 0, 0, . . . , 0 or 9, 9, . . . , 9). The loop  806  is continued until the number {{αa}b} becomes (K, D)-consistent. [The probability of a number {{αa}b} to be not (K,D)-consistent is extremely low. Namely, this probability is measured as at most 2·10 −D . The probability of the need for the second run of the loop  806  is measured as at most (2·10 −D ) 2 ]. The block  808  is then entered, this block represents the generation of a number γ which is the rounding of the (K, D)-consistent number {{αa}b} to K decimal places.  
      In a similar manner block  805  represents the routine that can be used by the first communicating party for computing the number {{αb}a}. The loop  807  is used in the case when the number {{αb}a} is not (K,D)-consistent (that is, in the case when the sequence of the digits d K+1 , d K+2 , . . . , d K+D  of the number {{αb}a} is either 0, 0, . . . , 0 or 9, 9, . . . , 9). The loop  805  is continued until the number {{αa}b} becomes (K,D)-consistent. [The probability of a number {{αb} a} to be not (K,D)-consistent is extremely low. Namely, this probability is measured as at most 2·10 −D . The probability of the need for the second run of the loop  807  is measured as at most (2·10 −D ) 2 ]. The block  809  is then entered, this block represents the generation of a number γ′ which is the rounding of the (K,D)-consistent number {{αb}a} to K decimal places.  
      By the design, the numbers γ and γ′ are equal, and thus comprise the common secret key in possession of both communicating parties.  
      The security of the system of the present invention comes from the built-in geometric density of certain sequences of irrational numbers in the semi-open interval [0, 1) of the real line. In other words, security of the proposed system is guaranteed by the obvious mathematical fact that there is no any a priori known general distribution pattern for members of certain sequences of irrational numbers.  
      More precisely, let β 2 , β 2 , . . . be a sequence of irrational numbers (or more generally, of irrational elements of a compact Lie group) and let γ be an irrational number computed with the precision of K decimal places. Then any algorithm that recognizes γ as an element of the sequence β 1 , β 2 , . . . and identifies the index n such that γ=β n  must work at least C·10 K  units of time where C is an a priori given constant.  
      Apparently, approaches that are the closest to the present invention are developed in U.S. Pat. No. 5,696,826 entitled METHOD AND APPARATUS FOR ENCRYPTING AND DECRYPTING INFORMATION USING A DIGITAL CHAOS SIGNAL by Gao, in U.S. Pat. No. 6,493,449 entitled METHOD AND APPARATUS FOR CRYPTOGRAPHICALLY SECURE ALGEBRAIC KEY ESTABLISHMENT PROTOCOLS BASED ON MONOIDS by Anshel et al, and in U.S. patent application Ser. No. 10/605,935 entitled GEOMETRY-BASED SYMMETRIC CRYPTOSYSTEM METHOD by Berenstein and Chernyak.  
      The idea of using fractional parts of multiples of given irrational numbers is not new in cryptography. This idea is used to obtain a uniform distribution of numbers in the unit interval. It was used, for example, in the patent by Gao. However, this is perhaps the only similarity between these previous works and the system of the present invention. In the system hereof, fractional parts of multiples of given irrational numbers are never used for obtaining a uniform distribution of numbers, but rather for creation of a deterministic (non-random) keys.  
      The idea of using infinite groups for key establishment an exchange is relatively new. It is presented in the patent by Anshel et al. However, the present invention is the first where continuous, topological groups are used for key establishment and exchange. In patent application by Berenstein and Chernyak the geometric continuity is utilized for constructing private encryption systems.  
      An embodiment of the system hereof deals with a publicly chosen real number α and a pair of secret integers a and b, where the first integer a is generated by the first communicating party and the second integer b—by the second communicating party. Absolute value of each of these integers is bounded by a publicly available constant 10 N  that may be arbitrarily big. Thus the keys created and distributed by the system hereof can be of any given in advance size. The present invention combines the idea of Diffie-Hellman protocol of key establishment with the idea of the geometric cryptosystem developed in the patent application Ser. No. 10/605,935 by the authors Arkady Berenstein and Leon Chernyak.  
      In this embodiment the real number α can be chosen essentially at random from the infinite set of known real numbers. This choice can include such numbers as, for example, √m, where m is any natural number that is not a complete square, or, more generally, α can be any irrational real root of an algebraic equation with integer coefficients. Also α can be chosen, for example, as π n  or e n , where n is any natural number, or, more generally, α can be any polynomial with integer coefficients evaluated at a given transcendental number. Another possible source of irrational numbers may include, for example, sin(n), cos( ), In(n), where n is any natural number; or, more generally, α (can be any transcendental function evaluated at a given natural number.  
      Let {x} be the fractional part of a real number x. By definition, for each real number x, the fractional part {x} is given by: 
 
 {x}=x−[x],  
 
 where [x]is the integer part of x, that is, [x]is the greatest integer that is less or equal x. If the numbers a and b are integers having at most N decimal digits each (that is, |a|&lt;10 N  and |b|&lt;10 N ) and each of the numbers {αa} and {αb} is rounded to D+N+K decimal places after dot (where D, N, and K are natural numbers each greater than 1), then the created and distributed key, which is {αab}, will have K correct decimal places after the dot. These K correct digits serve as the encryption/decryption key of major cryptosystems. 
 
      The security of the system of the present invention is based on the following well-known paradigm from Number Theory. Let β 1 , β 2 , . . . be a sequence of irrational numbers (or more generally, of irrational elements of a compact Lie group) and let γ be an irrational number computed with the precision of K decimal places. Then any algorithm that recognizes γ as an element of the sequence β 1 , β 2 , . . . and identifies the index n such that γ=β n  must work at least C·10 K  units of time where C is an a priori given constant.  
      In particular, the security of the system hereof is based on the fact that there cannot be any a priori known general distribution pattern of fractional parts of multiples of each taken at random irrational number α. Therefore, the security level of the system hereof can be measured as a number of operations needed for reconstruction of the number a out of a given fractional number {αa} calculated with the precision of D+N+K decimal places. The above implies that the only way to reconstruct the number a out of a given fractional number {αa} is to list all possible numbers {αL}, where L is any integer between −10 N  and 10 N . The number of such numbers L is 2·10 N −1.  
      For a cryptanalyst, the only alternative to listing all possible numbers a is to list all possible shared K-digits keys. The number of such keys is 10 K . Therefore, the security level of the system hereof can be measured as the minimum of the two numbers 2·10 N −1 and 10 K .  
      The processor time required for creation and distribution of one geometric key is at most quadratic in N and K, or, more precisely, is at most N(2D+3N+2K) units of processor time. This speed is several orders of magnitude higher than in existing key establishment systems.  
      In creating geometric key establishment system in accordance with an embodiment hereof, a first step is to choose publicly available parameters of the system: a real number α and natural numbers D, N, K, each greater than 1, where D stand for the size of the error control buffer, N stands for the maximum number of decimal places in each secret parameter a and b, and K stands for the key length.  
      An embodiment of the geometric key establishment system hereof relies on the concept of (K, D)-consistent numbers. An infinite decimal fraction δ=0. d 1  d 2  d 3  . . . is said to be (K, D)-consistent if the sequence of the digits d K+1 , d K+2 , . . . , d K+D  is neither 0, 0, . . . , 0 nor 9, 9, . . . , 9.  
      To implement the key creation and key distribution of this example, the first communicating party, call it Alice, chooses a secret integer a between −10 N  and 10 N  (i.e., a has at most N decimal places), calculates the number β, which is the fractional part {αa} rounded to D+N+K decimal places, and sends so calculated rounding β of {αa} to the second the first communicating party, call it Bob. [It is assumed in this example that Alice and Bob share the publicly available parameters α and D, N, K]. Simultaneously and independently Bob chooses a secret integer b between −10 N  and 10 N  (i.e., b has at most N decimal places), calculates the number β′, which is the fractional part {αb} rounded to D+N+K decimal places, and sends so calculated rounding β′ of {αb} to Alice. Upon receiving β from Alice, Bob multiplies β by b, computes the fractional part {βb} with the precision of K+D decimal places after the dot. If {βb} is (K, D)-consistent, Bob computes the number γ that is the rounding of {βb} to the K digits after the decimal dot. This number γ is the geometric key in possession of Bob. Upon receiving β′ from Bob, Alice multiplies β′ by a, computes the fractional part {β′a} with the precision of K+D decimal places after the dot. If {β′a} is (K, D)-consistent, Alice computes the number γ′ that is the rounding of {β′a} to the K digits after the decimal dot. Then Alice computes the number γ′ that is the rounding of {β′a} to the K digits after the decimal dot. This number γ′ is the geometric key in possession of Alice. In this case (which is the general case), the geometric key γ is equal to the geometric key γ′. In those (extremely rare) cases when {βb} is not (K, D)-consistent, the geometric key has to be redistributed because otherwise it may happen that γ≠γ′. In order to avoid such a situation, Alice and Bob choose new secret numbers a 1  and b 1  respectively (while keeping the same α and D, N, K) and repeat the above steps until they get a new geometric key γ 1 =γ′ 1  (provided that the new number {β 1  b 1 } is (K, D)-consistent).  
      The probability of the need for such redistribution is extremely low and is measured as at most 2·10 −D . The probability of the need for the second key distribution is measured as at most (2·10 −D ) 2 .  
      The embodiment of the system hereof is based on the following mathematical argument.  
      The definition of the fractional part {x} of x implies that {x} always belongs to the semi-open interval [0, 1), and that 
 
 {x+c}={x} 
 
 for any integer c. In its turn this implies that 
 
 {{x}d}={xd} 
 
 for any integer d. Therefore, 
 
{{α a}b}={αab}={αba}={{αb}a}.  
 
      Since −10 N &lt;a&lt;10 N , and −10 N &lt;b&lt;10 N ; and both {αa} and {αb} are computed with D+N+K correct decimal places after dot, it is asserted that the left hand side {{αa}b} and the right hand side {{αb}a} are equal in the first K decimal places after dot if {αa} and {αb} are both (K, D)-consistent. In order to prove the assertion, the following notation is introduced. Let {αa} be rounded to D+N+K correct decimal places, i.e., to the number 
 
β={α a}+θ· 10 −(D+N+K) , 
 
 where |θ|&lt;0.5, and let {αb} be also rounded to N+D+K correct decimal places, i.e., to the number 
 
β′={α b}+θ′· 10 −(D+N+K) , 
 
 where |θ′|&lt;0.5. Then {{αa}b} is calculated as {βb} and {{αb}a} is calculated as {β′b}. Furthermore,  
           {     β   ⁢           ⁢   b     }     =       {       (       {     α   ⁢           ⁢   a     }     +     θ   ·     10     -     (     D   +   N   +   K     )             )     ⁢   b     }     =       {         {     α   ⁢           ⁢   a     }     ⁢   b     +     θ   ·     10     -     (     D   +   N   +   K     )         ·   b       }     =     
     ⁢     =       {         {     α   ⁢           ⁢   a     }     ⁢   b     +     θ   ·     10       -   D     -   K           }     =       {       {       {     α   ⁢           ⁢   a     }     ⁢   b     }     +       θ   1     ·     10       -   D     -   K           }     =     {       {     α   ⁢           ⁢   ab     }     +       θ   1     ·     10       -   D     -   K           }                 ,       
          where θ 1  is some number such that |θ 1 |&lt;0.5.        

      Similarly  
           {       β   ′     ⁢   a     }     =       {       (       {     α   ⁢           ⁢   b     }     +       θ   ′     ·     10     -     (     D   +   N   +   K     )             }     ⁢   a     }     =       {         {     α   ⁢           ⁢   b     }     ⁢   a     +       θ   ′     ·     10     -     (     D   +   N   +   K     )         ·   a       }     =     
     ⁢     =       {         {     α   ⁢           ⁢   b     }     ⁢   a     +       θ   1   ′     ·     10       -   D     -   K           }     =       {       {       {     α   ⁢           ⁢   b     }     ⁢   a     }     +       θ   1   ′     ·     10       -   D     -   K           }     =     {       {     α   ⁢           ⁢   ba     }     +       θ   1   ′     ·     10       -   D     -   K           }                 ,       
          where θ′ 1  is some number such that |θ′ 1 |&lt;0.5.        

      Let γ be the rounding of {βb} to K decimal places after dot and let γ′ be rounding of {β′a} to K decimal places after dot. The above calculation of {βb} and {β′a} ensures that, if γ≠γ′, then necessarily one of these numbers, say {βb}, has all digits equal 9 at each i-th decimal place after dot, when i=K+1, K+2 . . . ., K+D, and, at the same time, the second of these numbers, i.e., {β′a}, has digits equal 0 at each i-th decimal place after dot, when i=K+1, K+2, . . . , K+D. In other words, γ≠γ′ implies that both {βb} and {β′a} are not (K, D)-consistent. Therefore, if {βb} is (K, D)-consistent, γ=γ′. Similarly, γ=γ′ if {β′a} is (K, D)-consistent. This proves the assertion.  
      In creating geometric key establishment system in accordance with an embodiment hereof (and with the following small numbers for ease of illustration), a first step is to choose publicly available parameters of the system: an real number α and integer parameters D, N, K greater than 1 each. Take, for example, α=√2, N=K=8, D=2. Next, suppose that Alice chooses the number a=48 176 925. Alice calculates the number {αa} with the precision D+N+K=18 by β={αa}={√2·48 176 925}=0.728431422183990298 and sends this number β to Bob. Suppose that at the same time Bob chooses the number b=19082791. Bob calculates the number {αb} with the precision D+N+K=18 by β′={αb}={√2·19082791}=0.840131236839417426 and sends this number β′ to Alice. Upon receiving the number from β Alice, Bob multiplies β by b, computes the fractional part {βb} with the precision K+D=10 decimal places after dot: 
 
{β b}={ 0.728431422183990298·19082791}=0.5873698504. 
 
      Since this number is (K, D)-consistent, i.e., it has digits 0 and 4 at 9 th  and 10 th  places after dot, the first K=8 digits of this number is the geometric key in possession of Bob: 
 
γ=0.58736985. 
 
      Upon receiving the number β′ from Bob, Alice multiplies β′ by a, computes the fractional part {β′a} with the precision K+D=10 decimal places after dot: 
 
{β a}={ 0.84013123683941742596·48 176 925}=0.5873698504. 
 
      Since this number is (K, D)-consistent, i.e., it has digits 0 and 4 at 9 th  and 10 th  places after dot, the first K=8 digits of this number is the geometric key in possession of Alice: 
 
γ′=0.58736985. 
 
      Thus, γ=γ′ is the geometric key shared by Alice and Bob. This key can be used in any major symmetric cryptosystem.  
      In a further embodiment of the invention the real number α is replaced by the 2-dimensional real vector g=(g 1 , g 2 ) in order to further enhance the security level of the proposed system.  
      A 2-dimensional embodiment of the system hereof works with the semi-open unit square and integer 2×2 matrices A and B of the form:  
         A   =         [           a   0           -     a   1                 a   1           a   0           ]     ⁢           ⁢   and   ⁢           ⁢   B     =     [           b   0           -     b   1                 b   1           b   0           ]         ⁢               
 
 where a 0 , a 1 , b 0 , b 1  are arbitrary integers. This structure of A and B guarantees their commutation: A·B=B·A. 
 
      Absolute values of each integer a 0 , a 1 , b 0 , b 1  are bounded by a publicly available constant 10 N  that may be arbitrarily big. Thus the keys created and distributed by the system hereof can be of any given in advance size.  
      In this embodiment the vector g=(g 1 , g 2 ) has coordinates g 1  and g 2  which are arbitrary real numbers, that is, g is an arbitrary point of plane.  
      Let {x} be the fractional part of a real number x. By definition, for each real number x, the fractional part {x} is given by: 
 
 {x}=x−[x],  
 
 where [x]is the integer part of x, that is, [x]is the greatest integer that is less or equal x. 
 
      If the numbers a 0 , a 1  and b 0 , b 1  are integers having at most N decimal digits each (that is, |a 0 |&lt;10 N , |a 1 |&lt;10 N  and |b 0 |&lt;10 N , |b 1 |&lt;10 N ) and each coordinate of the vectors 
 
{ g●A }=({ g   1   a   0   +h   2   a   1   }, {−g   1   a   1   +g   2   a   0 }) and { g●B }=({ g   1   b   0   +g   2   b   1   },{−g   1   b   1   +g   2   b   0 }) 
 
 is rounded to D+N+K decimal places after dot (where D, N, and K are natural numbers each greater than 1), then the created and distributed key, which is the vector {g●A●B}, in each of its coordinates will have K correct decimal places after the dot. These 2K correct digits serve as the encryption/decryption key of major cryptosystems. 
 
      The security of this two-dimensional embodiment is further enhanced even in comparison with the high security of the one-dimensional embodiment.  
      In creating geometric key establishment system in accordance with the 2-dimensional embodiment hereof, a first step is to choose publicly available parameters of the system: a real vector g=(g 1 , g 2 ) and natural numbers D, N, K, each greater than 1, where D stand for the size of the error control buffer, N stands for the maximum number of decimal places in each secret parameter a and b, and K stands for the key length.  
      This embodiment of the geometric key establishment system hereof relies on the concept of (K, D)-consistent vectors. An infinite decimal fraction δ=0. d 1  d 2  d 3  . . . is said to be (K, D)-consistent if the sequence of the digits d K+1 , d K+2 , . . . d K+D  is neither 0, 0, . . . , 0 nor 9, 9, . . . , 9. We say that a vector (×1,x 2) is (K,D)-consistent both x 1  and x 2  are (K,D)-consistent numbers.  
      To implement the key creation and key distribution of this example, the first communicating party, call it Alice, chooses a pair of secret integers (a 0 , a 1 ) each between −10 N  and 10 N  (i.e., each of these integers has at most N decimal places). Then Alice calculates the vector (y 1 , y 2 ), where γ 1  is {g 1 a 0 +g 2 a 1 } rounded to D+N+K decimal places and y 2  is {−g 1 a 1 +g 2 a 0 } rounded to D+N+K decimal places; and sends so calculated vector (y 1 , y 2 ) to the second communicating party, call it Bob. [It is assumed in this example that Alice and Bob share the publicly available parameters g=(g 1 , g 2 ) and D, N, K.] 
      Simultaneously and independently Bob chooses a pair of secret integers (b 0 , b 1 ) each between −10 N  and 10 N  (i.e., each of these integers has at most N decimal places). Then Bob calculates the calculates the vector (z 1 , z 2 ) where z 1  is {g 1 b 0 +g 2 b 1 } rounded to D+N+K decimal places and z 2  is {−g 1 b 1 +g 2 b 0 } rounded to D+N+K decimal places; and sends so calculated so calculated vector (z 1 , z 2 ) to Alice.  
      Upon receiving the vector (y 1 , y 2 ) from Alice, Bob calculates the vector (k 1 , k 2 ) by the formula: 
 
( k   1   , k   2 )=({ y   1   b   0   +y   2   b   1   }, {−y   1   b   1   +y   2   b   0 }). 
 
      If the vector (k 1 , k 2 ) is (K, D)-consistent then Bob calculates the geometric key (s 1 , s 2 ) by rounding each coordinate of (k 1 , k 2 ) to K decimal places. Otherwise, he restarts the protocol.  
      Upon receiving the vector (z 1 , z 2 ) from Bob, Alice calculates the vector (k′ 1 , k′ 2 ) by the formula: 
 
( k′   1   , k′   2 )=({ z   1   a   0   +z   2   a   1   }, {−z   1   a   1   +z   2   a   0 }). 
 
      If the vector (k′ 1 , k′ 2 ) is (K, D)-consistent then Alice calculates the geometric key (s′ 1 , s′ 2 ) by rounding each coordinate of (k′ 1 , k′ 2 ) to K decimal places. Otherwise, she restarts the protocol.  
      The mathematical argument presented below proves that the geometric key (s 1 , s 2 ) in possession of Bob to the geometric key (s′ 1 , s′ 2 ) in possession of Alice.  
      In those (extremely rare) cases when (k 1 , k 2 ) is not (K, D)-consistent, the geometric key has to be redistributed because otherwise it may happen that (s 1 , s 2 )≠(s′ 1 , s′ 2 ). In order to avoid such a situation, Alice and Bob choose new pairs of secret numbers (a′ 0 , a′ 1 ) and (b′ 0 , b′ 1 ) respectively (while keeping the same g=(g 1 , g 2 ) and D, N, K) and repeat the above steps until they get a new geometric key (s 1 , s 2 )=(s′ 1 , s′ 2 ) (provided that the new vector (k 1 , k 2 ) is (K, D)-consistent).  
      The probability of the need for such redistribution is extremely low and is measured as at most 4·10 −D . The probability of the need for the second key distribution is measured as at most (4·10 −D ) 2 .  
      The embodiment of the system hereof is based on the following mathematical argument.  
      Proposition. Let be P=(P 1 , P 2 , . . . , P n ), Q=(Q 1 , Q 2 , . . . , Q n ) and L=(L 1 , L 2 , . . . , L n ) be n-tuples of natural numbers. Let α and β be n×n matrices with natural coefficients such that: 
 
 Q   −1   ·α≦L   −1   , P   −1   ·β≦L   −1 . 
 
      Then for any real vector g=(g 1 , g 2 , . . . ,g n ) any n×n matrices A and B with integer coefficients such that A·B=B·A and 
 
|A ij |&lt;α ij , |B ij |&lt;β ij  
 
 (for all i=1,2, . . . , n, j=1,2, . . . ,n) one has: either at least one coordinate of [{([{g·A}] p )·B}] L  equals 0, or at least one coordinate of [{([{g·B}] Q )·A}] L  equals 0, or 
 
{([{ g·A}]   p )· B }−{([{ g·B}]   Q )· A}=θ·L   −1 , 
 
      Proof. By definition, one has: 
 
[{ g·A}]   P   ={g·A}+θ   1   ·P   −1   , [{g·B}]   Q   ={g·B}+θ   2   ·Q   −1 , 
 
 where −½≦θ 1 ≦½ and −  1 / 2 ≦θ 2 ≦½. Therefore, 
 
([{ g·A}]   P )· B= ({ g·A}+θ   1   ·P   −1 )· B={g·A}·B+θ   1   ·P   −1   ·B={g·A}·B+E   1 , 
 
 where E 1 =θ 1 ·P −1 ·B. 
 
      Similarly, 
 
([{ g·B}]   Q )· A =({ g·B}+θ   2   ·Q   −1 )· A={g·B}·A+θ   2   ·Q   −1   ·A={g·B}·A+E   2 , 
 
 where E 2 =θ 2 ·Q −1 ·A. 
 
      By the assumptions, one has: 
 
| E   1 |=|θ 1   ·P   −1   ·B|≦ ½·| P   −1   ·B|&lt; ½ ·P   −1 ·β≦½ ·L   −1  
 
and 
 
| E   2 |=|θ 2   ·Q   −1   ·A |≦½ ·|Q   −1   ·A |&lt;½ ·Q   −1 ·α≦½ ·L   −1 . 
 
      In its turn, this implies that either |([{g·A}] P )·B| is not greater than L −1  or: 
 
{([{ g·A}]   P )· B}={{g·A}·B+E   1   }={{g·A}·B}+E   1   ={g·A·B}+E   1 , 
 
      Similarly, this implies that either |([{g·B}] Q )·A| is not greater than L −1  or: 
 
{([{ g·B}]   Q )· A}={{g·B}·A+E   2   }={{g·B}·A}+E   2   ={g·B·A}+E   2 . 
 
      Since A·B=B·A, one has: 
 
{([{ g·A}]   P )· B }−{([{ g·B}]   Q )· A}=E   1   −E   2   =θ·L   −1 , 
 
 where −1&lt;θ&lt;1.□
 
      We say that a vector x=(x 1 ,x 2 , . . . , x n ) is (K, D)-consistent if: 
 
(− c, −c, . . . , −c )≦ x−[x]   K ≦( c, c, . . . , c ) 
 
 where c=½−1/(2D). 
 
      Corollary. In the notation of the Proposition, if L=D·K and one the vectors {([{g·A}] P )·B} and {([{g·B}] Q )·A} is (K, D)-consistent then 
 
[([{ g·A}]   P )· B]   K =[([{ g·B}]   Q )· A]   K . 
 
      For the 2-dimensional embodiment of the system hereof the Corollary is applied with n=2, K=(K,K). Therefore, the Corollary guarantees that (s 1 , s 2 )=(s′ 1 , s′ 2 ) in the protocol.  
      In creating a geometric key establishment system in accordance with the two-dimensional embodiment hereof (and with the following small numbers for ease of illustration), a first step is to choose publicly available parameters of the system: a vector g=(g 1 , g 2 ) and integer parameters D, N, K greater than 1 each. Take, for example, g 1 =√2, g 2 =√3, N=K=8, D=2. Next, suppose that Alice chooses a pair of secret integers (a 0 , a 1 )=(48176925, 18034725). Alice calculates the vector 
 
( y   1   , y   2 )=({ g   1   a   0   +g   2   a   1   }, {−g   1   a   1   +g   2   a   0 }) 
 
 each coordinate of which rounded to D+N+K=18 decimal places:  
         (       y   1     ,     y   2       )     =       (       {         2     ·     48176925       +       3     ·     18034725         }     ,     {         -     2       ·     18034725       +       3     ·     48176925         }       )     =       (       {       68132460.728431422183990297539596     +     31237060.000532620547511774721314       }     ,     {       -     25504952.688669116604000035676723       +     83444881.852435233704474767836253       }       )     =     (       0.728964042731502072     ,     0.163766117100474732)                 
 
 and sends this vector (y 1 , y 2 ) to Bob. Suppose that at the same time Bob a pair of secret integers (b 0 , b 1 )=(19082792, 27045821). Alice calculates the vector 
 
( z   1   , z   2 )=({ g   1   b   0   +g   2   b   1   }, {−g   1   b   1   +g   2   b   0 }) 
 
 each coordinate of which rounded to D+N+K=18 decimal places:  
         (       z   1     ,     z   2       )     =       (       {         2     ·     19082792       +       3     ·     27045821         }     ,     (         -     2       ·     27045821       +       3     ·     19082792         }       )     =     (       {       26987143.25434479912512475172839     +     46844736.104413300451707772339473       }     ,     {         -     38248566.863715063905876737732694       +     33052365.294268911065907204826118})       =     (       0.358758099664220248     ,     0   ⁢     .430553847160030467         )                   
          and sends this vector (z 1 , z 2 ) to Alice. Upon receiving the vector (y 1 , y 2 ) from Alice, Bob calculates the vector (k 1 , k) by the formula: 
 
( k   1   k   2 )=({ y   1   b   0   +y   2   b   1   }, {−y   1   b   1   +y   2   b   0 }) 
 
 with the precision K+D=10 decimal places after dot:  
         (       k   1     ,   K     )     =       (       {         0.728964042731502072     ·     19082792       +           0.1637661171   2     ⁢   00474732       ·     27045821         }     ,     {         -     0.728964042731502072       ·     27045821       +       0.163766117100474732     ·     19082792         }       )     =       (       {       13910669.202924365887545024     +     4429189.088964478616694972       }     ,     {       -     19715431.015152556100441112       +     3125114.749276002412011744       }       )     =     (       0.2918888445     ,     0.7341234463)                 
       

      Since this vector is (K,K, D)-consistent (i.e., its first coordinate has digits 4 and 5 at 9 th  and 10 th  places after dot, and its second coordinate has digits 6 and 3 at 9 th  and 10 th  places after the dot), the vector (k 1 , k 2 ), being rounded to the first K=8 digits of each coordinate, constitutes the geometric key in possession of Bob: 
 
(0.29188884, 0.73412345). 
 
      Upon receiving the vector (z 1 , z 2 ) from Bob, Alice calculates the vector (k′ 1 , k′ 2 ) by the formula: 
 
( k′   1   , k′   2 )=({ z   1   ·a   0   +z   2   ·a   1   }, {−z   1   ·a   1   +z   2   ·a   0 }) 
 
 with the precision K+D=10 decimal places after dot:  
         (       k   1   ′     ,     k   2   ′       )     =       (       {         0.358758099664220248     ·     48176925       +       0.430553847160030467     ·     18034725         }     .     {       -     0.358758099664220248       ·     48176925       }       )     =       (       {       17283862.0606656640713774     +     7764920.231223180463966575       }     ,     {       -     6470103.6689668045121118       +     20742760.403090250806373975       }       )     =     (       0.2918888445     ,     0.7341234463       )             
 
      Since this vector is (K, D)-consistent (i.e., its first coordinate has digits 4 and 5 at 9 th  and 10 th  places after dot, and its second coordinate has digits 6 and 3 at 9 th  and 10 th  places after the dot), the vector (k′ 1 , k′ 2 ), being rounded to the first K=8 digits of each coordinate, constitutes the geometric key in possession of Alice: 
 
(0.29188884, 0.73412345). 
 
      Thus, the vector (0.29188884, 0.73412345) is the geometric key shared by Alice and Bob. This key can be used in any major symmetric cryptosystem.  
      The invention has been described with reference to a particular preferred embodiment, but variations within the spirit and scope of the invention will occur to those skilled in the art. For example, it will be understood that the public information g=(g 1 , g 2 ), D, N, K of the system can be stored on any suitable media, for example a “smart card,” which can be provided with a microprocessor capable of performing arithmetic operations so that the keys can be distributed to and/or from the smart card.