CTNF 17/989,780 CTNF 93321 2182 DETAILED ACTION 07-03-aia AIA 15-10-aia The present application, filed on or after March 16, 2013, is being examined under the first inventor to file provisions of the AIA. Information Disclosure Statement 06-49-09 AIA The information disclosure statement filed 01/19/21 citation 14 fails to comply with 37 CFR 1.98(a)(3)(i) because it does not include a concise explanation of the relevance, as it is presently understood by the individual designated in 37 CFR 1.56(c) most knowledgeable about the content of the information, of each reference listed that is not in the English language. It has been placed in the application file, but the information referred to therein has not been considered. Claim Objections Claims 1-21 are objected to because of the following informalities. Claims 1, 3, 20, and 21 comprise more than one sentence and does not end in a period. See MPEP 608.01(m). Claims 2-19 inherit the same deficiency as claim 1 based on dependence. Claim 17 lines 3-4 recite “the paring computation”. This limitation lacks antecedent basis and should possibly recite “the pairing computation”. Appropriate correction is required. Claim Rejections - 35 USC § 112 07-30-02 AIA The following is a quotation of 35 U.S.C. 112(b): (b) CONCLUSION.—The specification shall conclude with one or more claims particularly pointing out and distinctly claiming the subject matter which the inventor or a joint inventor regards as the invention. The following is a quotation of 35 U.S.C. 112 (pre-AIA), second paragraph: The specification shall conclude with one or more claims particularly pointing out and distinctly claiming the subject matter which the applicant regards as his invention. 07-34-01 Claims 1-21 are rejected under 35 U.S.C. 112(b) or 35 U.S.C. 112 (pre-AIA), second paragraph, as being indefinite for failing to particularly point out and distinctly claim the subject matter which the inventor or a joint inventor (or for applications subject to pre-AIA 35 U.S.C. 112, the applicant), regards as the invention. Claim 1 lines 5-8 recites “the elliptic curve being expressed by: a polynomial r(x) =φ k (T(x))/h 2 (x), a polynomial p(x) = h 1 (x)r(x) + T(x), and a polynomial t(x) = T(x) +1 which are expressed with using a cyclotomic polynomial φ k (x) having a degree d and indicated by Formula 1, a polynomial T(x). a polynomial h 1 (x). and a polynomial h 2 (x)”. It is unclear which polynomial is expressed as a cyclotomic polynomial having degree d and indicated by Formula 1, the polynomial φ k (p(x)), the polynomial r(x), the polynomial p(x), the polynomial t(x), all of the polynomials, some of the polynomials. For purposes of examination, Examiner interprets as the polynomial φ k (p(x)). Furthermore, claim 1 recites variables r(x), x, and c, which are undefined. Claims 2-19 inherit the same deficiency as claim 1 based on dependence. Claims 20 and 21 each recite substantially the same limitation and are rejected for the same reasons. Claim 5 recites “the polynomial t(x) = x +1”, whereas claim 1 line 6 recites that “a polynomial t(x) = T(x) + 1”. It is unclear which equation defines t(x). For purposes of examination, Examiner interprets as for claim 5 the equation t(x) = x +1 is substituted for the claim 1 equation t(x) = T(x) + 1. Claims 6-15 similarly recite different equations for t(x), r(x) and p(x) than is recited in claim 1 and are rejected for the same reason. Claim Rejections - 35 USC § 101 07-04-01 AIA 07-04 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 1-21 are rejected under 35 U.S.C. § 101 because the claimed invention is directed to a judicial exception (i.e., a law of nature, a natural phenomenon, or an abstract idea) without significantly more. Regarding claim 1, under the Alice framework Step 2A prong 1, the claim recites Mathematical concepts. The claim recites mathematical calculations, mathematical relationships, and mathematical equations for calculating a final exponentiation computation. Specifically, the claim recites the following: a final exponentiation computation comprising decompose an exponent portion of a final exponentiation computation portion of pairing computation in an elliptic curve into an easy part and a hard part with using a polynomial φ k (p(x)), the elliptic curve being expressed by: a polynomial r(x) =φ k (T(x))/h 2 (x), a polynomial p(x) = h 1 (x)r(x) + T(x), and a polynomial t(x) = T(x) +1 which are expressed with using a cyclotomic polynomial φ k (x) having a degree d and indicated by Formula 1, a polynomial T(x), a polynomial h 1 (x), and a polynomial h 2 (x); and an embedding degree k, and compute the hard part obtained by decomposition, with using a power of a polynomial p(x) i for each integer i of i = 0,..., d – 1, a power of λ d-i (x) where λ d-i (x) = c d , a power of λ i where λ i = T(x)λ i+1 (x) + c i+1 for each integer i of i = 1,..., d – 2, a power of h 1 (x), a power of h 2 (x), and at least one of multiplication and inverse element computation. [Formula 1] ϕ k x = ∑ c i x ⅈ For these reasons, claim 1 recites mathematical concepts. Under the Alice framework Step 2A prong 2 analysis, additional elements not reciting Mathematical equations and mathematical calculations thereof include: a final exponentiation computation device comprising processing circuitry. These additional elements do no more than generally link the mathematical relationships and mathematical calculations to a computer in a manner that in effect merely recites “apply it” in a computation device comprising processing circuitry. For this reason the claim is not integrated into a practical application. Moreover, under the Alice Framework Step 2B analysis, the claim, considered individually and as an ordered combination does not include additional elements that are sufficient to amount to significantly more than the abstract idea. As discussed in the Step 2A prong 2 analysis, the claim merely generally links the additional element to the math. For these reasons claim 1 elements considered individually and as an ordered combination does not amount to significantly more than the abstract idea. Claims 2-19 are rejected for at least the reasons cited with respect to the claim 1 analysis. Under the Step 2A prong 1 analysis, claims 2-18 merely further mathematically limit the claim 1 mathematical elements recited. Claims 2-18 contain no further additional elements that would require further consideration under Step 2A prong 2 or Step 2B. Claim 19 recites the following further additional element: a cryptographic processing device which performs a cryptographic process with using a result of the pairing computation computed by the pairing computation device according to claim 17. Under the step 2A prong 2 and step 2B analysis, this additional element merely generally links the mathematical concepts to a technological environment or field of use. For these reasons claim 19 is neither integrated into a practical application nor amounting to significantly more than the abstract idea. Claim 20 is directed a method that would be practiced by the apparatus of claim 1. All steps performed by the method of claim 20 are performed by the apparatus of claim 1 as configured. The claim 1 analysis applies equally to claim 20. Claim 21 is directed to a non-transitory computer-readable recording medium with a final exponentiation computation program which causes a computer to function as a final exponentiation computation device that performs the steps of the apparatus of claim 1 as configured. All steps performed by the non-transitory computer-readable recording medium of claim 21 are performed by the apparatus of claim 1 as configured. The claim 1 analysis applies equally to claim 21. Furthermore, under the step 2A prong 2 and step 2B analysis the further additional element of a non-transitory computer-readable medium with a final exponentiation program merely comprises instructions to implement the math on a computer. For these further reasons, claim 21 is neither integrated into a practical application nor amounting to significantly more than the abstract idea. Allowable Subject Matter Claims 1-21 would be allowable if rewritten to overcome the rejections under 35 USC 101, 35 USC 112(b) and the claim objections. The following is a statement of reasons for the indication of allowable subject matter. Applicant claims apparatus, a method, and a non-transitory computer-readable recording medium for final exponentiation, wherein the apparatus as in claim 1 comprises: a final exponentiation computation device comprising processing circuitry to decompose an exponent portion of a final exponentiation computation portion of pairing computation in an elliptic curve into an easy part and a hard part with using a polynomial φ k (p(x)), the elliptic curve being expressed by: a polynomial r(x) =φ k (T(x))/h 2 (x), a polynomial p(x) = h 1 (x)r(x) + T(x), and a polynomial t(x) = T(x) +1 which are expressed with using a cyclotomic polynomial φ k (x) having a degree d and indicated by Formula 1, a polynomial T(x), a polynomial h 1 (x), and a polynomial h 2 (x); and an embedding degree k, and to compute the hard part obtained by decomposition, with using a power of a polynomial p(x) i for each integer i of i = 0,..., d – 1, a power of λ d-i (x) where λ d-i (x) = c d , a power of λ i where λ i = T(x)λ i+1 (x) + c i+1 for each integer i of i = 1,..., d – 2, a power of h 1 (x), a power of h 2 (x), and at least one of multiplication and inverse element computation. [Formula 1] ϕ k x = ∑ c i x ⅈ The primary reason for indication of allowable subject matter is the specific mathematical relationships, and equations used in performing the pairing computation in combination with the remaining limitations and as highlighted above. R. Barbulescu et al., A taxonomy of pairings, their security, their complexity , Cryptology ePrint Archive, Report 2019/485, 2019 (hereinafter “Barbulescu”) discloses evaluation of various pairing friendly elliptic curves, including Miller’s algorithm (abstract, section 4). Barbulescu further discloses the final exponentiation computation including calculation of the easy part and the hard part (section 5). Barbulescu does not however, teach or suggest the specific polynomials claimed as highlighted above. Y. Takahashi, et al., An Implementation and Evaluation of Pairing Library ELiPS for BLS Curve with Several Techniques , 2019 34 th International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC), 2019 (hereinafter “Takahashi”) discloses a pairing library including efficient implementation of Miller’s algorithm, and elliptic curve calculations (Introduction, section 3.4, 3.6.2). Takahashi further discloses the final exponentiation computation including calculating a hard part, an easy part and cyclotomic squaring (section 3.5). Takahahsi does not however, teach or suggest the specific polynomials claimed as highlighted above. US 20150023496 A1 Yonemura et al., (hereinafter “Yonemura”) discloses a pairing computation apparatus including a Muller function computation on an elliptic curve and a final exponentiation to raise an element on an extension field to a power of a value determined by a loop parameter of the Miller function (abstract). Yonemura further discloses an embedding degree of a freeman cure with respect to a cyclotomic polynomial ([0082]). Yonemura does not however, teach or suggest the specific polynomials claimed as highlighted above. US 20140359290 A1 Mccusker et al., (hereinafter “Mccusker”) discloses an authentication method for calculating an entity using a reconstructed secret wherein results provide an input to a pairing calculation (abstract). Mccusker further discloses performing final exponentiation including a cyclotomic subgroup for the hard part of the final exponentiation of the pairing ([0333]). Mccusker does not however, teach or suggest the specific polynomials claimed as highlighted above. D. Hayashida et al., Efficient Final Exponentiation via Cyclotomic Structure for Pairings over Families of Elliptic Curves , Cryptology ePrint Archive, Paper 2020/875, 07/12/2020, disclosure by inventors of aspects of the claimed invention. Conclusion Any inquiry concerning this communication or earlier communications from the examiner should be directed to EMILY E LAROCQUE whose telephone number is (469)295-9289. The examiner can normally be reached on 10:00am - 1200pm, 2:00pm - 8pm ET M-F. Examiner interviews are available via telephone, in-person, and video conferencing using a USPTO supplied web-based collaboration tool. To schedule an interview, applicant is encouraged to use the USPTO Automated Interview Request (AIR) at http://www.uspto.gov/interviewpractice. If attempts to reach the examiner by telephone are unsuccessful, the examiner’s supervisor Andrew Caldwell can be reached on 571-272-3702. The fax phone number for the organization where this application or proceeding is assigned is 571-273-8300. Information regarding the status of an application may be obtained from the Patent Application Information Retrieval (PAIR) system. Status information for published applications may be obtained from either Private PAIR or Public PAIR. Status information for unpublished applications is available through Private PAIR only. For more information about the PAIR system, see http://pair-direct.uspto.gov. Should you have questions on access to the Private PAIR system, contact the Electronic Business Center (EBC) at 866-217-9197 (toll-free). If you would like assistance from a USPTO Customer Service Representative or access to the automated information system, call 800-786-9199 (IN USA OR CANADA) or 571-272-1000. /EMILY E LAROCQUE/Primary Examiner, Art Unit 2182
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