Patent Publication Number: US-5835598-A

Title: Public key method of encoding data

Description:
FIELD OF THE INVENTION 
     The present invention relates to a computerized method for encoding data by a microprocessor, which data may then be transmitted and decoded by another microprocessor. The method of encoding data is based on the difficulty of factoring large natural numbers, where exactly two codes are associated with each user: a publicly known key for encoding the data and a private key known only to the respective user for decoding the data. A user A (who wants to encode data to make it accessible only to user B) encodes the data with the help of user B&#39;s public key, and user B retrieves the data with the help of the private key. 
     BACKGROUND OF THE INVENTION 
     Public key encoding methods were introduced by articles by Diffie and Hellmann (W. Diffie, M. E. Hellmann, &#34;New directions in cryptography,&#34; IEEE Transactions on Information Theory, vol. IT-22, November 1976, pages 644-654) and by Rivest, Shamir and Adleman (R. Rivest, A. Shamir and L. Adleman, &#34;A method for obtaining digital signatures and public-key cryptosystems,&#34; Communications of the ACM, vol. 27, no. 2, February 1978, pages 120-126, the so-called RSA method). These methods use two keys, one for encoding and another for decoding. Anyone can encode a message with the public key, but only someone who knows the private key can decode the message. 
     There are various applications of public key methods. Separating the functionality into a public key and a private key greatly facilitates key management. Public key methods can also be used for replacing keys for other encoding processes (such as DES). The above-mentioned examples of applications are described in detail by Beutelspacher (A. Beutelspacher, Cryptology, Vieweg-Verlag 1994). 
     SUMMARY OF THE INVENTION 
     While these known methods have provided the best possibility of achieving reliable data privacy so far, an even greater level of security is desired. The object of this invention is thus to achieve such data privacy. 
     This object is achieved by a computerized public key method in which the public key is a number n that is the product of two large primary numbers p and q, and the private key includes these two primary numbers. The data set m is encoded by breaking it down into blocks m 1  . . . m k  that are interpreted as being numbers smaller than n, and the encoded data set includes the coefficients a k-1 , . . . , a 0  of the polynomial P(x)=x k  +a k-1  x k-1  + . . . +a 1  x+a 0  :=(x-m 1 )· . . . ·(x-m k )mod n. Decoding is accomplished by first determining the zero positions of the polynomials P(x) mod p in the finite field GF(p) and P(x) mod q in the finite field GF(q) and then combining them with the help of the Chinese remainder theorem to yield solutions of the equation P(x)=0 mod n, where the blocks m 1 , . . . m k  of data set m are contained among the solutions of the equation P(x)=0 mod n calculated in this way. 
     Advantageous embodiments of this process in the sense of achieving even greater data privacy are: (1) that, to facilitate finding data blocks m 1 , . . . , m k  among the solutions of equation P(x)=0, before these data blocks are interpreted as numbers smaller than n, they are provided with additional information that establishes their sequence and makes it possible to differentiate them from the other solutions; and (2) that the numbers used satisfy the conditions k≧2 and log 2  n≈500. 
     The process according to this invention, like the process described by Rivest, Shamir and Adleman, is based on the difficult mathematical problem of factoring numbers that are the product of two large primary numbers. The public key here is one such number and the private key includes the two primary numbers of this number. 
     The data privacy of the method described here can be defined mathematically with a greater precision than that of the most widely used RSA public key method. It also offers advantages in terms of speed in the encoding operation in comparison with the general RSA method. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The following figures are provided: 
     FIG. 1 shows a flowchart of the method of the present invention. 
    
    
     DETAILED DESCRIPTION 
     This process and possible refinements of it are explained in greater detail below with reference to practical examples. 
     Use of an integer n that is the product of two large primary numbers p and q and polynomials over the ring Z n  is essential to this invention. As shown in FIG. 1 at step 102, to encode a message m, it is broken down into blocks m 1 , . . . m k , so these blocks can be interpreted as numbers smaller than n. 
     To encode the message, the polynomial P(x)=x k  +a k-1  x k-1  + . . . +a 1  x+a 0  :=(x-m 1 )· . . . ·(x-m k ) mod n is formed, as shown at step 106. The numerical sequence a k-1 , . . . , a 1 , a 0  thus represents the encoded message m. 
     As shown at step 108, to decode the message, the zero positions of the polynomial P(x) mod p in the finite field Z p  and of P(x) mod q in the finite field Z q  are calculated--for example, with the help of the probabilistic algorithm described by Ben-Or (Ben-Or, &#34;Probabilistic algorithms in finite fields,&#34; Proc. IEEE FOCS 1981, pages 394-398). One obtains with a high probability k solutions that can be combined with the help of the Chinese remainder theorem to yield k 2  solutions of the equation P(x)=0 mod n. 
     In order to be able to find the original blocks of plain text in the correct sequence, before the blocks mi are interpreted as a number smaller than n, they are preferably provided with redundant additional information that makes this possible, as shown at step 104. 
     It can be shown that the data privacy in the process according to this invention is equivalent to the difficulty of factoring the number n. If someone trying to break the code could find a solution to the equation P(x)=0 mod n, he could also factor the number n with a probabilistic method. For this method to be secure, the numbers that occur should at least meet the following conditions: 
     k≧2 and 
     log 2  n ≈500. 
     To provide a simple example, the parameters have been selected so they are very small. If p=1237 and q=5683, then n=7029871. To encode the message m=123456789101112, it is divided into three blocks m 1  =12345, m 2  =67891 and m 3  =01112. Then the blocks are provided with additional information, which in this example includes simply of placing the digits ii at the beginning of m 1  : 
     m 1  &#39;=1112345, m 2  &#39;=2267891 and m 3  &#39;=3301112. 
     Then the following is formed: 
     
         P(x)=x.sup.3 +348523x.sup.2 +3270693x+5041428 
    
     and the encoded message is: 
     (348523, 3270693, 5041428). 
     For decoding, two polynomials are formed: 
     
         P(x)mod p=x.sup.3 +926x.sup.2 +65x+653≈(x-796)(x-470)(x-282) mod p and 
    
     
         P(x)mod q=x.sup.3 +1860x.sup.2 +2968x+607≈(x-4972)(x-4160)(x-374) mod q. 
    
     The expanded Euclidian algorithm yields the notation: 
     -2683·p+584·q=1. (The variables p and q are known to the decoder microprocessor.) 
     The following table can be constructed with the help of this formulation by calculating the following numbers for each zero position z p  of P(x) mod p and z q  of P(x) mod q: 
     -2683·p·z q  +584·q·z p  mod n. 
     
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mddq\modp                                                       
           796          470      282                                      
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4972       3301112      3948974  5659557                                  
4160       57833771     6431633  1112345                                  
 374       1620029      2267891  3978474                                  
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     The data blocks m 1  &#39;=1112345, m 2  &#39;=2267891 and m 3  &#39;=3301112 can easily be differentiated from the other zero positions on the basis of their special structure and thus the message m can be recovered after removing the additional information. 
     The above described method is performed by an encoding microprocessor and a decoding microprocessor. In addition, the Chinese remainder theorem calculations could be performed not only by the decoder microprocessor itself, but also by a special apparatus associated with the decoder microprocessor, for example that described in U.S. Pat. No. 4,709,345, which is hereby expressly incorporated by reference herein. 
     Commonly-assigned U.S. Patent application Ser. No. 08/631,278, filed on Apr. 12, 1996, now pending is hereby expressly incorporated by reference herein.