Method and apparatus for improved pseudo-random number generation

[0001] The present invention is directed generally to cryptography, and more particularly to pseudo-random number sequencing.

[0002] Pseudo-random number generators (PRNGs) are used in a wide variety of cryptographic applications. In particular, random outputs of PRNGs are relied upon in generating, for example, (1) unpredictable session identifiers or cookies for online client-server sessions, (2) key generation for symmetric, asymmetric encryption, the Diffie-Hellman key exchange algorithm and digital signature algorithms (DSA), (3) generating nonces in challenge-response authentication mechanisms, (4) producing random padding in cryptographic padding mechanisms such as public key cryptosystems (PKCS-1) and (5) providing random variables for accomplishing secure transmissions via wireless transport layer security (WTLS) and wireless application protocols (WAPs).

[0003] Common PRNGs are based on the American National Standards Institute (ANSI) X9.17 standard and are typically used with the Digital Encryption Standard (DES) or 3-DES block ciphers. Other block ciphers, such as Rivest Cipher-5 (RC-5) may also be used. In order to accomplish secure communications, it is desireable that the outputs from the PRNG be unpredictable. If the output of a PRNG becomes predictable, it will, in turn become easier to decipher any communications from a cryptography system employing such a PRNG. Thus, the random nature of a PRNG is an important aspect in maintaining secure communications.

[0004] Recently, several studies have determined that PRNGs using the ANSI X9.17 standard may be vulnerable to certain cryptographic attacks. In particular, it has been discovered that if the internal key used by an ANSI X9.17 PRNG becomes known, the PRNG becomes vulnerable to permanent compromise attacks. If an attacker can force input seed values to an ANSI X9.17 PRNG in an adaptive attack, it may be possible to force the PRNG to generate outputs in a partially-predictable manner. In addition, if an internal state of an ANSI X9.17 PRNG becomes known, a backtracking attack may be performed to discover previous secret outputs of the PRNG. See, e.g., Kelsey, J., et al., “Cryptanalytic Attacks on Pseudo-Random Number Generators,” ESORICS '98 Proceedings, Springer-Verlag, 1998, pp. 77–110 and Kelsey, J. et al., “Yarrow-160: Notes and the Design and Analysis of the Yarrow Cryptographic Pseudo-random Number Generator,” Proceedings of the Sixth Annual Workshop on Selected Areas in Cryptography.

[0005] Various methods for random number generation have been previously disclosed. See, for example, U.S. Pat. Nos. 6,141,668; 6,065,029; 6,061,703; 6,044,388; 5,983,252; 5,966,313; 5,961,577; 5,872,725; 5,864,491; 5,828,752; and 5,046,036. However, none of these systems provide a sufficient solution to the possible attacks noted above. Accordingly, there is a need for a method and apparatus for pseudo-random number generation which addresses certain deficiencies in prior technologies.

[0006] According to certain embodiments of the present invention, a method and apparatus for seeding a PRNG is presented in which a plurality of state variables in an output buffer for use by the PRNG in determining a random number. The PRNG receives successive input entropy signals. The output buffer is cleared upon receipt of each of the successive input entropy signals and new state variables are calculated thereafter.

[0007] In a further embodiment of the present invention, a method and apparatus for seeding a PRNG in an initial state is provided for securely generating a random number. In this embodiment, an input seed is received. New state variables are then calculated by concatenating the input seed with a first constant, determining a first output based on a hash of the concatenated input seed and the first constant, concatenating the input seed with a second constant and determining a second output based on a hash of the concatenated input seed and the second constant. A key for generating a random number is then determined based on at least a portion of the first output. A counter variable for generating a random number is determined based on a portion of the second output. The key and the counter variable are then stored in an output buffer.

[0008] According to another embodiment of the present invention, a method and apparatus for generating state variables for a PRNG, after an initial state, is provided for securely generating a random number. In such an embodiment, first state variables are stored in an output buffer. The first state variable include a first key, a first seed value and a first counter variable. A new input seed is the received. The output buffer is then cleared in response to the new input seed. Second state variables are then determined based on the new input seed and the first state variables.

[0009] According to still another embodiment of the present invention, a method and apparatus for determining a random number using a PRNG in an initial state are provided in which state variables for the PRNG are stored in an output buffer, the state variables include a first key, a first seed value, and a first counter variable. A second counter variable is determined by summing the first counter variable with a constant. The second counter variable is then encrypted using the first key and a block cipher to generate a first encrypted result. The first encrypted result is concatenated with the first seed value to generate a second encrypted result. The second encrypted result is the encrypted using the first key and the block cipher to generate a random number.

[0010] According to still another embodiment of the present invention, a method and apparatus for generating a random number is provided in which a key and a counter variable are stored in an output buffer. The counter variable is not a timestamp variable relating to a particular time. A first random number is the generated based on at least the key and the counter variable.

[0011] According to a further embodiment of the present invention, a method and apparatus for determining a sequential output of random numbers using a PRNG in an initial state is provided. Initial state variables are stored in an output buffer. The initial state variables include a first key, a first seed value, and a first counter variable Prior to receiving further input seed, a second counter variable is determined by summing the first counter variable with a constant. The second counter variable is encrypted using the first key and a block cipher to generate a first encrypted result. The first encrypted result is then concatenated with the first seed value to generate a second encrypted result. The second encrypted result is then encrypted using the first key and the block cipher to generate a random number. A second seed value may then be determined by: (1) encrypting the second counter variable using the key and the block cipher to generate a third encrypted result; (2) performing an exclusive-or operation of the third encrypted result with the random number to determine a fourth encrypted result; and (3) encrypting the fourth encrypted result using the key and the block cipher to determine the second seed value for generating a subsequent random number. A third counter variable is then determined by (1) summing the second counter variable with the constant, (2) encrypting the third counter variable using the key and the block cipher to generate a fifth encrypted result, (3) XOR-ing the fifth encrypted result with the second seed value to generate a sixth encrypted result, and (4) encrypting the sixth encrypted result using the first key and the block cipher to generate a second random number.

[0012] It is an advantage of the present invention, therefore, to have a method and apparatus for seeding a PRNG and determining random numbers using a counter variable in place of a timestamp variable in order to improve the security of PRNGs in a cryptographic system.

[0013] It is a further advantage of the present invention to implement the PRNG associated with the invention in either hardware or software, or in a combination of both hardware and software.

[0014] It is another advantage of the present invention to implement the inventive method of seeding and determining random numbers by using small amounts of random access memory (RAM) and Read-Only Memory (ROM) so that the invention may be embodied in mobile terminal, such as wireless cellular, satellite telephones and other wireless devices capable of two-way wireless communications, e.g. personal digital assistants (PDA's).

[0022] The invention relates to a system and method for securing a PRNG against crypto-analytic attacks by which outputs from the PRNG may be guessed or determined. The PRNG may be enabled in hardware or software and may be employed by a mobile device, such as a mobile telephone using WAP/WTLS. The contemplated security features include: (1) reseeding the PRNG at each instance of input entropy in order to change the internal state of the PRNG, thereby generating a new internal key for each new seed input, and (2) replacing a timestamp variable with a running counter, the value of which is less likely to be known or determined outside the system. Specifically, input seeds are provided in sufficiently large increments to avoid iterative-guessing attacks. Upon the input of a new seed to the PRNG, two hash functions (employing either Secure Hash Algorithm-1 (SHA-1) or Message Digest-5 (MD5)) are used to generate a new internal key, a new seed variable and a new counter variable. These two hash functions each operate on the concatenation of a chosen constant value in base 16, the current internal key K, the current seed value S, the current counter variable T and an input seed comprising a byte array of arbitrary length. The constant values chosen for each of the two hash functions may not be the same.

[0023] The results of these hash functions are then used in the following manner. The first 128-bit segment of the results of the first hash function are used to generate the next internal key. The first 64-bit segment of the results of the second hash function are used to generate a new seed value. The second 64-bit segment of the results of the second hash function are used to generate a new counter variable. A random number O_i is generated from the results of encrypting the current counter variable and the previous seed value, according to a block cipher. The block cipher may be RC-5, although other block ciphers may also be used. Other available block ciphers include the Data Encryption Standard (DES), IDEA, Blowfish, CAST-n, MISTY, SKIPJACK and KASUMI. With come adjustments, the following block ciphers may likewise be used with the processes herein: Rivest Cipher 6 (RC-6), triple Data Encryption Satndard (3-DES), Advanced Encryption Standard (AES), and Twofish. Methods for encrypting using RC-5 are disclosed in U.S. Pat. Nos. 5,835,600 and 5,724,428, each being incorporated herein by reference. The output segments are received in an output buffer, which is purged upon the input of a new seed to the PRNG.

[0024] Referring now to FIGS. 1–6B, wherein similar components of the present invention are referenced in like manner, preferred embodiments of a method and apparatus for improved pseudo-random number generation are disclosed.

[0025] FIG. 1 discloses a cryptographic system 100 implemented in hardware and suitable for use with the present invention. The system 100 includes a central processing unit (CPU) 102, a PRNG 104, a persistent memory store 106, ROM 108, RAM 110, a clock 112, and an entropy input source 114. These elements of system 100 may communicate over a common data bus or in any other equivalent manner.

[0026] The CPU 102 may be any available cryptographic processor capable of handling, for example, 128-bit encryption processes. It is contemplated that CPU 102 may be an ARM-7 CPU core manufactured by ARM, INC. However, other processing systems may likewise be used.

[0027] The PRNG 104 may be a physical PRNG by which input entropy signals are received and a string of random bits are generated and output. Such physical PRNG hardware is commonly available and known to one of ordinary skill in the art. Alternatively, the PRNG may be software code stored in a persistent memory 106 of the system 100. The PRNG software may be implemented, for example, using ANSI-C programming code or JAVA programming languages to emulate a physical PRNG.

[0028] The persistent memory store 106 may be any memory device, such as semi-conductor memory device for storing binary instructions and data. The persistent memory store may be a CMOS storage device, or any other device in which such binary instructions and data may be maintained in the absence of power. Preferably, the persistent memory store 106 is suitable for operation with mobile terminals. The persistent memory store 106 may act as an output buffer for state variables and random numbers used by the system 100.

[0029] ROM 108 may be any memory device, such as an electronically eraseable and programmable read-only memory (EEPROM) device suitable for providing processing instructions upon power-up of the system 100.

[0030] RAM 110 may be any memory device, such as a Single In-Line Memory Module (SIMM) chip capable of temporary, power-dependent storage for storing processing instructions and data during operation of the system 100.

[0031] Clock 112 may be any device for providing clocking signals to synchronize the communication between the elements of system 100.

[0032] Input source 114 may be any device capable of providing input entropy signals to the PRNG 104. Accordingly, the input source 114 may detect system events, capture noise signals from a microphone or particular radio frequencies, generate or receive random bits from other devices or components, or retrieve random data from memory allocation tables stored in persistent memory store 106. The input source 114 may then transmit the input entropy signals received in any of these manners to the PRNG as an input seed.

[0033] Alternatively, the input entropy signals may be accumulated in an entropy accumulation pool as may be stored in persistent memory store 106. When a predetermined amount of entropy signals are stored in such pool, the accumulated signals may then be provided to the PRNG 104. Such process for providing accumulated signals is described further below in conjunction with FIG. 5B. The input entropy signals or accumulated entropy signals may be transmitted to the PRNG 104 at random or predetermined intervals in order to re-seed the PRNG. Such re-seeding is discussed further below in conjunction with FIG. 5A.

[0034] The system 100 is contemplated to be implemented within a mobile terminal, such as cellular telephone model nos. 6210, 6250, 7160 and 7190 manufactured by NOKIA CORPORATION.

[0035] Referring now to FIG. 2, therein is depicted an exemplary re-seeding process 200 for initializing a PRNG implemented in hardware and/or software. The process 200 begins by initializing an output buffer, such as persistent memory store 106, to store state variable for the PRNG 104 (step 102). A first constant C1 is then appended to the output buffer (step 204). The constant C1 may be, for example, 5555AAAA₁₆, as a binary number expressed in base-16.

[0036] State variables representing a first key K_o, a first seed value S_o, a first counter variable T_o and an input seed X may be appended to the output buffer (step 206). The state variables may each be set to be zero in an initial state of the PRNG. Methods for determining such state variables are described further below with respect to FIG. 3. The input seed X may be a byte array of arbitrary length which may be generated by input source 114.

[0037] The CPU 102 may then perform a cryptographic hash of the values in the buffer and may store the results as a first output A (step 208) in RAM 110 The hash may be a function such as a Secure Hash algorithm-1 (SHA-1) or a Message Digest-5 (MD-5) algorithm.

[0038] The output buffer may then be cleared upon receipt of new input seed X₁ (step 210). A second constant C2 may then be appended to the output buffer (step 212). The constant C2 may be, for example, AAAA5555₁₆, a binary number expressed in base-16.

[0039] New state variables may then be appended to the output buffer, including a second key K₁, a first seed value S₁, a first counter variable T₁ and the input seed X₁ (step 214). The state variables may each be set to be zero in an initial state of the PRNG. Methods for determining such state variables are described further below with respect to FIG. 3. The input seed X₁ may be a byte array of arbitrary length which may be generated by input source 114.

[0040] The CPU 102 may then perform a cryptographic hash of the values in the buffer and may store the results as a first output A (step 216) in RAM 110. The hash may be a function such as SHA-1 or MD-5.

[0041] A new key K may then be determined as the value of output A. The new seed value S may be determined as a portion of output B. The new counter variable T may be a second portion of output B (step 218). These new state variables may then be stored for use by the PRNG 104, after which process 200 ends.

[0042] It is preferable that process 200 is performed upon each new receipt of input entropy from the input source 114.

[0043] In mathematical terms, the above process 200 may be expressed as follows:

• • [0044] Let K₁=a 128-bit key used by a block cipher;
  • [0045] Let T₁=a 64-bit counter variable;
  • [0046] Let S₁=a 64-bit chaining variable or seed value;
  • [0047] Let X₁=an input seed of arbitrary length;
  • [0048] Let H(x) denote an SHA-1 or MD-5 hash of x,
  • [0049] Let x∥y denote a concatentation of two byte strings x and y.
  • [0050] Let C1 and C2 be constants (e.g. 5555AAAA₁₆ and AAAA5555₁₆, respectively)

[0051] Then output variables A and B may be determined as follows:

• • [0052] A=H(C1∥K₁∥S₁∥T₁∥X₁)
  • [0053] B=H(C2 ∥K₁∥S₁∥T₁∥X_i)

[0054] It is contemplated that A and B may be determined as 128 bit strings. In such a case, a new key K will be determined as the entire 128 bit string of A. A new seed value S may be determined as the first 64 bits of B (i.e. bits 1.0.64) and the new counter variable T may be determined as the second 64 bits of B (i.e. bits 65 . . . 18 of B).

[0055] FIG. 3 is a flow chart depicting an exemplary random number generation process 300 for generating a random number O without new input seed in accordance with certain embodiments of the present invention. The process 300 begins by adding a constant C to the current counter variable T and place the result in the output buffer (step 302). The constant C may be a 64-bit odd constant, such as 2⁶⁴log2 or B17217F7D1CF79AB₁₆. The addition of T and C may be performed in little endian fashion modulo 2⁶⁴.

[0056] The counter variable T is then encrypted with a block cipher using key K and stored as a first encrypted result (step 304). An exclusive-OR (XOR) operation is then performed on the first encrypted result and a previous seed value S. The result of the XOR operation is then encrypted using the block cipher and the current key K (step 306). The resulting value is the generated random number O. The first encrypted result from step 304 is then XOR-ed with the random number O and the result is encrypted to generate a current seed value S (step 308), after which process 300 ends. The current seed value S may then be used to generate subsequent random numbers.

[0057] In mathematical terms, the process 300 may be expressed as follows:

• • [0058] Let C=a 64-bit odd constant;
  • [0059] Let O_i=a 64-bit random number;
  • [0060] Let K₁=a 128-bit key used by a block cipher;
  • [0061] Let T₁=a 64-bit counter variable;
  • [0062] Let S₁=a 64-bit chaining variable or seed value;
  • [0063] Let x (+) y denote an XOR operation between byte strings x and y;
  • [0064] Let x [+] y denote the modulo 2ⁿ sum of x and y;
  • [0065] Let E_k(x) denote the encryption of x with key K using a block cipher. (It is preferred that the block cipher uses a 64-bit block size in 16 rounds with a 128-bit key.)

[0066] State variables and pseudo-random numbers then may be generated as follows:

• • [0067] T₁=T_i-1[+]C
  • [0068] O₁=E_k(E_k(T₁)(+) S_i-1)
  • [0069] S₁=E_k(E_k(T₁)(+) O_i)

[0070] FIG. 4 is a flow chart depicting an exemplary random number generation process for providing a random output in a series of rounds (3 rounds as shown) in accordance with certain embodiments of the present invention. As shown therein, T represents the counter variable, C represent a constant, E represents an encryption function, S represents a seed value, O represents a random number, [+] represent a modulo 2ⁿ sum and (+) represents and XOR operation.

[0071] FIGS. 5A–5B are flow charts depicting exemplary processes 500 and 510, respectively, for initiating re-seeding of a PRNG 104 in accordance with certain embodiments of the present invention. Referring to FIG. 5A, a process 500 for re-seeding upon each instance of new input entropy is shown. The process 500 begins upon receipt of a new entropy signal from input entropy source 114 (step 502). The PRNG is re-seeded by generating new state variables (step 504) as described above with respect to FIG. 2. The process 500 then ends.

[0072] FIG. 5B depicts an exemplary process 510 for determining when to transmit new input entropy to the PRNG 104 when entropy signals are accumulated. The process 510 begins at step 512 when the CPU 102 determines whether new input entropy is available. This may be done by searching an input entropy accumulation pool stored in persistent memory store 106. If there is no sufficient accumulation of input entropy (i.e. if a predetermined value of input entropy has not been stored), the process 510 continues to step 516 where entropy is further accumulated in the entropy pool. If, on the other hand, sufficient input entropy has been stored, the process 510 continues to step 518 where the PRNG 104 is re-seeded, where newly determined state variables are based at least in part on the accumulated input entropy signals. The process 510 then ends.

[0073] FIGS. 6A–6B are flow charts depicting exemplary processes for storing and retrieving PRNG state information from persistent storage in accordance with certain embodiments of the present invention.

[0074] FIG. 6A depicts an exemplary shutdown process 600 which may be performed by system 100. When a shutdown of system 100 is detected (step 602), the CPU 102 may direct the storage in PRNG state variables in persistent memory store 106 (step 604), after which process 600 ends.

[0075] FIG. 6B depicts an exemplary power-up process 610 for the system 100. Upon detection of a power-up condition (step 612), the CPU 102 determines whether the PRNG has been previously initialized (step 614), e.g. if previous state variables are stored in persistent memory store 106. If so, the process 610 continues to step 618 where the previous state variables are retrieved from persistent memory store 106 for use by the PRNG 104 in generating new random numbers. If no previous state variables are stored, the process continues to step 616 where new state variables are generated based on input entropy signals from input source 114, in accordance with process 200 above. The process 610 then ends.

[0076] The PRNG 104 as described herein may be bijective, e.g. it may be run backwards or forward in between seeding operations. The counter variable T, described above, does not include a timestamp value, i.e. denoting a particular time and/or date, which may be learned or guessed by an attacker by noting the particular time. Rather, the counter variable is a random variable that may be incremented by a constant between re-seeding processes. The counter variable may further be determined based on received input entropy upon re-seeding of the PRNG 104. The use of the counter variable, therefore, increases the security of the cryptographic system 100 in a manner not contemplated in previous technologies.

[0077] Although the invention has been described in detail in the foregoing embodiments, it is to be understood that the descriptions have been provided for purposes of illustration only and that other variations both in form and detail can be made thereupon by those skilled in the art without departing from the spirit and scope of the invention, which is defined solely by the appended claims.