WO1982002129A1

WO1982002129A1 - Rsa public-key data encryption system having large random prime number generating microprocessor or the like

Info

Publication number: WO1982002129A1
Application number: PCT/US1981/001644
Authority: WO
Inventors: Milgo Inc Racal; William J Miller; Nick G Trbovich
Original assignee: Milgo Inc Racal
Priority date: 1980-12-15
Filing date: 1981-12-11
Publication date: 1982-06-24
Also published as: FR2496303B1; SE8204697D0; GB2101855A; BE891490A; BR8108915A; GB2101855B; US4351982A; AU544169B2; FR2496303A1; CH660822A5; SE8204697L; CA1173538A; JPS57502145A; NL8120500A; AU8006982A

Abstract

A public-key data encryption system employing RSA public-key data encryption including a message encryptor (56) or (110) capable of encrypting messages using a non-secret encryption key, a transmitter-receiver (70) or (76) coupled to the message encryptor (56) or (110) which transmits or receives an encrypted message to or from a remote location, the transmitter-receiver also being coupled to a decryptor (54) or (82) capable of decrypting a received encrypted message using a decryption key which is a secret input to the decryptor (54) or (82), and an encryption-decryption key generator (52) or (106), including a microprocessor or other large-scale integrated circuit or circuits (50) or (108) formed to generate a sequence of prime numbers beginning with a selected known prime number having a length relatively short with respect to the desired length of the last in the sequence of prime numbers, which is constructed to form the sequence of prime numbers in the form hP + 1 where P is the preceding prime number in the sequence, and to test hP + 1 for primality by first determining if hP + 1 has a GCD of 1 with x, wherein x is a composite number consisting of the product of all known prime numbers less than or equal to a pre-selected known prime number and if the GCD is not equal to 1, incrementing h to form a new hP + 1 to be tested for a GCD equal to 1, and when a GCD is found to be 1, performing the primality tests to determine whether 2hP (Alpha) 1 (Mod (hP + 1)) and 2h (Alpha) 1 (Mod (hP + 1)) and if either 2h (Alpha) 1 (mod (hP + 1)) or 2h (Alpha) 1 (mod (hP + 1)) further incrementing h and so on until a prime is found in this manner and then determining if the length of the prime number is of or greater than the desired length. If the hP + 1 which has been determined to be prime is not of the desired length, hP + 1 is placed in the sequence of prime numbers and a new h selected to be used to find the next prime number in the sequence in accordance with the above described procedure by forming a new hP + 1 in which P is the previously determined prime number in the sequence of prime numbers. When a prime number in the sequence of prime numbers is found which is of the desired length it is input into the encryption-decryption key generator (52) or (106) for generating the RSA public-key encryption and decryption keys.

Description

RSA PUBLIC-KEY DATA ENCRYPTION SYSTEM HAVING LARGE RANDOM PRIME NUMBER GENERATING MICROPROCESSOR OR THE LIKE

FIELD OF THE INVENTION The present invention relates to RSA public- key encryption systems.

BACKGROUND AND SUMMARY OF THE INVENTION

The present invention relates to public- key encryption systems, which employ the RSA so-called "trap-door, one-way permutation" data encryption. More particularly, the present invention relates to the method and apparatus employing a currently commercially available microprocessor for generating the large random prime numbers satisfying the requirements for the so- called "trap-door, one-way permutation," incorporated into an RSA public-key data encryption system.

Public-key data encryption, -as originally suggested by Diffie and Hellman, "New Directions in Cryptography," I.E.E.E. Transactions on Information Theory (Nov. 1976) (the disclosure of which is hereby incorporated by reference), and perfected by Rivest, Shamir, and Adelman, "A Method for Obtaining Digital Signatures in Public-Key Crypto Systems," MIT Technical Memo LCS/TM82 (Aug. 1977) (the disclosure of which is hereby incorporated by reference), is by now well-known.

OMPI IPO The basic reason for public-key encryption system is to ensure both the security of the information transferred along a data line, and to guarantee the identity of the transmitter and ensure the inability of a receiver to "forge" a transmission as being one from a subscriber on the data line. Both of these desired results can be accomplished with public-key data encryption without the need for the maintenance of a list of secret keys specific to each subscriber on the data line and/or the periodic physical delivery or otherwise secure transmis¬ sion of secret keys to the various subscribers on the data line. Through the use of the so-called "open trap¬ door, one-way permutations" data can be sent from a transmitter to a receiver in an encrypted form using a publicly-known publicly transmitted encryption key, but at the same time not allowing an eavesdropper on the line to be able to decrypt the message within a period of time so. large as to guarantee the security of the encrypted message. This method of public-key encryption developed by Rivest, Shamir & Adelman, and now generally referred to as RSA, is based upon the use of two extremely large prime numbers which fulfill the criteria for the "trap¬ door, one-way permutation." Such a permutation function enables the sender to encrypt the message using a non- secret encryption key, but does not permit an eaves¬ dropper to decrypt the message by crypto-analytic techniques within an acceptably long period of time. This is due to the fact that for a composite number composed of the product of two very large prime numbers, the computational time necessary to factor this compos¬ ite number is unacceptably long. Another method of public-key encryption has been suggested for the trans- ittal of NBS standard keys in "Electronics" magazine of

OMPI WIPO June 5,-1980 at 96-97. This does not use the RSA scheme, but rather employs a public-key encryption scheme similar to earlier proposals by, e.g., Hellman, prior to RSA, and has certain security problems not associated with the RSA scheme.

Generally, the RSA system has the following features. Assuming that the receiver of the message is located at Terminal A, Terminal A will have first com¬ puted two very large random prime numbers p, q. The product of p and q is then computed and constitutes the value n. A large random integer e is then selected which has the property that the greatest common divisor of. e and the product of (p - 1) and (q - 1)- is 1, i.e., GCD [e, (p - 1) (q - 1)] = 1 In other words, e is a large random integer which is relatively prime to the product of (p - 1) and (q - 1). An integer d is then computed which is the "multiplica¬ tive inverse" of e in modulo (p - 1) (q - 1). That is to say: e*d 3 i [mod (p - 1) (q - 1)].

Terminal A transmits n and e to another terminal. Terminal B, in plain text without encryption, or a public list of n and e for every terminal, including Terminal A, is made publicly known. Terminal B responds by encrypting and transmitting a message M into an encrypted transmission C as follows:

C ≡ E (M) = M^e (mod n) . It will be understood that each character transmitted along the data network is encoded as a number prior to any encryption, and upon decryption the identical number will result which corresponds to the identical character.

It will be further understood that the message M to be encrypted may be .a single binary number of.

O PI I e.g., 25 bytes in length, i.e., 336 bits, with each group of, e.g., 3 or 4 bits representing the encoding of a specific character within the message M. When the encrypted message C is received and decrypted using the RSA scheme, the identical 25 byte binary number is repro¬ duced, from which the encoded data characters of the message M can be decoded as is well known in the art.

Terminal 3 then sends the encrypted message C. Terminal A then performs an operation upon the received encoded message C as follows:

(C)^d (mod n) Due to the particular nature of the selected large random prime numbers this "open trap-door, one-way permutation" results in the identical message K. However, an eavesdropper on the line who receives or otherwise knows the publicly transmitted n and e cannot decode the message sent by terminal B without the number d. Thus, the transmission from Terminal B to Terminal A, after receipt of or knowledge of at Terminal B of n and e computed at or for Termi¬ nal A, is totally secure. In addition, because the sender of the message and the intended receiver of the message each have a unique n and e, the sender and receiver can each guarantee the authentication of the other by an encrypted "signature," encrypted using their separate n's and e's and decrypted using their separate d's. In this way, both sender and receiver can guaran¬ tee the authentication of the origin and receipt of the particular message. This is extremely important in applications such as encrypted electronic mail used for business transactions where proof of transmission and receipt are vital.

In the past, however, the use of such an RSA public-key encryption system has been limited to

O PI TPO transmitting and receiving terminals which have access to large scale digital data computers. This is due to the fact that the generation of the required large random prime numbers has only been practical on large scale digital computers. This is because the random numbers p and q must be extremely large. For example, as explained in Rivest, Shamir & Adelman in the above- noted technical memo, there exists a factoring algo¬ rithm, the Schroeppel algorithm, for factoring a number n. For a n of e.g. , 50 digits in length, the Schroeppel algorithm can be used to factor n in 3.9 hours, on a large scale digital computer. Factoring n is the easiest technique for use by a cryptoanalyst to break the RSA encryption code. If the length of n is increased to 100 digits, the computational time necessary for complete factoring with the Schroeppel algorithm increases to 73 years. Approximately a 1,000 year period is generally accepted as being a totally secure computational decryption time for an encrypted message. This requires a n of approximately 120 digits in length. Since n is derived from the product of two large prime numbers p and q, the product of p and q will have a number of digits equal to the sum of the digits in p and q. Therefore, p and q must be large random prime numbers each having approximately

60 digits in order for about a 1,000 year cryptoanalysis time using, e.g., the Schroeppel algorithm. A good rule of thumb is that for each 2.5 digits in a decimal number there will be one byte of eight binary bits, thus sixty digits would translate to 24 bytes.

The method of finding large random primes outlined in the RSA scheme requires the evaluation of a-^**^5""^***- (mod P) for 100 random a < P. If for any a, a^***⁵-^*-^*- (mod P) is not 1, then another P must be chosen and another iteration of 100 modulo exponentiations begun. For the processing of a digital binary representation of decimal numbers on a commercially available micro¬ processor, each exponentiation requires a multiprecision multiply and divide for each bit in the exponent and an additional multiply and divide for each 1 bit. Thus, for each p and q, which must be on the order of about 184 bits (23 bytes) in order for the product p x q to be of the required approximately 50 bytes in length, each exponentiation takes an average of 92 seconds on a microprocessor, for example, an Intel 8085 microprocessor.

In testing the RSA scheme, it has been found that most of the time, whenever 3P-***- (mod P) is 1, then aP~l (mod P) is also 1 for each choice of o < a < P.

Therefore, the first value of a was always chosen to be 3, and the remaining 99 were chosen as random. Approxi¬ mately 120 value P have to be tested before one is found which works. The time to find such a P using commer- cially available microprocessors averages approximately 3 hours. Since 4 separate primes are required for RSA implementation, 12 hours are needed to find the 4 prime numbers with a probability of one-half. For the recom¬ mended probability of 2~1°^*-*, the time necessary increases to 1200 hours.

Public-key encryption has tremendous utility in both unique signatures for message authentication and for transmitting on open channels the periodic changes in encryption keys, e.g., the NBS standard keys. In the latter application, the need for a master key in which to encrypt the periodic changes of the standard key is avoided. Thus, the need to transmit over a secure channel or to physically transport the master key by courier or the like is avoided. Without public-key

OMP encryption, each subscriber must have the master key. Though the master key does not change often, as each new subscriber comes on the data encryption line, a master key must be sent in some secure manner to that sub- scriber. Each such transfer, even over a secure channel or by physical hand delivery, could be compromised, thus necessitating changing the master^'key for all sub¬ scribers, when and if compromise is discovered, and putting the master key in the hands of each subscriber in a secure manner. Public-key encryption enables the standard keys, which change periodically, to be sent over open channels to each subscriber with a publicly- known public-key, which though publicly known, is not capable of decryption by anyone other than the indi- vidual subscriber. The utility of a public-key data encryption system for message authentication and trans¬ mission of standard keys is more fully described in Hellraan, "The Mathematics of Public-Key Cryptography," Scientific American, Vol. 241 (1979) , the disclosure of which is hereby incorporated by reference.

These advantages of public-key encryption will enable the expanded use of encryption using, e.g., the NBS standard keys, for message transfers by electronic means in business applications, where security and transmission and receipt authentication are crucial.

Presently, however, in order to come "on-line" in such a data encryption system, using RSA public-key encryption for the transfer and receipt of the standard keys, or signatures, a large start-up time or access to a large- scale digital computer is needed. Another alternative of hand delivery of the large randomly generated prime numbers, unique to each subscriber, exists. However, this also requires a possibly compromisable physical transfer by some secure means, which cannot alvays be

OMP guaranteed secure. This also requires the same central location which generates and provides the "secret" public-key decryption key also, at least at some time prior to providing this decryption key to a subscriber, know this key. This is another possible avenue of compromise.

It is, therefore, much more preferable for each subscriber to be able to generate its own large randomly selected prime numbers. Currently, in order to do this, access to a large-scale digital computer is needed, or some twelve to twenty hours of computational time, on a currently commercially available micro¬ processor, is needed. Even with twelve to twenty hours on the microprocessor, using the exponentiation tech- nique suggested by Rivest, Shi ir and Adelman, the numbers generated have only a 50—percent probability of being prime. The only way to check the primarity is to try encrypting and decrypting a message using the generated primes in the RSA scheme. For the recommended probability of 2^~ 0, 1200 hours of computation time, approximately, are needed on currently commercially available microprocessors.

There thus exists a tremendous need for an RSA public-key data encryption system in which a subscriber, having only a microprocessor of the kind currently com¬ mercially available, can come on-line in a relatively short period of time by generating the required prime numbers in a few hours, rather than dozens of hours. The use of a GCD routine according to the present invention for eliminating composite numbers without exponentiation, along with a unique method of forming a sequence of primes, enables this time to be decreased to about 2 hours, because the number of required exponentiations is dropped from 120 to 20 for

OMPI each P tested. And only 2 are needed as opposed to standard RSA's 4 (see line 30, page 5). In addition, because the sequence of primes is generated in the form of (hP + 1), finding an hP + 1 which is prime and of a sufficient length as one of the RSA large random prime numbers p or q, the value of p - 1 (or Q - 1), i.e., hP, will also have a large prime factor, satisfying the RSA requirements. Thus, only hP + 1 must be tested for primality, eliminating one of the required tests of primality in the RSA scheme for each of p and q. There- fore, two large random numbers of the form hP + 1 must be tested for primality according to the present inven¬ tion, rather than four numbers according to the sug¬ gested procedure in the RSA scheme. The GCD routine eliminates most nonprimes. The GCD routine involves the use of a precomputed composite number equal to the product of the first selected number, e.g. , 34 of the known prime numbers, i.e., less than or equal to 139, in a determination of whether the GCD of that composite number and the number being tested is equal to 1. The GCD equaling 1 is a necessary but not sufficient test of the primality of the number being tested. Thus, if the GCD does not equal 1, then the number can be eliminated as a choice of a prime number without the need for the further tests for primarily. The further tests for pri- mality according to the present invention are the Ξuler identities, which constitute a determination of whether both of the following relationships hold true:

2hP s 1 (mod hP + 1) 2^h ≡ 1 (mod hP + 1) The reason that most nonprimes are found by the GCD routine is that a random sample of odd integers has one- third of the integers divisible by 3, one-fifth by 5, etc. Tests by the applicants have shown that about 140 hP + 1 random numbers of 23 bytes long must be chosen in accordance with the present invention before a prime number is found. Using a sample of the size of 140, 1/139 or 1 of the sample numbers should be divisible by 139, which is the largest prime not greater than 140. If the sample number is divisible by any of the primes of up to 139, then a composite number which has that prime as a factor will have a GCD with respect to the sample number that is not equal to 1. By using a com- posite number equal to the product of all of the 34 primes up -to 139, the GCD test checks whether the sample is divisible by one of those primes.

The present invention relates, therefore, to a method and apparatus employing a commercially available microprocessor for selecting the large random prime numbers necessary for RSA public-key encryption. More particularly, the present invention relates to a large prime number generating system for use in a public-key encryption system to determine the large random prime numbers by generating a sequence of prime numbers hP + 1, where P initially is a randomly selected known prime number of a short length relative to the approx¬ imately 23 byte size of the desired randomly selected prime number, and wherein the successive numbers in the sequence have the relationship of hP + 1 to the pre¬ ceding prime number P in the sequence, with h initially being selected to be of a byte length approximately one- half that of the byte length of P, and the values of hP + 1 being initially checked by the GCD routine to eliminate the necessity of checking the number hP + 1 with the exponentiation modulo (hP + 1) tests of pri¬ marily for a large number of the value of n = hP + 1, as h is incremented to h + 2 until the value of n is determined to be prime for a given n in the sequence.

OMPI The unique method and apparatus employing the GCD elimination along with the generation of a sequence of primes hP + 1 until a prime of sufficient length is reached, which prime hP + 1 is also in the precise form 5 needed for RSA public-key encryption, and is guaranteed to be a prime, rather than probabilistically selected as prime as with pure exponentiation prime derivation, makes the generation of prime numbers of suitable length on a microprocessor commercially feasible for GSA

10. systems.

The problems enumerated in the foregoing are not intended to be exhaustive, but are rather among many which tend to impair the effectiveness of previously- known methods and apparatus for generating large random 5 prime numbers for use in RSA public-key encryption.

Other noteworthy problems may also exist; however, those presented above should be sufficient to demonstrate methods and apparatus for generating random prime numbers for RSA public-key encryption appearing in prior 0 art have not been altogether satisfactory.

Examples of the more important features of the present invention have been summarized broadly in order that the detailed description thereof that follows may be better understood, and in order that the contribution 5 to the art may be better appreciated. There are, of course, features of the invention that will be described hereinafter and which will also form the subject of the appended claims. These other features and advantages of the present invention will become more apparent with 0 reference to the following detailed description of a preferred embodiment thereof in connection with the accompanying drawings, in which:

OMPI WIPO BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 shows a flow chart of the steps carried out by the microprocessor in order to select the necessary large random prime numbers for use in RSA public-key encryption;

Figure 2 is a block diagram of the public-key encryption system according to the present invention.

DETAILED DESCRIPTION

Turning to Figure 1, a flow chart for a commer- cially available microprocessor, e.g., an Intel 8085 microprocessor, for the purpose of determining the very large random prime numbers necessary for RSA public-key encryption is shown. It will be understood by those skilled in the art that other large-scale integrated circuits or combinations thereof which are not micro¬ processors, as that term has been come to be understood in the art, could be hard wired together or along with other circuit components to perform the same function as the microprocessor herein described and without the need for a main frame programmed digital computer or mini¬ computer. For example, a large-scale integrated circuit chip has been known to be useful for performing the expo¬ nentiation operation in the RSA scheme. This same chip could be used in a circuit to perform the same, or a similar, exponentiation operation as part of the method and apparatus of the present invention. Other well- known circuit components, e.g., registers, shifters, adders and multipliers, can be hard wired together, as is known in the art pursuant to the teaching of the present application, for carrying out the method of the present invention on an apparatus, which, though not commonly referred to as a microprocessor, performs the same function and at the same time is not a main frame

OMPI WIPO programmed digital data computer or minicomputer. These components, including the exponentiator chip, could all be on the same large-scale integrated circuit, custom designed to carry out only the method of the present invention, or on a number of separate large-scale inte¬ grated circuit chips, each connected electrically to one or more of the others fnd each custom designed to carry out one or more of the steps of the process of the present invention. This single large-scale integrated circuit chip or group of large-scale integrated circuit chips could have a control means hard wired on the chip or be associated with a separate control means, e.g. , logic sequencer or finite state machine, hard wired to produce a sequence of control signals which control signals might also be effected by the outcome of inter¬ mediate steps in the process. Based upon the disclosure in the present invention, one skilled in the art would be able to set up the control means of a commercially available microprocessor or be able to assemble other large-scale integrated circuit components with asso¬ ciated integral or external control means in the manner of the apparatus of the present invention for the purpose of carrying out the process of the present invention. When used in the attached claims, "large scale integrated circuit" or "large scale integrated circuit means" is intended to cover both commercially available microprocessors and an assemblage of one or more large-scale integrated circuit components, hard wired to perform a specific function or sequence of functions and with an associated integral or external control means, and not to include a software programmed main frame digital computer or minicomputer.

Turning now to the preferred embodiment using a microprocessor means, beginning at start 10 the

iURE

OMPI -14-

micrcprocessor is constructed to choose, in block 12, a prime number P at random from the known prime numbers having a byte length of, e.g., two eight-bit bytes, i.e., of a decimal of length of approximately five. Step A, at 13 in Figure 1, then indicates that the micro¬ processor is constructed to choose, in block 14 of Figure 1, a random h with one-half the bytes of P, i.e., initially one byte. Also, h is selected to oe an even number and 2 _ • h < P. In step B, at 15 in Figure 1, the value N is set to equal hP + 1 in block 16 in Figure 1. A greatest common divisor (GCD) routine is then performed within the microprocessor in diamond 17, wherein the relationship GCD L(hP + 1), (x)] = 1 is determined to be true or false, where x is a composite number consisting of the product of the first selected number, e.g., thirty-four, of the prime numbers, i.e., 2, 3, 5, 7, 11, ... 139. If GCD equals one as deter¬ mined in block 17, step D at 19 in Figure 1 is carried out in the microprocessor, as indicated by block 20, wherein the current prime number in the sequence of prime numbers being computed.

It can be seen that the sequence of prime numbers generated by the microprocessor makes a rela¬ tively rapid progression to the 23 byte length. This is due to the fact that after each P is found in the sequence, by virtue of being determined to be a prime number, the succeeding prime number in sequence is selected using the current P and an h having one-half the byte length of that current P. Thus, for example, the product hP of the first sampled number will be of three bytes in length, i.e., the sum of two bytes for P and one byte for h. For th next prime number in the sequence, the product hP will have a four-byte length, i.e., the sum of three bytes for the first P determined

OMPI in accordance with the above method, i.e., the current P at that time, and one byte for h. The next product hP in the sequence will have a byte length^' of six (four bytes for the current P and two for h), with the next value of hP being of nine bytes (six bytes for the current P and three for h), the next value of hP being 13 (nine bytes for the current P and four for h) and the next successive prime number in the sequence will have the value of hP of 19 bytes (13 for the current P and six for h). After this point, selecting the value of h to be one-half the byte length of the current P will result in an increase in the byte length to approxi¬ mately 27 bytes. However, it is necessary to achieve only a 23-byte length for the utilization of the composite number of two random prime numbers generated according to the present invention in an RSA system having a cryptoanalytically secure computation time of approximately 1,000 years. Therefore, the micro¬ processor is set up to reduce the size of the randomly selected h, i.e., the byte length thereof, when the current P is determined to be of greater than or equal to 19 bytes, but not of the desired 23 or more bytes in length. There is achieved by the operations performed in diamonds 32 and 34, in which the microprocessor is shown to be constructed to proceed from diamond 32 to diamond 34 if the current P in the sequence is deter¬ mined to be of 19 or more bytes, and, in diamond 34, if the current P in sequence is determined to be not of 23 or more bytes, the microprocessor proceeds to the step in block 36 where he is limited to a, e.g., four byte length for use in step B, rather than the carrying out of step A in block 14, where h is normally chosen to be one-half the byte length of the current P. Having found a P in the sequence of P's produced according to the

OMPI /,, WIPO icroprocessor flow chart shown in Figure 1, which satis¬ fies the requirements of diamond 34, i.e., that P be of 23 or more bytes in length, the microprocessor proceeds to the stop command 40 and the existing current P at the time of the stop command is the output as one large random prime number useful for carrying out RSA public- key encryption. This large random prime number is an input into the remainder of the full system discussed below for generating and employing encryption/decryption keys. Turning now to Figure 2, a public-key data encryption system according to the present invention is shown. A first terminal, terminal A, is shown to . include a large random prime number generating micro¬ processor 50, an encryption/decryption key generating microprocessor 52, and an encryption/decryption micro¬ processor 54. It will be understood that the micro¬ processors 50, 52 and 54 are shown separately for purposes of discussion and illustration, but could be a single microprocessor formed to perform the various functions, as is indicated by the microprocessors 50, 52 and 54 being included within the dotted lines 56. The large random prime number generating microprocessor 50 is constructed to perform the above-noted steps to generate^' a large random prime number p-j_ and to repeat those steps to generate a second large random prime number.q*j_, which are shown schematically to be inputs on lines 58 and 60, respectively, to the encryption/ decryption key generating microprocessor 52. The encryption/decryption key generating microprocessor 52 computes the value of n^*j_ as the product of p]_ and q , the value of e*j_ as having a greatest common divisor (GCD) with ( ^*ι_ - 1) (q^*j_ - 1) = 1; and the value of d as the multiplicative inverse in modulo (p*_j_ - 1) (q - 1) of

OMPI . IP e^*j_, i.e., dχ*eχ ≡ 1 [mod (p - 1) (q - 1)]. The values of d]_, ex and n^*j_, are shown schematically to be the inputs to the encryption/decryption microprocessor 54 on lines 62, 64 and 66, respectively.

The encryption/decryption microprocessor 54 provides the values of e^* _j_ and nx on line 68 to a transmitter-receiver 74 for Terminal A. It will be understood that the values of ex and n could be supplied directly from the encryption/decryption key generating microprocessor 52, as shown by the dotted lines 72 and 74, respectively. The values of e and n are transmitted from the transmitter-receiver 70 at Terminal A to a transmitter-receiver 76 at Terminal B as shown schematically on line 78.

The transmitter-receiver 76 at Terminal B provides the values of ex and nx on line 80 to an encryption/decryption microprocessor 82 at Terminal B. A message M is input into the encryption/decryption microprocessor 82 at Terminal B, which generates the encrypted message C equal to m^el (mod nx) and provides the encrypted message to the transmitter-receiver 76 at Terminal B on line 86. The encrypted message C is trans¬ mitted from Terminal B through the transmitter-receiver 76 to Terminal A as indicated by line 88 where it is received on the transmitter-receiver 70 of Terminal A, and relayed to the encryption/decryption microprocessor 54 at Terminal B. The encryption/decryption micro¬ processor 54 performs a decryption operation on the encrypted measure C of C^*******-- (mod nx) = M, i.e., (M-31 (mod nχ))^l (mod nx) = M. In the case where the message M is the standard encryption/decryption keys, e.g., the NBS standard keys, these are input on line 102 to an encryptor/deσryptor 104, which may be any suitable encryptor/decryptor, e.g., that shown in the co-pending application of Miller, one of the present applicants, and assigned to the assignee of the present applicants, Serial No. 108,039, filed on December 28, 1979. It will also be understood that the message M could be an authenticating signature, specific to Terminal B, when the system is used for message receipt and transmission authentication.

Terminal B is also shown to have a large random prime number generating microprocessor 108 and an encryption/decryption key generating microprocessor 106. AS with the corresponding elements at Terminal A, these microprocessors 106, 108, along with encryption/ decryption microprocessor 82, may be an angle micro¬ processor formed to perform. the various functions, as is indicated by the inclusion of the microprocessors 106, 108 and 82, within the dotted lines 110. The large random prime number generating microprocessor 108 gener¬ ates two large prime numbers P2 and q2 which are input into the encryption/decryption key generating micro¬ processor 106 which generates the values 2/ <≥2 and n2, specific to Terminal B, and inputs them into the encryption/decryption microprocessor 82. These values of d2, 2 and n2 can then be used by Terminal B in the same manner as the values of bχ_r ex and nx are described above with respect to Terminal A. -It will further be understood that the neces¬ sity to transmit the values of ex and nx for Terminal A or e2 and n2 for Terminal B may be eliminated by having a publicly-known list of values of e and n for each sub¬ scribing data terminal. Since these can be publicly- known, there is no need to securely transmit or securely ship the list of e and n values for the various sub¬ scribers on the data transmission line. SUMMARY OF THE SCOPE AND ADVANTAGES OF THE PRESENT INVENTION

It will be appreciated that in constructing a public-key data encryption system according to the present invention, certain significant advantages are provided.

In particular, for the first time, subscribers on an encrypted data line who do not have access to a large-scale digital computer, can come on-line with values of e and n specific to that subscriber, securely generated in-house by the subscriber, in a relatively short period of time using a microprocessor to generate the large random prime numbers p and q. In the past, microprocessors were unable to generate the required large random prime numbers, having both the desired cryptoanalytically secure length and the recommended probability of being prime, in less than approximately 1200 hours. Thus, a new subscriber's data encryption system, including the large random prime number gener- ating microprocessor according to the present invention can be installed and an encrypted test message developed to check the proper operation of the system and the primality of the generated large random prime numbers, in only a few hours. The necessity for access to a large-scale digital computer or for the secure trans¬ mission from a central location of values of d, e and n for a new subscriber, which transmission is always subject in some degree to possible compromise, are thereby eliminated by the present invention. The present invention vastly expands the access to public- key data encryption systems for subscribers who need to use microprocessors because of the inability to access securely a large-scale digital computer. In addition, the system according to the present invention enables

✓^*"^' a user, with the application only of currently available microprocessors or other large-scale integrated circuit means, to periodically change the values of d, e and n, which can then be used in the system. The values of e and n are then either directly transmitted on public transmission lines to another terminal for the purpose of initiating the transmission_^ by that terminal of an encrypted message using the values of e and n so trans¬ mitted, or are then included in an updated ersion of the publicly-known list of values of e and n for the various subscribers on the date line. Further, the microprocessor constructed according to the present invention will generate large random prime numbers p and q, which are guaranteed to be both prime and of a desired cryptoanalytically secure length. Still fur¬ ther, only two numbers of the desired length must be checked for primality rather than the typical four numbers for RSA public-key data encryption.

The foregoing description of the invention has been directed to a particular preferred embodiment in accordance with the requirements of the patent statutes and for purposes of explanation and illustration. It will be apparent, however, to those of ordinary skill in this art, that many modifications and changes in both the apparatus and method of the present invention may be made without departing from the scope and spirit of the invention. For example, the exponential tests of pri¬ mality may be performed in the order set forth in the flow chart of the microprocessor or may be done in reverse order, so long as a determination is made that 2hP [mod (hp + 1)] is equal to one, and 2ⁿ [mod (hP + 1)] is not equal to one. In addition, the 23 byte length for the finally-selected key is chosen to give a

OMP cryptoanalytically secure computational time of approxi¬ mately 1,000 years, but may be varied as desired for longer or shorter cryptoanalytically secure computa¬ tional times. In addition, the step of reducing the byte length of the randomly chosen h, after P reaches a certain byte length, less than the desired cryptoanalyti¬ cally secure byte length for P, is used for optimizing the reduction in total computation time necessary to achieve a P having the desired cryptoanalytically secure byte length, as illustrated in Figure 1. However, this step may be eliminated by, e.g., in the example shown in Figure 1, in which h is initially chosen to be one-half the byte length of the previously determined P, the ulti¬ mate byte length for P -could be desirably set at 27. In this- event, upon reaching a 19 byte length for P, the next successive h would be selected at a byte length of 9, thus achieving the byte length desired, i.e., 27. Further, the selection of h at one-half the byte length of the current prime number P in the sequence of prime numbers is a convenient way to relatively rapidly increase the total byte length of hP + 1, in order to achieve the desired cryptoanalytically secure byte length of 23, e.g., as is shown in Figure 1. It will be understood that h could be selected to be, e.g. , an equal byte length (providing h < P) or a one-third byte length of the current P in the sequence of prime numbers, or of any other desired fraction of P, so long as h < P. The initial P in the sequence of prime numbers _is shown in the example of Figure 1 to be randomly selected. This adds additional randomness to the ultimately determined P having the desired crypto¬ analytically secure length. However, it is not neces¬ sary that the initial P be a randomly selected one, since H is also chosen at random for each generation of a prime number in the sequence of prime numbers.

OMPI wrpo

Claims

WHAT IS CLAIMED IS:

1. In a public-key data encryption/decryption system having a plurality of terminals, a terminal comp ising: a transmitter-receiver capable of receiving an encrypted message encrypted using a non-secret encryption key specific to the terminal;

An encryptor/decryptor, coupled to the transmitter-receiver, adapted to decrypt the encrypted message using a secret decryption key; an encryption/decryption key genera¬ tor coupled to the encryptor/decryptor and adapted to generate the non-secret and secret keys by use of a pair of large randomly selected prime numbers; a large random prime number genera¬ tor, including a large-scale integrated circuit, coupled to the key generator and adapted to provide the key-generator with the pair of random large prime numbers having a desired length, the large-scale integrated circuit being constructed to perform the following operations in select- ing each of the pair of large random prime numbers:

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OMPI starting with a P, which is a known prime number having a length which is relatively short in com¬

30 parison to the desired length, form¬ ing a sequence of prime numbers in the form hP + 1, wherein P is the current prime number in the sequence of prime numbers, by selecting at

35 random an h and forming hP + 1 and testing hP + 1 for primality by deter¬ mining first if the greatest common divisor (GCD) of (hP + 1), (x) is one, wherein x is a composite number

40 consisting of the product of all known prime numbers less than or equal to a selected known prime number, and if the GCD is equal to one, determining if both 2^nP = 1 [mod

45 (hP + 1)] and 2^ f l [_raod (hP + 1)], and, if GCD is not equal to one, incrementing h to form a new hP + 1, and if GCD is equal to one, but either 2^hP ψ- 1 [mod hP + 1)] or

50 2*¹ = 1 [mod hP + 1)], incrementing h to form a new hP + 1, but if both 2h a l [mod hP + 1)] and 2^P £ l [mod hP + 1)], determining if the length of hP + 1 is greater than or

55 equal to the desired length, and if greater than or equal to the desired length then setting hP + 1 as an output as a large random prime number to the encryption/decryption key

60 generator, and if not greater than or

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OMPI equal to the desired length, then placing hP + 1 in the sequence of prime numbers and repeating the above steps using hP + 1 as the new P and commencing by selecting a new h as above set forth.

2. The public-key data encryption system of Claim 1 wherein the large-scale integrated circuit is constructed to select the initial h, for the formation of each succeeding prime number in the sequence of prime numbers, to have a length of about one-half that of the current prime number in the sequence of prime numbers.

3. The public-key data encryption system of Claim 1 wherein the large-scale integrated circuit is constructed to further determine if the length of hP + 1 is within a pre-selected length of the desired length, if not greater than or equal to the desired length, and if within the pre-selected length and not greater than or equal to the desired length, to select the next initial h with a selected length less than about one- half of the length of the current prime number in the sequence of prime numbers, but of a length sufficient to make the next prime number in the sequence of prime numbers greater than or equal to the desired length.

4. The public-key data encryption system of Claim 3 wherein the desired length is 23 bytes each of eight bits.

5. The public-key data encryption system of Claim 4 wherein the pre-selected length is 4 bytes each of eight bits.

OM

6. The public-key data encryption system of Claim 5 wherein the selected length is 4 bytes.

7. The public-key data encryption system of Claim 1 wherein: the non-secret key is (e, n) such that tue encrypted form of a message M is M^e (mod n) ; the secret key is d, such that the decrypted form of the encrypted form of the message M is (M^e (mod n) )^<-^■- (mod n) ; n is the product of two large random prime numbers p and q, each having a desired length, such that n has a suitable cryptoanalytically secure length to make a cryptoanalytically secure for a desired cryptoanalytic computational time; e is an integer selected to have a GCD of [(e),. (p - 1) (q - 1)] equal to 1; and e*d 3 I [mod (p - 1) (q - 1)] .

8. The public-key data encryption system of Claim 7 wherein the desired cryptoanalytic computational is approximately 1,000 years.

9. The public-key data encryption system of Claim 8 wherein p and q are at least 23 bytes in length and n is at least 46 bytes in length.

10. In a public-key data encryption system having a plurality of terminals, a terminal comprising: a transmitter-receiver means for receiving encrypted messages, encrypted using a non-secret encryption key specific to the terminal;

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OMPI an encrypter/decrypter means, for decrypting the encrypted message using a secret decryption key; an encryption/decryption key gen¬ erating means, coupled to the encrypter/ decrypter means, for generating the non- secret and secret keys by use of a pair of large randomly selected prime numbers; a large random prime number gen¬ erating means, including a large-scale integrated circuit means, coupled to the encryption/decryption key generating means, for generating separately each of the pair of large randomly selected prime numbers, each having a desired length, the large-scale integrated circuit means including: a random prime number select¬ ing means for selecting a prime number P from the known prime numbers having a length relatively short in comparison to the desired length; a prime number sequence generating means for generating a sequence of prime numbers starting with the prime number P selected by the prime number selecting means, the prime number sequence gener¬ ating means including: a selecting means for selecting a random h and forming the quantity hP + 1, wherein P is the current prime in the computed sequence of prime numbers and is initially the P selected by the prime number select¬ ing means;

OMP a greatest common divisor (GCD) testing means for testing whether the GCD of (x), (hP + 1) is equal to 1,

45 wherein x is a composite number formed from the product of known prime numbers less than or equal to a selected known prime number; a primality testing means,

50 responsive to the determination by the GCD testing means that the GCD is equal to 1, for determining if both 2h a x [_mod (hP + 1)] and 2^hP f 1 [mod (hP + !)} ;

55 an incrementing means, respon¬ sive to either the determination by the GCD testing means that the GCD is not equal to 1 or the determination by the primarily testing means that

60 either 2^hp f 1 [mod (hP + 1)] or 2^hP s 1 [mod (hP + 1)], for incrementing h and forming a new hP + 1 and initi¬ ating the determination by the GCD testing means with respect to the new

65 hP + 1; a length determining means, responsive to the determination by the primarily testing means that both 2^P 2 1 [mod (hP + 1)] and 2-^-P f 1

70 [mod (hP + 1)3, for determining whether the length of hP + 1 is greater than or equal to the desired length; and.

OMPI a sequence placement means,

75 responsive to the determination by the length determining means, that the length of hP + 1 is not greater than or equal to the desired length, for placing hP + 1 in the sequence of

80 prime numbers and for initiating the selection by the selecting means of a new h to form a new hP + 1, wherein P is the hP + 1 previously determined to be prime but not greater than or

85 equal to the desired length; and, an output means, coupled to the prime number sequence generating means and responsive to the determination by the length determining means that' a respective

90 hP + 1 is greater than or equal to the desired length, for selecting the respective hP + 1 as the output to the encryption/decryption key generating means as one of the necessary large randomly

95 selected prime numbers having the desired length.

11. The public-key data encryption system of Claim 10 wherein: the selecting means includes a means for selecting each initial h which is even 5 and which has a length of about one-half of the length of the current prime number in the sequence of prime numbers.

OMPI

12. The public-key data encryption system of Claim 10 wherein: the length determining means includes a means for determining if the respective hP + 1 is within a pre-selected length of the desired length, if not greater than or equal to the desired length, and if within the pre-selected length but not greater than or equal to the desired length, for initiating the selection by the selecting means of new initial h having a length less than approximately one-half of the length of the current prime number in the sequence of prime numbers, but sufficient to make the next prime number in the sequence of prime numbers greater than or equal to the desired length.

13. The public-key data encryption system of Claim 12 wherein the desired length is 23 bytes.

14. The public-key data encryption system of Claim 13 wherein the pre-selected length is 4 bytes.

15. The public-key data encryption system of Claim 14 wherein the selected length is 4 bytes.

16. The public-key data encryption system of Claim 10 wherein: the non-secret encryption key is (e, n) such that the encrypted form of a message M is M^e (mod n);

OMPI the secret decryption key is d, such that the decrypted form of the encrypted form of the message M is (M^e (mod n))^e (mod n) ; n is the product of two large randomly selected prime numbers p and q, each having a desired length, such that n has a suitable cryptoanalytically secure length to make n cryptoanalytically secure for a desired cryptoanalytic computational time; e is an integer selected to have a GCD of [(e), (p - 1) (q - 1)3 equal to 1; and e*d ≡ 1 [mod (p - 1) (q - 1)3.

17. The public-key data encryption system of Claim 16 wherein the desired cryptoanalytic computa¬ tional time is approximately 1,000 years.

18. The public-key data encryption system of Claim 17 wherein p and q are at least 25 bytes in length and n is at least 46 bytes in length.

19. The public-key data encryption system of Claims 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 or 18 wherein P is initially randomly selected from the known prime numbers having a length relatively short in comparison to the desired length.

20. The public-key data encryption system of Claims 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 or 18 wherein the selected known prime number is 139.

OMPI

21. In a public-key data encryption system employing RSA public-key data encryption, a large random prime number generating means for generating large randomly selected prime numbers necessary for RSA public- key data encryption, each having a desired length, the random prime number generating means including a large- scale integrated circuit means for performing the fol¬ lowing operations in selecting each of the necessary large random prime numbers: starting with a P, which is a known prime number having a length which is rela¬ tively short in comparison to the desired length, forming a sequence of prime numbers in the form hP + 1, wherein P is the current prime number in the computed sequence of prime numbers, by selecting at random an h and forming hP + 1 and testing hP + 1 for primality by determining first if the greatest common divisor (GCD) of [(hP + 1), (x)3 is one, wherein x is a composite number consisting of the product of known prime numbers less than or equal to a selected known prime, and if the GCD is equal to 1, determining if both 2^nP = 1 [mod (hP + 1)3 and 2^h ≠ 1 [mod (hP + 1)1, and if the GCD is not equal to 1, or if the GCD is equal to 1, but either 2^nP 1 [mod (hP + 1)3 or 2^h ≠.1 [mod (hP + 1)3, incrementing h to form a new hP + 1 , but if both 2h^p ≡ l [mod (hP + 1)3 and 2^ f 1 [mod (hP -f 1)3, determining if hP + 1 is greater than or equal to the desired length, and if greater than or equal to the desired length, then setting hP + 1 as

OMPI one of the necessary large randomly selected prime numbers, and if not greater than or equal to the desired length, then placing hP + 1 in the sequence of prime numbers and repeating the above steps using hP + 1 as the new P and commencing by selecting a new h as above set forth.

22. In a public-key data encryption system employing RSA public-key data encryption, a large random prime number generating means for generating large randomly selected prime numbers necessary for RSA public- key data encryption, each having a desired length, the random prime number generating means including a large- scale integrated circuit means for generating the random prime numbers, the large-scale integrated circuit means having: a random prime number selecting means for selecting a prime number P from the known prime numbers having a length relatively short in comparison to the desired length; a prime number sequence generating means for generating a sequence of prime numbers starting with the prime number P selected by the prime number selecting means, the prime number sequence gener- ating means including: a selecting means for selecting a random h and forming the quantity hP + 1, wherein P is the current prime number in the sequence of prime numbers and is initially the P selected by the prime number select- ing means;

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OMPI a greatest common divisor (GCD) testing -means for testing whether the

30 GCD [(x), (hP + 1)3 is equal to 1, wherein x is a composite number formed from the product of all known prime numbers less than or equal to a selected known prime number;

35 a primality testing means, responsive to the determination by the GCD testing means that the GCD is equal to 1, for determining if both 2h^p 2 i [mod (hP + 1)3 and 2^ f \

40 [mod (hP + 1)3 ; an incrementing means, respon¬ sive to either the determination by the GCD testing means that the GCD is not equal to 1, or the determination

45 by the primality testing means that either 2ⁿ ≠ 1 [mod (hP + 1)3 or 2*^~* ψ x [mod (hP + 1)3 , for incre¬ menting h and forming a new hP + 1, and initiating the determination by

50 the GCD testing means with respect to the new hP + 1; a length determining means, responsive to the determination by the primality testing means that both

55 2hP ^» [mod (hP + 1)3 and 2¹* ψ 1 [mod (hP + 1)3, for determining whether the length of hP + 1 is greater than or equal to the desired length;

OMPI sequence placement means, respon¬ sive to the determination by the length determining means, that the length of hP + 1 is not greater than or equal to the desired length, for placing hP + 1 in the sequence of prime numbers and for initiating the selection by the selecting means of a new h to form a new hP + 1 wherein P 'is the hP + 1 previously determined to be prime but not greater than or equal to the desired length; and, an output means, responsive to the determination by the length determining means that a respective hP + 1 is greater than or equal to the desired length, for selecting the respective hP + 1 as one of the large randomly selected prime numbers necessary for the RSA public-key data encryption.

OMPI ^WIPO