DETAILED ACTION
The present application, filed on or after March 16, 2013, is being examined under the first inventor to file provisions of the AIA .
Receipt is acknowledged of certified copies of papers required by 37 CFR 1.55.
The IDS filed 1/13/2025 has been considered.
The preliminary amendment filed 1/13/2025 has been placed of record in the file.
Claims 1-13 are presented for examination.
Claim Rejections - 35 USC § 101
35 U.S.C. 101 reads as follows:
Whoever invents or discovers any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof, may obtain a patent therefor, subject to the conditions and requirements of this title.
Claims 1-13 are rejected under 35 U.S.C. 101 because the claimed invention is directed to an abstract idea without significantly more. The claims recite mathematical operations for determining a congruent, which falls within the mathematical concepts grouping of abstract ideas. This judicial exception is not integrated into a practical application because it is unclear what additional structure or function might impose any meaningful limits on practicing the abstract idea. Where the dependent claims recite some structural elements, these elements are generic computer elements and do not add a meaningful limitation to the abstract idea because they amount to simply implementing the abstract idea on a computer. Further, the claims do not include additional elements that are sufficient to amount to significantly more than the judicial exception because the use of the claimed computer or processor represents well-understood, routine, conventional activity.
Claim Rejections - 35 USC § 103
In the event the determination of the status of the application as subject to AIA 35 U.S.C. 102 and 103 (or as subject to pre-AIA 35 U.S.C. 102 and 103) is incorrect, any correction of the statutory basis (i.e., changing from AIA to pre-AIA ) for the rejection will not be considered a new ground of rejection if the prior art relied upon, and the rationale supporting the rejection, would be the same under either status.
The following is a quotation of 35 U.S.C. 103 which forms the basis for all obviousness rejections set forth in this Office action:
A patent for a claimed invention may not be obtained, notwithstanding that the claimed invention is not identically disclosed as set forth in section 102, if the differences between the claimed invention and the prior art are such that the claimed invention as a whole would have been obvious before the effective filing date of the claimed invention to a person having ordinary skill in the art to which the claimed invention pertains. Patentability shall not be negated by the manner in which the invention was made.
Claims 1-13 are rejected under 35 U.S.C. 103 as being unpatentable over Non-Patent Literature “Montgomery Reduction for Gaussian Integers” by Safieh and Freudenberger, hereinafter referred to as Safieh, in view of Shamee et al. (U.S. Patent Application Publication Number 2019/0020470), hereinafter referred to as Shamee.
Safieh disclosed techniques for modular arithmetic over integers. In an analogous art, Shamee disclosed techniques for data transformation using integers. Both systems are directed toward modular reduction.
Regarding claim 1, Safieh discloses a method for generating a cryptographic key and/or for encryption or decryption, the method comprising: determining a Gaussian integer congruent to a given Gaussian integer modulo a Gaussian integer modulus, wherein the norm of the Gaussian integer is smaller than the norm of the square of the Gaussian integer modulus (section 3, Montgomery reduction for Gaussian integers resulting in congruent) by: considering a real integer base raised to a first integer exponent having a norm larger than that of the real and larger than that of the imaginary part of the Gaussian integer modulus (section 3, Montgomery reduction using real part and imaginary part); considering a variable value candidate for the Gaussian integer congruent is considered that is first initialized with the given Gaussian integer (section 3, Montgomery reduction considering candidates); and decrementing the Gaussian integer, either fully or in truncated form, by a multiple of the Gaussian integer modulus (section 3, Montgomery reduction using bit shifts); evaluating the multiple of the Gaussian integer modulus by calculating an auxiliary product of a component-wisely down rounded quotient of the current value of the variable value candidate for the Gaussian integer congruent and the real integer base raised to the sum of the first integer exponent and the second integer exponent with a prefactor (section 3, Montgomery reduction with final reduction satisfying bound).
Safieh does not explicitly state wherein a second integer exponent is considered that is equal to or smaller than -2 and wherein a third integer exponent is considered, that is equal to or larger than the first integer exponent incremented by one and calculating a component-wisely down rounded quotient of this auxiliary product and the real integer base raised to the difference of the third integer exponent and the second integer exponent and multiplying this latter quotient with the Gaussian integer modulus. However, performing modular reduction in such a fashion was well known in the art as evidenced by Shamee. Since the inventions encompass the same field of endeavor, it would have been obvious to one of ordinary skill in the art before the effective filing date of the claimed invention to modify the system of Safieh by adding the ability that a second integer exponent is considered that is equal to or smaller than -2 and wherein a third integer exponent is considered, that is equal to or larger than the first integer exponent incremented by one and calculating a component-wisely down rounded quotient of this auxiliary product and the real integer base raised to the difference of the third integer exponent and the second integer exponent and multiplying this latter quotient with the Gaussian integer modulus as provided by Shamee (see paragraph 179, modular reduction that replaces n-th powers via add left shift until degree is less than n). One of ordinary skill in the art would have recognized the benefit that performing modular arithmetic in this way would assist in utilizing encryption that improves the security of a cryptosystem (see Shamee, paragraph 21).
Regarding claim 2, the combination of Safieh and Shamee discloses wherein such a second integer exponent is considered that is equal to or smaller than -3 and such a third integer exponent is considered, that is equal to or larger than the first integer exponent incremented by three (Shamee, paragraph 179, modular reduction that replaces n-th powers via add left shift until degree is less than n).
Regarding claim 3, the combination of Safieh and Shamee discloses wherein the Gaussian integer is decremented by a multiple of the Gaussian integer modulus in a truncated form such that the Gaussian integer is truncated via modulo reduction of the Gaussian integer modulo the real integer base raised to the difference of the third integer exponent and the second integer exponent and then the multiple of the Gaussian integer modulus is truncated via modulo reduction of the multiple of the Gaussian integer modulus modulo the real integer base raised to the difference of the third integer exponent and the second integer exponent and subtracted from the truncated Gaussian integer (Safieh, section 3, Montgomery reduction using bit shifts).
Regarding claim 4, the combination of Safieh and Shamee discloses wherein the prefactor includes the down rounded quotient of the real integer base raised to the sum of the first integer exponent and the third integer exponent and the Gaussian integer modulus (Safieh, section 3, Montgomery reduction with final reduction satisfying bound).
Regarding claim 5, the combination of Safieh and Shamee discloses wherein the norm of the determined Gaussian integer congruent is smaller than the norm of the given Gaussian integer (Safieh, section 6, smaller absolute value).
Regarding claim 6, the combination of Safieh and Shamee discloses wherein the norm denotes the absolute value (Safieh, section 3, Montgomery reduction using absolute value).
Regarding claim 7, the combination of Safieh and Shamee discloses wherein the norm denotes the Manhattan weight or the absolute square value (Safieh, section 4, precision reduction using Manhattan weight).
Regarding claim 8, the combination of Safieh and Shamee discloses wherein the method is conducted on a computer that stores numbers in a positional numeral system with a radix, wherein the radix is equal to the real integer base or where a radix raised to an integer power is equal to the real integer base (Safieh, section 2, binary representation).
Regarding claim 9, the combination of Safieh and Shamee discloses wherein the real integer base is an ordinary integer base (Safieh, section 2, ordinary integers).
Regarding claim 10, the combination of Safieh and Shamee discloses carried out on a processor with a word-size, the real integer base being equal to the word-size of the processor, the word-size equal to 16 or 32 or 64 or 128 (Shamee, paragraph 115, fixed word size).
Regarding claim 11, the combination of Safieh and Shamee discloses wherein the Gaussian integer is reduced using a final reduction (Safieh, section 3, Montgomery reduction with final reduction).
Regarding claim 12, the combination of Safieh and Shamee discloses wherein the down rounded fractions are evaluated involving bit shifting by an integer number of bits and involving bit truncation down to an integer number of bits (Safieh, section 3, Montgomery reduction using bit shifts).
Regarding claim 13, the combination of Safieh and Shamee discloses determining a
reduction of a given Gaussian integer modulo a Gaussian integer modulus by further reducing the Gaussian integer congruent with a final reduction (Safieh, section 3, Montgomery reduction with final reduction).
Conclusion
The prior art made of record and not relied upon is considered pertinent to applicant's disclosure.
Lablans (U.S. Patent Application Publication Number 2023/0125560) disclosed techniques for modifying cryptographic parameters using computations based on Gaussian integers.
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/Victor Lesniewski/Primary Examiner, Art Unit 2493
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