US 8532286

I'll search for this specific patent number across USPTO and litigation sources.

Let me dig deeper into litigation and any CAFC/Federal Circuit activity for this patent.

Let me check specifically for any Federal Circuit (CAFC) 2026 docket tied to this patent.

US Patent 8,532,286 — Verified Summary

Important scope note on your request: I searched for a CAFC (Federal Circuit) 2026 docket for this patent and did not find one. All confirmed 2025–2026 activity is in district court (W.D. Tex. and E.D. Tex.) plus an ex parte reexamination at the USPTO. I found no appeal docketed at the Federal Circuit for the '286 patent as of the search date. I flag this explicitly rather than infer an appeal exists.

I also note a discrepancy worth recording: the Google Patents legal-status record lists a reexamination event dated 2026‑03‑10 ("Request for reexamination filed," effective 2026‑01‑05), while the USPTO paperwork found in the litigation record shows the request was deposited January 5, 2026 (with an incomplete request December 23, 2025) and reexamination was ordered January 30, 2026. These are consistent in substance but differ in the surfaced dates.

Bibliographic Data (from the patent text and record)

Abstract (as issued)

A system and method providing an alternative way to produce a Montgomery reduction from below by storing a new precomputed value used to substantially replace the μ and n values used in Montgomery reduction with a single value. Modifying the reduction in this way reduces the number of multiplications and registers required. To avoid storing both μ and n, a modified reduction value, or a logical shift or signed version of it, is used in place of μ and n for the bulk of the low-order reduction.

Plain-Language Overview of the Independent Claims

Claim 1 — Method
Performed on a cryptographic apparatus for a Montgomery-style reduction: (a) obtain an operand; (b) compute a modified operand using a reduction value instead of the modulus used in standard Montgomery reduction, such that the reduction value replaces (rather than cancels) the least significant word of the operand — where the reduction value is a function of the modulus; and (c) output the modified operand. The novelty the applicant relied on during prosecution was precisely this "Reduction Value Limitation."

Claim 10 — Apparatus
Same three steps as claim 1, but framed as a cryptographic apparatus comprising a processor configured to operate as a Montgomery engine plus computer-executable instructions that, when executed, obtain the operand, compute the modified operand with the reduction value (function of the modulus, replacing rather than cancelling the LSW), and output it.

Claim 18 — Non-transitory computer readable medium
Same three steps as claim 1, embodied as computer-executable instructions on a non-transitory CRM that cause a cryptographic apparatus to perform the obtain / compute-via-reduction-value / output sequence.

Representative dependent claims add: reduction value = n′ = 2⁻ʷ mod n (or a shifted/signed version), w = word size, n = modulus (claims 2, 11, 19); successively applying the reduction value through the second-most-significant word and doing a standard Montgomery reduction on the most significant word (claims 3, 12, 20); storing precomputed μ, computing m, and overwriting the μ register with m (claims 4, 13, 21); pre-computing and storing n′ with cryptographic system parameters (claims 6, 14, 22); zeroing/modifying/shifting words (logical or physical) (claims 7, 15, 23); carry handling (claims 8, 16, 24); and multiplication or squaring as the cryptographic operation (claims 9, 17, 25).

2025–2026 Enforcement & Challenge Activity (grounded in search results)

• Malikie Innovations Ltd. v. MARA Holdings, Inc. (f/k/a Marathon Digital Holdings, Inc.), W.D. Tex. 7:25-cv-00222 — '286 patent asserted; MARA moved to dismiss the '286 claim for failure to plausibly plead the Reduction Value Limitation (citing Vervain v. Micron). Per a PTO § 290 report, a June 15, 2026 order dismissed the case (Doc. 99). Claim construction was issued March 12, 2026 (Doc. 80), construing inter alia "perform a cancellation thereof."
• Malikie Innovations Ltd. v. Core Scientific, Inc., E.D. Tex. 2:25-cv-00519 — '286 asserted against bitcoin-mining operations; later amended to add HPC/CoreWeave allegations. Core Scientific and CoreWeave moved to dismiss the '286 (with '960 and '062) as patent-ineligible abstract math under § 101.
• Malikie Innovations Ltd. v. Foundry Digital LLC, Fortitude Mining, LLC, Riot Platforms, Inc., and Cipher Digital/Cipher Mining Inc., W.D. Tex. 7:25-cv-00567 (filed Dec. 12, 2025) — '286 asserted; claim construction focused on '286 and '039.
• Ex parte reexamination, Control No. 90/015,830 — Request by Core Scientific, Inc. (deposited Jan. 5, 2026); reexamination ordered Jan. 30, 2026; patent owner filed a March 27, 2026 petition to terminate under 35 U.S.C. § 325(d); requester opposed April 10, 2026. Per the surfaced OPLA decision, the § 1.183 waiver petition was granted (waiving 37 CFR 1.540) and the § 325(d) petition was entered and considered. The outcome of the § 325(d) request itself is not established by the snippets I retrieved — I do not have authoritative confirmation of the final result.

Uncertainty / Caveats

• I have no authoritative confirmation of a CAFC appeal for this patent; treat any claim of one as unverified.
• The reexamination's final disposition (survival, amendment, or cancellation of claims) is not confirmed by the retrieved material.
• Docket documents referenced (Dkt. 99, 105, 115, 127) are drawn from secondary/aggregated sources (CourtListener, archive.org, PTO transmittal forms); I have not independently verified each filing's substance.

Sources: https://patents.google.com/patent/US8532286/en ; https://patents.justia.com/patent/8532286 ; https://www.courtlistener.com/docket/70243940/ (MARA); https://www.courtlistener.com/docket/70244989/ (Core Scientific); https://www.courtlistener.com/docket/72028096/ (Foundry Digital); https://archive.org/download/gov.uscourts.txwd.1172841950/gov.uscourts.txwd.1172841950.93.5.pdf (reexam 90/015,830 OPLA decision).

I'll research litigation involving US 8,532,286 across patent litigation databases.

Let me get details on each specific case and the reexamination.

Let me verify the outcomes and check the reexamination status.

Litigation Involving U.S. Patent No. 8,532,286

Important dating caveat: You stated the current date as April 26, 2026. Several of the docket entries and documents my searches returned are dated after that date (e.g., a May 21, 2026 filing and a September 1, 2026 joint motion). I am reporting exactly what the search results show rather than trimming them, and I flag the post-April-26 items below so you can reconcile them against your own timeline. I also note that I did not find any appellate (CAFC) decisions on this patent.

Ownership context (matters for identifying the correct plaintiff): U.S. 8,532,286 ("the '286 patent") was originally assigned to Certicom Corp., later to BlackBerry Limited, and in 2023 to Malikie Innovations Limited (recorded assignments; Google Patents "Current Assignee: Malikie Innovations Ltd"). The '286 patent is asserted by Malikie Innovations Ltd. together with co-plaintiff Key Patent Innovations Ltd. in a Bitcoin-focused campaign (per RPX/Mondaq, 1 Aug 2025).

Google Patents lists three litigation links for this family, which correspond to the three cases below:

• https://portal.unifiedpatents.com/litigation/Texas%20Western%20District%20Court/case/7%3A25-cv-00567
• https://portal.unifiedpatents.com/litigation/Texas%20Western%20District%20Court/case/7%3A25-cv-00222
• https://portal.unifiedpatents.com/litigation/Texas%20Eastern%20District%20Court/case/2%3A25-cv-00519

Case 1 — Malikie Innovations Ltd. et al. v. MARA Holdings, Inc. (f/k/a Marathon Digital Holdings, Inc.)

Details: The complaint asserts six patents — U.S. 7,372,960; 7,372,961; 8,532,286; 8,666,062; 8,788,827; and 10,284,370 — against MARA's Bitcoin protocol implementations, ASIC‑based mining rigs, mining computers, nodes, mining software, and digital wallets (complaint ¶¶ 6, 124, 129). The '286 patent was pleaded as at least independent claim 1.

Procedural history from the docket (CourtListener, Docket No. 70243940):

• Aug 7, 2025 — MARA's Rule 12(b)(6) motion to dismiss, arguing Malikie failed to plausibly plead the "Reduction Value Limitation" of claim 1 (Dkt. 38).
• Mar 2, 2026 — MARA sur‑reply re motion to stay pending ex parte reexamination.
• Mar 4, 2026 — Markman hearing held (Judge Gilliland).
• Mar 10, 2026 — Order denying MARA's motion to dismiss (Dkt. 78).
• Mar 12, 2026 — Claim construction order and memorandum.
• Jun 15, 2026 — Stipulation of dismissal; order dismissing the case. (This date post‑dates your stated current date.)

Third‑party commentary (PatSnap) reports the matter was resolved by a Notice of Voluntary Dismissal with Prejudice under Rule 41(a)(1)(A)(ii) — a self‑executing mechanism requiring no court approval — with each party bearing its own costs, and no merits ruling issued on any of the six patents. Because the search‑result summary of the docket shows the case terminating by dismissal rather than judgment, the litigation against MARA on the '286 patent is closed.

Case 2 — Malikie Innovations Ltd. et al. v. Foundry Digital LLC, et al.

Details: The complaint targets the defendants' Bitcoin mining operations, ASICs, software, mining‑pool participation (the Foundry USA Pool), and wallets; the '286 patent's Montgomery‑style reduction techniques are among the asserted subject matter. The docket shows:

• Aug 17, 2026 — Report & Recommendations on motions to dismiss (Fortitude Mining; Dkt. 140) and a sealed Report & Recommendation on Foundry Digital's improper‑venue motion (Dkt. 138).
• Aug 18, 2026 — Reply claim construction brief (Dkt. 141).
• Sept 1, 2026 — Joint motion to stay and notice of settlement as to Cipher Digital Inc.; the plaintiffs and Cipher reached a settlement of all matters in controversy and requested a 21‑day stay, with a joint stipulation of dismissal to follow (Dkt. 150).
• A scheduling order (Dkt. 63, filed April 9, 2026) sets a final pretrial conference for Sept 16, 2027 and jury selection Oct 4, 2027.

Many of these entries post‑date your stated current date of April 26, 2026.

Case 3 — Malikie Innovations Ltd. et al. v. Core Scientific, Inc.

Details: Asserted patents include U.S. 7,372,960; 8,532,286; 8,666,062; 8,788,827; and 10,284,370 (the amended complaint also adds U.S. 8,712,039). The complaint accuses Core Scientific's Bitcoin mining operations (e.g., Antminer S19J XP and S21 ASICs) and its HPC/colocation services (including for CoreWeave) (Compl. ¶¶ 46–60). The '286 patent was asserted at least as independent claim 1. The complaint seeks damages not discharged by Core Scientific's bankruptcy (Compl. p. 3).

Docket/CourtListener highlights:

• Feb 16, 2026 — Motion to compel briefing touching on claim‑accrual/bankruptcy damages issues (Dkt. 64).
• Mar 17–18, 2026 — Order granting motion to compel.
• Mar 24–25, 2026 — Memorandum order on the motion to transfer venue; second amended docket control order (Markman/claim construction hearing Nov 18, 2026; pretrial conference May 18, 2027; jury selection July 12, 2027).
• Mar 27, 2026 — Core Scientific filed a third‑party complaint against CoreWeave, Inc.
• Mar 31 / Apr 7, 2026 — Reply and sur‑reply on Core Scientific's motion to dismiss the amended complaint under 35 U.S.C. § 101.
• Apr 10, 2026 — Order adopting the venue memorandum order.

Related USPTO proceeding (not district‑court litigation, but arising from this dispute)

Ex parte reexamination of the '286 patent — Control No. 90/015,830. Requester: Core Scientific, Inc. An incomplete request was filed December 23, 2025; the complete request was deposited January 5, 2026; the Central Reexamination Unit ordered reexamination on January 30, 2026 (finding substantial new questions of patentability on each proposed ground). Patent owner (Malikie) filed a combined petition on March 27, 2026 seeking termination under 35 U.S.C. § 325(d) and suspension of the rules; the requester filed an opposition April 10, 2026. This is the reexamination referenced in the MARA stay motion and incorporated as an exhibit in the W.D. Tex. docket (7:25‑cv‑00222, Dkt. 93‑5; Google Patents also records a "Request for reexamination filed" dated 2026‑03‑10). Because the '286 patent is in reexamination while Case 3 remains pending, expect further stays/parallel‑proceeding activity.

Summary

• Three district‑court cases have asserted U.S. 8,532,286, all in the Malikie Innovations Ltd. / Key Patent Innovations Ltd. Bitcoin‑mining campaign:
  1. W.D. Tex. 7:25‑cv‑00222 v. MARA Holdings (filed May 12, 2025) — dismissed with prejudice (voluntary dismissal; no merits ruling).
  2. W.D. Tex. 7:25‑cv‑00567 v. Foundry Digital, Fortitude Mining, Riot Platforms, Cipher Digital, and Shenzhen MicroBT (filed Dec 12, 2025) — pending; Cipher settled.
  3. E.D. Tex. 2:25‑cv‑00519 v. Core Scientific (filed May 12, 2025) — pending.
• No CAFC appellate decision or PTAB IPR/PGR on the '286 patent appeared in my searches; the only post‑grant USPTO challenge found is the ex parte reexamination Control No. 90/015,830 (Core Scientific as requester).

Confidence and limitations: I am highly confident about the three case numbers, parties, jurisdictions, and filing dates (multiple corroborating sources: Google Patents, CourtListener dockets, PacerMonitor, and RPX/Mondaq). I am less certain about the final outcome of the MARA case — PatSnap describes a voluntary dismissal with prejudice and the docket shows a stipulation/order of dismissal, but I could not retrieve the signed dismissal order itself to confirm the "with prejudice" and "each party bears its own costs" language firsthand. I was also unable to verify the current disposition of the March 27, 2026 reexamination‑termination petition (my last search on its status was cut off). If you need the precise docket‑sheet endings or the reexamination decision, those should be pulled directly from PACER and USPTO PTAB/CRU records.

I'll verify the structured block against live sources before writing anything.

Found a reexamination thread — let me check whether any AIA trial (IPR/PGR/CBM) exists and whether there are appeals.

Now let me pin down the reexamination details and confirm there's truly no AIA trial on the '286 patent.

Proceedings overview

There are zero AIA trial proceedings on file for US 8,532,286. The canonical PTAB list supplied for this patent is empty (no IPR, PGR, or CBM has ever been instituted or even denied on this patent), so the defensive posture is not "hardened by surviving IPRs" and not "claims canceled" — it is untested at the PTAB, but under active attack in a different PTO forum. The real action on the '286 patent is an ex parte reexamination, Control No. 90/015,830, requested 2026-01-05 by Core Scientific, Inc., in which the CRU found a substantial new question of patentability (SNQ) on all proposed grounds covering claims 1–25, and where the patent owner's attempt to kill the proceeding outright under 35 U.S.C. § 325(d) was denied on 2026-05-20. That reexam is a live merits threat to every claim of the patent, including the independent claims. A defendant today should read the absence of IPRs as an absence of estoppel — all prior-art grounds remain available — but must immediately account for the reexam and for the § 325(d) / SNQ record it has already created.

Verification note: I searched for any IPR/PGR/CBM number tied to "8,532,286" or to the parties (Malikie, Key Patent Innovations, MARA, Core Scientific) and found none. The proceedings below are reported with that caveat — nothing here is an AIA trial.

Non-AIA proceeding of record (flagged)

Ex parte reexamination Control No. 90/015,830 — Requester: Core Scientific, Inc. v. Patent Owner: Malikie Innovations Ltd. (US 8,532,286)

• Type: Ex parte reexamination under 35 U.S.C. §§ 301–307 — not an AIA trial; handled by the Central Reexamination Unit (CRU), Art Unit 3992, with petitions routed to OPLA. No APJ panel exists; there are no judge names to report.
• Filed: 2026-01-05 (request deposited; an incomplete request was filed 2025-12-23). A request/reexam "RR" legal event in the structured patent data is keyed to effective date 2026-01-05, matching this proceeding.
• Status: Reexamination ordered and pending on the merits — claims 1–25 in scope. The CRU's 2026-01-30 order found that each proposed ground raises a substantial new question of patentability. Patent owner's 2026-03-27 combined petition (seeking rule suspension and summary termination under § 325(d)) was denied on 2026-05-20; the reexam "will proceed on the merits." I found no reexam certificate and no final office action on the public record as of 2026-09-28 — do not assume either.
• Requester identity caveat: The OPLA decision names Core Scientific, Inc. as requester (third-party correspondence address: King & Spalding LLP). MARA's own filings in W.D. Tex. 7:25-cv-00222 characterize the six reexaminations of the patents-in-suit as "MARA petitioned the Patent Office" and attach this very '286 decision as Exhibit E to its 2026-05-21 supplemental-authority notice. I cannot reconcile the two from the public record; treat the requester/real-party-interest question as open and worth pulling from the full file via USPTO Patent Center and PTAB/PTO reexam records.
• Petition grounds: Not fully published in the excerpted decision. The order confirms an SNQ was found on all proposed grounds against claims 1–25; the specific references and § 102/§ 103 bases were not disclosed in the documents I could retrieve. Separately, MARA's stay briefing in the parallel case represents that the PTO accepted the prior-art status of a "Brown" reference in the sibling '960/'062 reexams — a signal, not proof, that the requester group is running a common prior-art package across the six asserted patents, including the '286.
• Institution decision: No AIA-style institution decision (not applicable). The functional equivalent is the 2026-01-30 CRU order granting reexamination on SNQ grounds.
• Final Written Decision: None exists and none can exist in an ex parte reexam. Outcome at claim level to date: all 25 claims remain in force and contested; no claim has been canceled, confirmed, or amended.
• Settlement / termination: The patent owner attempted summary termination, not settlement. On 2026-05-20, OPLA denied the requests for summary termination as to five of the six patents-in-suit (including the '286), while granting the § 1.183 waiver to even entertain the § 325(d) petition. Requester filed an opposition (2026-04-10) rather than settling. The related district court case against MARA ended by stipulated dismissal on 2026-06-15 (Doc. 99, W.D. Tex. 7:25-cv-00222) — terms are not public, and the dismissal does not terminate the '286 reexam.
• Appeal: None. A reexam appeal would lie to the PTAB under § 134(b) and then to the Federal Circuit; neither has occurred. I found no CAFC docket involving US 8,532,286.
• Defensive value: This is currently the only adversarial PTO proceeding against the '286 patent, and it covers every claim. For a defendant, that means (a) there is no IPR final written decision to cite and no § 315(e)(2) estoppel, so your invalidity case is unconstrained; and (b) if you are in a district court case on this patent, the SNQ finding plus the pending reexam is a strong, well-documented basis for a stay motion — precisely the playbook MARA ran in W.D. Tex. (Motion to Stay Pending Ex Parte Reexamination, 7:25-cv-00222, Dkt. 54). Because the '286 patent's adjusted expiration is 2031-08-11, it has not expired, so unlike the expired patents in that case, claim amendments are still available to Malikie — a reexam win here may be a narrowed patent rather than a dead one.

Strategic summary

Claim status: everything is UNTESTED at the PTAB; everything is AT RISK in reexam. No claim of US 8,532,286 has been canceled, held unpatentable, or confirmed in any AIA trial — there simply are no AIA trials. Claims 1–25 are all live and all within the scope of Control No. 90/015,830, where the CRU found an SNQ on every asserted ground and OPLA refused to terminate. The independent claims (1, 10, 18) and the dependent claim sets that matter commercially are therefore all potentially exposed, but none has yet been adjudicated. If a demand letter or complaint asserts claim 1, you cannot say "claim 1 is dead." You can say the PTO has found substantial new questions of patentability as to every claim, and that the patent owner has already been rebuffed once in trying to shut that inquiry down.

Estoppel landscape: wide open, but not free. Because no IPR/PGR was ever instituted, 35 U.S.C. § 315(e)(2) estoppel does not attach and neither does § 325(e) PGR estoppel. A defendant being asserted against can raise any § 102/§ 103 ground, including art that was or could have been raised elsewhere — a materially better position than a defendant facing a patent that lost an IPR (where the surviving claims are narrowed and the patent owner has a favorable record to lean on). Two cautions: (1) § 315(b) timing — the patent has been in suit since 2025 (E.D. Tex. 2:25-cv-00519 against Core Scientific; W.D. Tex. 7:25-cv-00222 against MARA, filed 2025-05-12; and a third W.D. Tex. case, 7:25-cv-00567). If you were served more than one year ago, an IPR may be time-barred; check service dates before promising a petition. (2) § 325(d) — Malikie is affirmatively using § 325(d) to try to terminate PTO proceedings (it did so here, and per Law360 reporting in July 2026 the Office rejected a Unified Patents reexam request on a Malikie video-coding patent based on a Malikie filing under a new spring-2026 policy). Expect the same § 325(d) / "settled expectations" discretionary-denial arguments against any IPR you file. A fresh IPR petition should be built to distinguish the reexam's art and to pre-empt settled-expectations and § 325(d) attacks.

Pattern signals. (1) No repeat petitioner at the PTAB on this patent — because there is no PTAB activity at all on the '286. (2) The attack channel is ex parte reexamination, run as a portfolio campaign: MARA and/or Core Scientific reexamined all six patents-in-suit (the '286, '827, '062, '960, '370, '961), with reexam grants on five by mid-February 2026 and the sixth pending; Unified Patents filed and won institution on Control No. 90/016,454 against Malikie's '372,960 patent (granted 2026-09-17) and is therefore an active defensive aggregator in this family tree — though no Unified proceeding against the '286 itself has surfaced. (3) Malikie is a seasoned PTAB litigant on both sides: it is Patent Owner in at least IPR2026-00120 (Vivint/NRG v. Malikie, U.S. 11,119,756) and it litigates its reexam terminations hard. (4) The litigation is churning fast: the MARA case was dismissed by stipulation on 2026-06-15; Core Scientific has a pending motion for leave to supplement invalidity contentions (up to 2026-07-01) expressly to import the reexam art into its E.D. Tex. case.

Recommended next steps

• Do not represent that any claim of the '286 patent has been canceled. It has not. As of 2026-09-28 the honest statement is: no AIA trial has ever been filed; all 25 claims stand; a reexam is pending on all 25 claims with an SNQ finding on every ground.
• Pull the '830 file immediately from USPTO Patent Center / the reexam record and confirm: the requester's real party in interest (Core Scientific vs. MARA), the specific references and § 102/§ 103 grounds, whether any office action has issued, and the current response deadline. The OPLA decision is available at Dkt. 93-5 in W.D. Tex. 7:25-cv-00222; the docket is at CourtListener, 7:25-cv-00222, and the Core Scientific case is 2:25-cv-00519.
• If you are sued on the '286, move to stay pending the reexam early and cite the same record MARA used: the CRU's SNQ finding on every ground, the 2026-05-20 OPLA denial of summary termination, and the PTO's ~80% claim-cancellation/amendment rate in third-party-requested reexams. Marshall Division's order in the Core Scientific case allowing supplementation of invalidity contentions with reexam art (2:25-cv-00519, Dkt. 101-3) is useful persuasive authority for keeping your invalidity case synchronized with the PTO.
• On the IPR track, verify your § 315(b) window before committing; if it is open, a petition would be the first-ever AIA trial on this patent and would face § 325(d) and settled-expectations discretionary-denial arguments from Malikie. If the window is closed, your leverage is the reexam plus district-court invalidity — which, given the absence of any § 315(e)(2) estoppel, is a relatively unconstrained position.
• Docket-watch two triggers: (i) any reexam certificate, final rejection, or Malikie amendment in Control No. 90/015,830 (an amendment narrows the patent and changes your non-infringement analysis; a certificate canceling claims ends the case as to those claims), and (ii) any § 134(b) appeal or CAFC appeal from that reexam. There is no AIA trial deadline to track — no institution date, no one-year FWD clock, no oral hearing — because no AIA trial exists.
• One unresolved gap I will not paper over: the identity of the defendant in W.D. Tex. 7:25-cv-00567 and the exact grounds/references in the '830 reexam are not established by the sources I could reach. Both should be confirmed from primary sources before any of this is used in a filing or an opinion letter.

I'll reconstruct the chain from the authoritative patent record and verify key facts with targeted searches.

Now let me try to pin down the recorded correspondent and verify the reel/frame entries.

Ownership Reconstruction — US 8,532,286 B2

Patent: System and method for reducing the computation and storage requirements for a Montgomery-style reduction
App. 12/838,999 · filed 2010‑07‑19 · priority US 61/226,427 (2009‑07‑17) · issued 2013‑09‑10 · PTA 388 days · Google Patents status: Active, adjusted expiration 2031‑08‑11
Family: EP2276194B1 (granted 2014‑11‑26), CA2710064C (granted 2016‑02‑16)
Family ID: 42989673 · Google Patents family has litigation flag: yes

Source discipline note. I could not query assignmentcenter.uspto.gov or assignment.uspto.gov directly with the tools available in this session, and I could not retrieve the correspondent of record for any recording. Every reel/frame below comes from the Google Patents legal‑events rendering of the same USPTO assignment records (which reproduces the REEL/FRAME string), and is cross‑checked against PTAB and district‑court filings. Where a field is blank in that rendering — specifically the correspondent — I say so rather than guess. Verify each entry at USPTO Assignment Center (search 8532286) or the legacy Assignment Search.

Inventors

• Lambert is Certicom's most prolific cryptography inventor of this era. The same single inventor appears on the sibling modulus‑reduction family that was swept into the same 2023 sale — e.g. US 8,862,651 "Method and apparatus for modulus reduction" (app. 12/609,772, Certicom, filed 2009‑10‑30) and US 2011/0270906 "Method and apparatus for providing flexible bit‑length moduli on a block Montgomery machine" (app. 12/770,996, filed 2010‑04‑30, co‑inventor Yuri Peehev). Employer identity is inferred from (a) the recording of 2010‑08‑05, Reel 024793/0989, in which Lambert assigns to Certicom, and (b) the repeated Certicom/Cambridge pairing on the neighbouring filings.
• Departure pattern: none. There is no evidence Lambert left Certicom/BlackBerry within 12 months of filing; he continued to file in the same technology for the same assignee through at least 2010‑04‑30. The classic "inventors flee → fire‑sale" tell is not present here. This is a straightforward single‑inventor employee assignment.

Original assignee

Certicom Corp., Mississauga, Ontario, Canada — named on the issued patent as (73) Assignee: Certicom Corp.

• Primary business. Elliptic‑curve and embedded cryptography. Certicom sold licensable crypto toolkits (Security Builder / ECC Core family), VPN and smart‑card security software, and supplied the ECC stack used across Research In Motion's BlackBerry devices and into content‑protection applications. The patent's own specification names the AACS copy‑protection 160‑bit curve and "the popular ARM architecture" as deployment targets, which is consistent with a licensable software/hardware engine business rather than a consumer product.
• Product embodying the claims. The claims recite a computational method performed on a cryptographic apparatus (claim 1, claim 10) and are implemented in the "block Montgomery machine" described at FIGS. 1–2. Certicom's business model was precisely to ship this class of engine as library/engine IP, and BlackBerry shipped devices incorporating Certicom crypto. I flag this as reasonably inferred from the specification and Certicom's known product line, but not verified against product documentation in this pass.
• Status — operating, but not independent. Certicom was acquired by Research In Motion Limited (bid announced January 2009; ~US$107M reported at the time — verify against RIM's FY2009 filings). RIM was renamed BlackBerry Limited effective 2013‑07‑09 (the change‑of‑name recorded against this patent on 2013‑07‑29). Certicom Corp. continued to exist as a BlackBerry affiliate and held record title to this patent until 2019‑09‑30, when it assigned upstream to BlackBerry Limited. No bankruptcy, no insolvency — BlackBerry divested the portfolio as a monetization decision, not a distressed sale.

Assignment timeline

Five recorded events touch this patent (one of which is a name change, not a transfer of this patent). Correspondent is not exposed in the available rendering of every entry — marked [not retrievable in this pass].

• 2009‑07‑17 — provisional priority date; no assignment associated with it.

• Executed 2010‑07‑19 / recorded 2010‑08‑05 — Reel 024793/0989

  • Conveyance: Assignment (ASSIGNMENT OF ASSIGNORS INTEREST)
  • Assignor: LAMBERT, ROBERT JOHN
  • Assignee: CERTICOM CORP. (Canada)
  • Correspondent: [not retrievable in this pass] — the rendering shows only the reel/frame and an "EFFECTIVE DATE: 20100719" field.
  • Context: Standard employee‑inventor assignment to the operating company, filed with the initial application. This is the only assignment that ties the patent to an entity that practiced the technology.

• Executed 2013‑07‑09 / recorded 2013‑07‑29 — Reel 030919/0108

  • Conveyance: Change of Name only (ASSIGNOR: RESEARCH IN MOTION LIMITED → BLACKBERRY LIMITED)
  • Assignor: RESEARCH IN MOTION LIMITED
  • Assignee: BLACKBERRY LIMITED (Ontario)
  • Correspondent: [not retrievable in this pass]
  • Context: Corporate name change of the parent, recorded broadly across the RIM/BlackBerry docket. Important caveat: this record does not move title to US 8,532,286. The owner of record at this date was still Certicom Corp., so there is a six‑year gap (2010‑08‑05 → 2019‑10‑02) in which no assignment of this patent to BlackBerry appears in the record. Note also that the reel for this name change is patent‑specific: on a sibling BlackBerry patent the same RIM→BlackBerry change of name recorded at Reel 034045/0741, not 030919/0108.

• Executed 2019‑09‑30 / recorded 2019‑10‑02 — Reel 050610/0937

  • Conveyance: Assignment (ASSIGNMENT OF ASSIGNOR'S INTEREST)
  • Assignor: CERTICOM CORP.
  • Assignee: BLACKBERRY LIMITED (Ontario)
  • Correspondent: [not retrievable in this pass]
  • Context: Internal corporate reorganization. Six years after the RIM name change, Certicom Corp. finally passed record title on this patent up to the public parent — the step that had been missing since 2010.

• Executed 2023‑05‑11 / recorded 2023‑06‑16 — Reel 064104/0103

  • Conveyance: Assignment (ASSIGNMENT OF ASSIGNOR'S INTEREST)
  • Assignor: BLACKBERRY LIMITED
  • Assignee: MALIKIE INNOVATIONS LIMITED (Ireland)
  • Correspondent: [not retrievable in this pass]
  • Context: Transfer‑to‑asserter / portfolio sale. Part of the announced BlackBerry→Malikie transaction covering roughly 32,000 patents, reported at approximately US$200M, with Malikie stated by IAM and JUVE Patent to be an Irish NPE under Key Patent Innovations Ltd. acting to commercialize the former BlackBerry portfolio.

• Executed 2023‑05‑11 / recorded 2023‑06‑19 — Reel 064270/0001

  • Conveyance: Nunc pro tunc Assignment (corrective recording of the same 2023‑05‑11 transfer)
  • Assignor: BLACKBERRY LIMITED
  • Assignee: MALIKIE INNOVATIONS LIMITED (Ireland)
  • Correspondent: [not retrievable in this pass]
  • Context: Documentation clean‑up within three days of the first recording — a nunc pro tunc ("now for then") instrument back‑dating the Malikie acquisition to 2023‑05‑11. Two recordings of the same sale on consecutive business days is a bulk‑conveyance artifact, not two separate transfers.

Related structure worth flagging (not a record against this patent). In the same BlackBerry→Malikie transaction, sibling patents moved through an intermediate escrow conduit: BLACKBERRY LIMITED → OT PATENT ESCROW, LLC at Reel 063471/0474, then OT PATENT ESCROW, LLC → MALIKIE INNOVATIONS LIMITED at Reel 064015/0001, with further OT Patent Escrow recordings at Reels 064806/0669 and 064807/0001 (source: a 37 CFR 3.73(c) chain‑of‑title statement filed 2023‑12‑27 in a PTAB proceeding over US 8,115,731 / app. 11/240,501). OT Patent Escrow, LLC is listed at 200 West Madison, 37th Floor, Chicago, IL 60606 — a Chicago business address consistent with an IP escrow vehicle, not a manufacturing operation. I did not find an OT Patent Escrow record specifically naming US 8,532,286; if one exists it is likely a collateral/escrow recording that does not appear in this patent's visible legal‑events set.

Timeline diagram

timeline
    title Ownership of US 8532286
    2009 : Provisional filed Jul 17
    2010 : Application filed Jul 19
         : Inventor assigns to Certicom
    2013 : RIM renamed BlackBerry
         : Patent issued Sep 10
    2019 : Certicom assigns to BlackBerry
    2023 : BlackBerry sells to Malikie Innovations
         : Nunc pro tunc recording
    2025 : Suits filed in Texas districts
         : Core Scientific case E D Tex
    2026 : Ex parte reexam requested

NPE / troll-pattern signals

1. Shell‑entity transfer — PRESENT.
Transfer from an operating company to a licensing‑only vehicle is documented at Reel 064104/0103 and Reel 064270/0001, both executed 2023‑05‑11: assignor BlackBerry Limited (a public company that manufactured handsets and security software), assignee Malikie Innovations Limited. The evidence is functional, not onomastic: Malikie's stated purpose is commercializing the portfolio; the PTAB mandatory notice in IPR2026‑00120 (Vivint Smart Home / NRG Energy v. Malikie, US 11,119,756) states that "Malikie is a subsidiary of KPI. KPI is the beneficiary of a trust pursuant to which Malikie owns and holds the '756 Patent," and identifies both Malikie and Key Patent Innovations Ltd. as real parties in interest — a trust‑held, licensing‑only structure. Caveat I will not paper over: Malikie is an Irish limited company, not a Delaware/Texas LLC with an "IP/Holdings" suffix, and I have no evidence of a registered‑agent service address for it. The signal rests on the operating→licensing transfer plus the trust/NPE structure, not on the name.

2. Known asserter in the chain — PRESENT.
Malikie Innovations Ltd. / Key Patent Innovations Ltd. is the current owner of record (Reels 064104/0103; 064270/0001) and is a documented high‑frequency plaintiff: IAM reports BlackBerry patents "now been asserted against 19 companies"; five new US suits were filed in May 2025; the E.D. Tex. §1782 order (Case 2:25‑mc‑0006) confirms litigation against Xiaomi in Munich and Delhi; UPC actions against Nintendo (ACT_53365/2024, ACT_52355/2024) and against Oppo and Hisense followed. The Google Patents page itself carries Unified Patents litigation links for three 2025 cases. Precision note: Malikie/KPI does not match any of the enumerated legacy names (Acacia, Marathon, IV, IPNav, Wi‑LAN, Mosaid/Conversant, Vringo, Pendrell, Innovatio, MPHJ, Lumen View, Round Rock, Document Generation Corp, Spangenberg entities). The match is to the modern high‑frequency asserter category, supported by IAM/JUVE reporting and the IPR record — not to that list.

3. Repeat correspondent across the chain — UNCLEAR (not verifiable in this pass).
The correspondent of record is not exposed for any of the five recordings above, so I cannot state that one attorney or firm appears on multiple links. One lead, flagged as a lead, not a finding: in the PTAB chain‑of‑title statement filed 2023‑12‑27 on a sibling BlackBerry→Malikie patent (US 8,115,731, recorded via Reels 063471/0474, 064066/0001, 064015/0001, 064806/0669, 064807/0001), the submission shows USPTO Customer Number 89653 with the correspondence name "MOFFAT‑RIM, 11 HOLLAND AVE" — i.e. the Ottawa IP firm Moffat & Co. operating a docket dedicated to the RIM/BlackBerry portfolio — and is signed /Richard J. Botos/, Reg. No. 32,016. That is a single repeat‑player filing contact for the post‑sale Malikie portfolio at the USPTO, and Moffat & Co. likewise handled the pre‑sale BlackBerry recordings. But this is a §3.73(c) filing in a different patent's IPR, not an assignment‑recordation correspondent for US 8,532,286, and one firm doing the paperwork for both an operating company and its divested portfolio is entirely unremarkable. Do not treat this as an NPE signal without pulling the actual reel records.

4. Cascading transfers — NOT PRESENT as described (partial structural analogue only).
There is no chain of sequentially created LLCs and no shared‑address cluster of assignees. The real cadence is slow: 2010 (inventor→Certicom) → 2019 (Certicom→BlackBerry, a 9‑year internal step) → 2023 (BlackBerry→Malikie). The only sub‑24‑month clustering is the 2023‑06‑16 and 2023‑06‑19 pair (Reels 064104/0103 and 064270/0001), which is one transaction documented twice — the second being an explicitly corrective nunc pro tunc instrument. The OT Patent Escrow, LLC conduit used for sibling patents in the same deal (Reels 063471/0474, 064015/0001, 064806/0669, 064807/0001) supplies at most one intermediate hop, and no escrow recording tied to this patent was found.

5. Pre‑litigation transfer — NOT PRESENT (fails the 6‑month test).
The Malikie acquisition executed 2023‑05‑11 and was recorded in June 2023. The first assertion of this patent in the current campaign appears in the 2025 filings — Malikie Innovations Ltd. et al. v. Core Scientific, Inc., E.D. Tex. 2:25‑cv‑00519‑JRG‑RSP, and the W.D. Tex. cases 7:25‑cv‑00567 (Foundry Digital, Riot Platforms, Fortitude Mining, Cipher Digital) and 7:25‑cv‑00222 — a lag of roughly 24 months. That is ordinary portfolio‑acquisition‑to‑assertion latency, not a standing‑setting transfer arranged within six months of suit. (The '286 patent was folded into the Core Scientific case by amended complaint/ infringement contentions; the complaint also asserts US 8,712,039 and US 8,666,062 against bitcoin‑mining ASICs and HPC hardware.)

6. Bankruptcy fire‑sale — NOT PRESENT.
No Chapter 7 or Chapter 11 proceeding by Certicom Corp. or BlackBerry Limited. The 2023 transfer was a solvent, negotiated divestiture of a large patent portfolio, and BlackBerry continued reporting licensing revenue after it (IAM: $22.2M in a recent reporting period, plus a new Sisvel pool membership). No Nortel/Kodak/Polaroid‑style auction.

7. Privateering — UNCLEAR.
The structural preconditions are visible: an operating company sold a large portfolio to a trust‑held NPE, that NPE now sues BlackBerry's former licensees and competitors, and BlackBerry still earns licensing revenue. The §1782 proceeding (Case 2:25‑mc‑0006) even forced the production of BlackBerry's confidential A‑B (Apple–BlackBerry) licence agreements for use against Xiaomi in Germany and India — the classic posture in which a divesting operating company's earlier royalty terms become the NPE's comparables. But I have not seen the 2023 sale agreement and cannot state whether BlackBerry retained a revenue share or a direction right over assertions, which is the finding that would move this from structure to proof. Marked unclear, not present.

8. Defensive aggregator (anti‑NPE) — NOT PRESENT.
The chain terminates at Malikie Innovations Limited as the current assignee of record, with no onward transfer to RPX, AST, LOT Network, Unified Patents, or OIN. The patent is live and being asserted, not neutralized. (Unified Patents appears on this patent's page only as a litigation‑data provider, not as an owner.)

Additional post‑acquisition event worth recording (not an assignment): a request for ex parte reexamination was filed effective 2026‑01‑05 and docketed 2026‑03‑10 (Google Patents legal‑event code RR). This is consistent with defendants challenging the asserted claims in the 2025–2026 campaign; it is not an ownership event. Relatedly, the German Federal Patent Court has issued a preliminary adverse opinion on the validity of a different Malikie patent (EP2879321) asserted against Xiaomi.

Verdict

NPE — high confidence.

Two independent strong signals are documented rather than inferred. (1) Shell‑entity transfer: the '286 patent moved from BlackBerry Limited — an entity that manufactured products and practiced the claimed crypto engine through its Certicom subsidiary — to Malikie Innovations Limited, at Reel 064104/0103 and Reel 064270/0001, both executed 2023‑05‑11, in the ~32,000‑patent sale; Malikie is not a manufacturer, and the parallel PRAB record in IPR2026‑00120 confirms it is a KPI subsidiary holding patents as beneficiary of a trust. (2) Known asserter: Malikie/KPI is plaintiff on 2:25‑cv‑00519 (E.D. Tex., Core Scientific), 7:25‑cv‑00567 and 7:25‑cv‑00222 (W.D. Tex.), plus UPC and German/Indian actions, and is reported by IAM as having asserted BlackBerry‑derived patents against 19 companies. A 2026 ex parte reexamination request and a 24‑month acquisition‑to‑suit lag complete the picture of an assertion‑driven chain rather than an operational one.

The two signals that would deepen the case are unresolved: the correspondent of record is not retrievable from the sources available to me (signal 3), and the privateering question turns on sale‑agreement terms I have not seen (signal 7). Neither gap undermines the verdict, but both are the right targets for the next pass.

Verify: USPTO Assignment Center — search 8532286 · Legacy Assignment Search · Google Patents US8532286B2 legal events

I'll search the USPTO and related sources for US 8,532,286 and its cited prior art.

Let me gather details on the remaining key references, including the non-patent literature.

Prior-Art Analysis — US 8,532,286 B2

Scope note. Per your instruction I searched the USPTO/Google Patents record for the specific number 8,532,286 (application 12/838,999) and did not substitute any neighboring numbers. All citations below are the references literally listed on the '286 record (20 examiner/third-party U.S. patent documents plus 6 non-patent citations, with 7 additional documents appearing in the extended "Patent Citations (27)" list). Where a search result contradicted what I would otherwise assume, I followed the record.

Two facts drive everything below and build on the earlier sections of this analysis:

1. Claim 1 as issued is narrower than claim 1 as published. The pre-grant publication (US 2011/0013768 A1, Jan. 20, 2011) recited "obtaining a modified reduction value." The issued claim 1 recites obtaining an operand, then "computing a modified operand using a reduction value, instead of a modulus used in performing a standard Montgomery reduction, to perform a replacement of a least significant word… rather than perform a cancellation thereof, the reduction value being a function of the modulus." This amendment (the "Reduction Value Limitation" you already identified) post-dates all of the §102(b) art and most of the §102(e) art, so each reference must be read against the issued language.
2. All references here are cited art, not adjudicated invalidating art. The ex parte reexamination 90/015,830 found a substantial new question of patentability (SNQ) on every proposed ground (per the Jan. 30, 2026 CRU order reflected in the OPLA decision), but the merits have not been decided in anything I retrieved. So the §102 mappings below are analyst candidate mappings, not holdings.

1. The §102 framework applied here

Because application 12/838,999 was filed July 19, 2010, pre-AIA §102/§103 governs (AIA first-inventor-to-file applies only to applications filed on/after Mar. 16, 2013). Relevant dates:

A reference is easy §102(b) art only if published before Jul. 19, 2009. Otherwise it can only reach the '286 via §102(e) (U.S. patent/application publication with an earlier effective U.S. filing date) — and even then only if it is "by another."

To anticipate under §102, a single reference must disclose every element of the claim, including: (i) an operand of a cryptographic operation; (ii) a reduction value that is a function of the modulus; (iii) use of that value instead of the modulus used in a standard Montgomery reduction; (iv) a replacement of the LSW rather than a cancellation; and (v) outputting the modified operand. The "replacement rather than cancellation" contrast is the element most references fail.

2. Most relevant references — detailed §102 candidates

(a) US 2010/0332578 A1 — Gopal, Vinodh et al. (Intel Corporation) → granted as US 8,392,494 B2

• Citation: U.S. Patent Application Pub. No. 2010/0332578 A1, "Method and apparatus for performing efficient side-channel attack resistant reduction"; U.S. Patent 8,392,494 B2 (issued Mar. 5, 2013).
• Dates: Filed Jun. 26, 2009; published Dec. 30, 2010; priority Jun. 26, 2009.
• Description: Discloses side-channel-resistant modular reduction using modified Barrett and modified Montgomery reduction. Folding reduces a 1024-bit value to 640 bits, and "the least significant 'zeroed' 128-bits can be eliminated." Expressly discusses Montgomery reduction of (a·b) mod m with respect to C where (a·b)C⁻¹ mod m.
• §102 candidate mapping: This is the closest structural reference to claim 1 / claim 10 / claim 18 because it is a modified Montgomery reduction that eliminates low-order bits without recomputing μ per iteration, and it is a printed publication before the Jul. 19, 2010 filing with a §102(e) date (Jun. 26, 2009) that precedes even the Jul. 17, 2009 provisional. The fatal gap for anticipation is the "replacement … rather than … cancellation" element: Gopal zeroes/cancels the low bits (adds a multiple of the modulus and discards), which is exactly the standard cancellation the '286 distinguishes itself from. Best characterized as strong §103 art against claims 1, 3, 7, 9, 10, 15, 17, 18, 23, 25, not a clean §102 reference. (This appears to be among the reexam grounds Core Scientific advanced.)

(b) US 2009/0268900 A1 — Lambert, Robert J. ("Signed montgomery arithmetic") → granted US 8,781,112 B2

• Citation: U.S. Pub. No. 2009/0268900 A1; US 8,781,112 B2.
• Dates: PCT/US filing Oct. 29, 2008; priority Nov. 2, 2007 (US 60/984,851); published Oct. 29, 2009.
• Description: Same inventor, same assignee (Certicom). Applies Montgomery reduction to signed/negative quantities, using signed versions of Montgomery values and avoiding conditional add/subtract.
• §102 candidate mapping: Its publication (Oct. 29, 2009) postdates the provisional, so it is not §102(b) art; its only path is §102(e) via the Nov. 2, 2007 effective filing date. But it names the same sole inventor, so it is not "by another" and is likely disqualified as §102(e) prior art against at least claims 2, 11, 19 (which expressly recite "a shifted or signed version of n′"). Its real significance is that it supplies the "signed version" concept the dependent claims recite. This is an important non-anticipation point to flag.

(c) US 2010/0146028 A1 — Certicom Corp. ("Method and apparatus for modulus reduction") → granted US 8,862,651 B2

• Citation: U.S. Pub. No. 2010/0146028 A1; US 8,862,651 B2 (issued Oct. 14, 2014), inventor Robert John Lambert.
• Dates: Filed Oct. 30, 2008; published Jun. 10, 2010.
• Description: Performs modulus reduction without precomputation by substituting an equivalent higher-order word value for each lowest-order word — expressly contrasted with Montgomery reduction's need to precompute and store μ = (−1/n) mod R. The substance is the equivalence a₀V ≡ a₀ (mod p) where p = V−1.
• §102 candidate mapping: The most conceptually similar Certicom document: it "eliminat[es] the lowest order digital word" by substitution rather than cancellation, and it removes the storage/multiplication burden of μ. Against claim 1 (and mirrors 10/18), this reference bears on the "reduction value … instead of a modulus" and "avoids storing μ" elements. Again, same-assignee/same-inventor status raises "by another" questions under §102(e); and it substitutes fixed power-of-two forms rather than a value that is a function of the modulus per claim 1. Strong §103 art; not a clean §102 reference.

(d) Knezevic, Batina & Verbauwhede, "Modular Reduction without Precomputational Phase" (ISCAS 2009)

• Citation: M. Knezevic, L. Batina, I. Verbauwhede, Modular Reduction without Precomputational Phase, 2009 IEEE Int'l Symposium on Circuits and Systems (ISCAS), Taipei, pp. 1389–1392, ISBN 978-1-4244-3827-3; DOI 10.1109/ISCAS.2009.5118024.
• Dates: Conference May 24–27, 2009; IEEE Xplore added Jun. 26, 2009. Before Jul. 19, 2009 → §102(b)-eligible printed publication.
• Description: Shows Barrett and Montgomery reduction of integers without a precomputational phase, using four special sets of moduli (e.g., m = bⁿ − Δ, m = bⁿ⁻¹ + Δ), and notes applicability to NIST ECC primes (P-192, P-224, P-384, P-521).
• §102 candidate mapping: Directly addresses eliminating the μ precompute — the motivation of the '286. But it achieves this by constraining the modulus, not by defining a new reduction value n′ = 2⁻ʷ mod n that replaces the LSW. Fails the "replacement rather than cancellation" and "function of the modulus" elements. Best framed as §103 art combinable with a modified-Montgomery reference; it independently cannot anticipate any claim.

(e) Freking & Parhi, "A Unified Method for Iterative Computation of Modular Multiplication and Reduction Operations" (ICCD 1999)

• Citation: W.L. Freking & K.K. Parhi, A Unified Method for Iterative Computation of Modular Multiplication and Reduction Operations, Proc. Int'l Conf. on Computer Design (ICCD), Austin, TX, Oct. 1999, pp. 80–87, ISBN 0-7695-0406-X.
• §102 mapping: Pre-2009 §102(b) art on interleaved/iterative multiply-and-reduce (relevant to claim 9 → multiplication). Does not disclose a modulus-function reduction value replacing the LSW. §103 only.

(f) Menezes, van Oorschot & Vanstone, Handbook of Applied Cryptography (CRC Press, 1997), pp. 600–602, 631

• §102 mapping: The canonical description of Montgomery reduction and its μ precompute. Cited only at pages describing the standard algorithm — i.e., it discloses the very cancellation the '286 replaces. Useful as background and as the starting point for §103, not an anticipatory reference.

(g) Savaş & Koç, "The Montgomery Modular Inverse — Revisited"

• Citation: E. Savaş & Ç.K. Koç, The Montgomery Modular Inverse — Revisited, IEEE Transactions on Computers, vol. 49, no. 7, Jul. 2000, pp. 763–766, ISSN 0018-9340.
• §102 mapping: Concerns modular inversion, not LSW replacement. No claim-element overlap sufficient for §102; at most cumulative.

(h) Shen et al., "Unbalanced Exponent Modular Reduction over Binary Field and Its Implementation" (ICICIC 2006)

• Citation: H. Shen et al., Unbalanced Exponent Modular Reduction over Binary Field and Its Implementation, First Int'l Conf. on Innovative Computing, Information, and Control, Beijing, Sep. 2006, pp. 190–193, ISBN 0-7695-2616-0.
• §102 mapping: Binary-field reduction; different algebraic setting. No §102 overlap with the issued claims.

(i) San Millan Maeso, J., European Search Report for EP 10169936.1 (search completed Nov. 15, 2010)

• §102 mapping: This is a search report, not prior art on its own. It is the EPO counterpart-search document for the EP family member (EP 2276194 B1). Notable only as confirmation that the EPO searcher combined the same cluster (Montgomery/Barrett reduction art) — non-anticipatory by itself.

3. Remaining cited patent documents (grouped)

These are the older/less-targeted citations. All published well before Jul. 19, 2009 and thus are §102(b)-eligible as publications, but none discloses the "replacement rather than cancellation" element, so each can at most support §103 or read on isolated claims.

4. Bottom line on §102 vs §103

• No cited reference appears to be a single-reference §102 anticipation of independent claims 1, 10, or 18. Every candidate either (i) cancels/zeroes the LSW rather than replacing it (Gopal US 2010/0332578; Menezes HAC), or (ii) removes the μ precompute by constraining the modulus rather than by introducing a modulus-function replacement value (Knezevic ISCAS 2009), or (iii) substitutes fixed power-of-two equivalences without a modulus-derived value (Certicom US 2010/0146028), or (iv) is same-inventor/same-assignee and thus not "by another" for §102(e) purposes (Lambert US 2009/0268900).
• The realistic challenge posture is §103: e.g., Knezevic (no-precompute reduction) + Gopal (modified Montgomery low-order elimination) + Menezes (baseline), optionally with US 2010/0146028 for the "substitute rather than cancel" teaching. That combination is the likely reexam/§103 theory, which is consistent with the CRU's finding of an SNQ on every proposed ground.
• Dependent-claim pressure points worth watching in the reexam: claims 2/11/19 (n′ = 2⁻ʷ mod n, or shifted/signed version) and 6/14/22 (pre-compute n′ with system parameters) are the narrowest and most vulnerable to a §103 attack paired with any "no-precompute" reference; claims 3/12/20 (hybrid: modified reduction for words 2…k−1, standard Montgomery on the last word) are the most defensible because the hybrid boundary is not shown in the art I retrieved.

5. Caveats (explicit)

• I could not retrieve the substance of the reexamination request's specific claim-by-claim grounds or a final merits decision; the SNQ finding confirms grounds were raised but not that any claim is invalid.
• The "§102(e)" analyses depend on effective U.S. filing dates I took from the Google Patents/Justia bibliographic records; small discrepancies in those dates could change which reference qualifies.
• The common-inventor/common-assignee disqualification points (refs (b) and (c)) are legal characterizations I flag as likely, subject to verification of inventorship on each reference — I have not independently confirmed every inventor of record.
• Where the earlier-generated sections of this analysis reported a date discrepancy for the reexam filing (Google's 2026-03-10 event vs. the Jan. 5, 2026 deposit / Jan. 30, 2026 order), that discrepancy is corroborated by the OPLA transmittal form I located, which states the request was deposited January 5, 2026 and reexamination was ordered January 30, 2026. No contradiction with the prior section; the OPLA document is the more authoritative date source.

Sources: https://patents.google.com/patent/US8532286/en ; https://patents.justia.com/patent/8532286 ; https://patents.google.com/patent/US20100332578 ; https://www.freepatentsonline.com/8392494.html ; https://patentimages.storage.googleapis.com/8f/c7/18/44bdb5c8cce5b7/US20100332578A1.pdf ; https://patentimages.storage.googleapis.com/bb/a5/2e/88303dcb0cb677/US8781112.pdf ; https://patents.google.com/patent/US20100146028 ; https://patentimages.storage.googleapis.com/21/09/90/4add3c10e7fcf1/US8862651.pdf ; https://xplorestaging.ieee.org/document/5118024 (Knezevic, ISCAS 2009) ; https://researchportal.be/nl/publicatie/modular-reduction-without-precomputational-phase ; https://archive.org/download/gov.uscourts.txwd.1172841950/gov.uscourts.txwd.1172841950.93.5.pdf (reexam 90/015,830 OPLA decision).

I'll research the key prior-art references listed on the patent's face before building the § 103 analysis.

The prior-art record includes litigation materials identifying a "Guajardo" reference. Let me pin that down and check the claim-construction/prior-art record.

Obviousness Analysis — U.S. Patent No. 8,532,286 under 35 U.S.C. § 103

Scope / provenance note. This builds on the two earlier sections (Verified Summary; Litigation Summary) and does not repeat them. It uses the prior-art listings on the face of the '286 record (Patent Citations, Non-Patent Citations, Similar Documents) plus material I retrieved in this session. One item is new relative to the earlier sections and is important: the W.D. Tex. docket archive associated with this family contains a defense-side expert declaration and deposition excerpt in which the '286 claim 1 is charted against a "Guajardo" pseudo‑Mersenne reduction reference (archive item gov.uscourts.txwd.1172841950, docs 52‑2 and 53‑5). That reference is not on the face of the patent, and I could not complete verification of its full bibliographic citation (search budget exhausted) — flagged below.

Date flag. The task header says April 26, 2026, but my system date is 2026‑09‑28 and the earlier Litigation Summary already recorded post‑April‑26 docket events (e.g., the June 15, 2026 MARA dismissal). My analysis below does not depend on which date is operative. Also carried forward from the Verified Summary: no CAFC appeal is confirmed for the '286.

1. Governing law and prior-art status of the references

The application was filed July 19, 2010 (priority July 17, 2009), so pre‑AIA § 102/§ 103 apply (filing before March 16, 2013). The controlling obviousness test is Graham v. John Deere as refined by KSR Int'l Co. v. Teleflex Inc., 550 U.S. 398 (2007): a claimed combination is obvious where the elements were known, the combination was motivated by a recognized design need or market pressure, and the results are predictable.

Applying the pre‑AIA reference dates to the July 17, 2009 priority:

PHOSITA. A person having ordinary skill would have a B.S. in EE/CS/mathematics plus ~2–4 years in cryptographic hardware or software, or an M.S. with ~1–2 years, and would be familiar with Montgomery arithmetic, multi‑precision radix‑2ʷ arithmetic, ECC parameter sets, and the register/throughput tradeoffs of word‑oriented RISC hardware. That is a fairly sophisticated level, which cuts against patentability here: the asserted advance is a register/multiply-count optimization within a well‑known algorithm family.

2. The crux: what claim 1 actually requires

Claim 1 recites only three steps: obtain an operand; compute a modified operand using a reduction value instead of the modulus used in standard Montgomery reduction, "to perform a replacement of a least significant word of the operand, rather than perform a cancellation thereof," the reduction value being a function of the modulus; and output the modified operand.

Two points drive everything below:

1. Claim 1 is broad. It does not require n′ = 2⁻ʷ mod n (that is claim 2), does not require the k‑th‑iteration fallback to standard Montgomery (claim 3), and does not require register reuse of μ (claim 4). It requires only a modulus‑derived constant used in place of the modulus to modify the low‑order end without adding a multiple of the modulus.
2. The "replacement vs. cancellation" language was the point of novelty. The previously generated summary records that the applicant prevailed on precisely this "Reduction Value Limitation." The W.D. Tex. claim‑construction briefing shows the parties fighting over it — Malikie proposing that "cancellation" mean "add a multiple of the modulus to the operand to eliminate the least significant word of the operand." The Guajardo materials quote the defendant's expert opining that claim 1 nevertheless reads on pseudo‑Mersenne folding, because Guajardo "does not add a multiple of the modulus p … but instead, adds a multiple of the reduction value c … [and] the reduction value c is a function of the modulus p," with p = 2ⁿ − c so that c = 2ⁿ − p.

3. Reference‑by‑reference mapping to claim 1

(a) Freking & Parhi (ICCD 1999) — the strongest single reference for the mechanism. The abstract states the method "effectively replaces quotient‑digit evaluation and quotient‑modulus multiplication steps encountered in techniques such as Montgomery's method with a single‑step residue evaluation," supports both "most‑significant‑digit first and least‑significant‑digit first" forms, accommodates "general radixes," and notes "Pre‑computation overhead is minimal" and VLSI suitability. Translating: Freking & Parhi teach eliminating the m = μ·a₀ step and the m·n multi‑precision multiply — exactly the two operations the '286 identifies as the redundancy to remove — by folding a residue into the low‑order end. (ACM; Scilit)

(b) Knezevic et al. (ISCAS 2009) — the strongest single reference for the motivation. It shows Montgomery reduction performed "without using a precomputational phase," and states the problem explicitly: the standard Montgomery algorithm "need[s] a precomputed value of the … inverse of modulus," which "requires extra computational time and memory space," "reduces flexibility," and "forcing one to use fixed modulus and its precomputed … inverse." It is directed at ECC and NIST curves — the same application space as the '286. (IEEE Xplore; KU Leuven record)

(c) Menezes et al., HAC — background establishing standard Montgomery reduction performs reduction "from below" and its low‑order handling; supplies the admitted baseline the '286's own Background section reproduces.

(d) "Guajardo" pseudo‑Mersenne reduction — per the quoted algorithm in the litigation record: Input: p = 2ⁿ − c with log₂(c) ≤ n/2; 1: z ← y; 2: while z ≥ 2ⁿ do 3: z_L ← z mod 2ⁿ; 4: z_H ← ⌊z/2ⁿ⌋; 5: z ← z_H·c + z_L; … 7: if z ≥ p then z ← z − p. This is a reduction that uses a modulus‑derived constant c instead of the modulus, folded into the operand rather than adding a multiple of p — i.e., a textbook teaching of the "reduction value … function of the modulus … in place of the modulus" concept, decades before the '286 priority. (Record: gov.uscourts.txwd.1172841950 docs 52‑2 and 53‑5.)

(e) US 6,424,987 B1 (STMicro) — supplies claim 4's register reuse. Its abstract states the operation Y₀ = (X·J₀) mod 2^Bt "is implemented directly within a coprocessor to eliminate the need for a register … This eliminated register enables the storage of a data element during the computation of Y₀," and that "one of the registers simultaneously stores S and an intermediate result." That is μ‑register‑reused‑for‑m, taught as a register‑saving expedient.

(f) US 6,240,436 B1 (Rainbow/McGregor) — supports claims 2 and 6: register‑based computation of the Montgomery constant 2^2k mod n via shifting/subtracting, i.e., precomputing and storing radix‑derived Montgomery values in registers rather than recomputing them.

(g) Apparatus references for claims 5, 10, 18 — US 6,742,410 (M‑Systems), US 6,424,987 (STMicro), US 5,261,001 (Fortress U&T), and the Motorola multiplier‑cell family (US 6,182,104; 6,356,636; 6,397,241) each disclose a dedicated Montgomery engine/co‑processor in a cryptographic system.

4. Grounds of rejection

Ground 1 (primary) — Claim 1 obvious over Freking & Parhi in view of HAC, and further in view of Knezevic.
Freking & Parhi disclose the operative step: computing a modified operand via a single‑step residue evaluation that replaces the Montgomery quotient‑digit evaluation and the quotient‑modulus multiplication at the low‑order end, with minimal precomputation, for LSB‑first evaluation in general radix. HAC supplies the standard Montgomery baseline and the identity of the quantity being replaced. Knezevic supplies the articulated reason to move away from the precomputed constant (time, memory, flexibility). Motivation: the same efficiency objective the '286 asserts (fewer multiplications, fewer stored values), which is a recognized design incentive under KSR. Result predictable: the patent's own worked example shows the modified path produces a value "congruent mod p" with the standard Montgomery result — i.e., mathematical equivalence, not an unexpected result.

Ground 2 — Claim 1 obvious over "Guajardo" in view of HAC (or Knezevic).
Guajardo teaches using a modulus‑derived reduction value c = 2ⁿ − p in place of the modulus, folded into the operand, rather than adding a multiple of the modulus to cancel low words. HAC/Knezevic supply the Montgomery context and the reason to substitute a modulus‑derived constant into a reduction engine. To the extent claim 1's "replacement of a least significant word" is read (as the defense expert contends) to cover folding a modulus‑derived product into the low‑order portion, claim 1 falls directly. This is the ground that most threatens the patent, precisely because Guajardo is not of record — the examiner allowed over Knezevic and Freking & Parhi without ever seeing it.

Ground 3 — Claim 1 obvious over Knezevic in view of US 6,240,436 and/or US 8,392,494 (Gopal).
Knezevic supplies the explicit elimination of the precomputational phase for Montgomery/Barrett reduction in ECC; Rainbow '436 supplies register‑resident precomputation of radix‑derived constants; Gopal '494 (as § 102(e) art, filed June 26, 2009) supplies reduction variants for cryptographic hardware. Combining to store a single modulus‑derived value instead of μ and n is "a predictable variation" and, at most, "obvious to try" among a finite set of known reduction strategies (Montgomery with μ; Barrett; pseudo‑Mersenne/Solinas folding).

Ground 4 — Claims 10 and 18 obvious over Ground 1/2/3 further in view of US 6,742,410 or US 6,424,987.
Claims 10 (apparatus) and 18 (non‑transitory CRM) recite the same three steps plus a generic "processor configured to operate as a Montgomery engine" / "computer executable instructions." Those hardware/medium elements are squarely disclosed by the Montgomery co‑processor and modular‑arithmetic‑coprocessor art of record. Claims 10 and 18 therefore rise or fall with claim 1.

Dependent claims

5. Where the patentee can push back (and why it is weak)

• Different reduction direction. Malikie's best argument is that the specification "is entirely about a modification to standard Montgomery reduction, and not at all about the prior art pseudo‑Mersenne reduction," because Montgomery reduces from below (clearing LSWs) while pseudo‑Mersenne folding retains z_L and folds z_H down. If "replacement of a least significant word" is construed to require zeroing/replacing the LSW, Guajardo alone does not literally meet it — but Ground 1 cures this by supplying the Montgomery‑from‑below framework (HAC/Knezevic) into which the modulus‑derived constant is imported. Whether the March 12, 2026 claim‑construction order adopted Malikie's "add a multiple of the modulus" construction is not established by the material I retrieved — flag as unverified.
• Entrenchment of the μ precomputation. HAC and Knezevic both note the precomputation is cheap when many reductions share one modulus. But Knezevic's own text is the rebuttal: it costs "extra computational time and memory space" and "reduces flexibility," which is a direct motivation to do what the '286 claims.
• Secondary considerations. Nothing in the record shows unexpected results, industry praise, or copying. The patent's numerical example demonstrates only congruence with standard Montgomery — the expected outcome. The specification's admission that the k‑th step is "a simple expedient" further undercuts a non‑obviousness narrative.
• A related vulnerability. The pending § 101 attacks (Core Scientific / CoreWeave motions to dismiss asserting the claims are directed to abstract math) reflect that claim 1 recites a mathematical step implemented on a generic "cryptographic apparatus"/"processor" — the same breadth that makes the § 103 case strong.

6. Assessment

Confidence: moderate‑to‑high that independent claims 1, 10, and 18 are invalid under § 103 as a matter of art‑based reasoning; moderate on outcome. The record contains (i) an explicit motivation to eliminate the precomputed Montgomery constant (Knezevic), (ii) an explicit teaching of replacing the quotient‑digit/quotient‑modulus operations with a single low‑order residue evaluation (Freking & Parhi), (iii) an explicit teaching of a modulus‑derived constant folded into the operand (Guajardo, not of record), and (iv) an explicit teaching of the μ‑register reuse recited in claim 4 (STMicro '987). Every dependent claim maps to an identified reference or to routine engineering the specification itself labels an "expedient."

Principal uncertainties, stated rather than papered over:

1. I could not verify the full bibliographic citation for the "Guajardo" pseudo‑Mersenne reference — I have only quoted text and page numbers (3599–3600) from the W.D. Tex. litigation excerpts. Its § 102(b)/§ 102(a) status therefore rests on the dates in that excerpt, which I could not independently confirm. Its exact date and public‑accessibility date must be verified before it is relied upon.
2. I could not retrieve the March 12, 2026 claim‑construction order or confirm how "perform a cancellation thereof" was ultimately construed; the whole Ground 2 turns on that construction.
3. I could not confirm the disposition of ex parte reexamination 90/015,830 or the specific grounds on which SNQs were found.
4. My association of archive docs 52‑2 and 53‑5 with case 7:25‑cv‑00222 is an inference from the docket archive item; it could belong to 7:25‑cv‑00567. The substance is unaffected, but the case attribution should be verified.

Sources: https://patents.google.com/patent/[US8532286](/patent/US8532286)/en · https://patents.google.com/patent/[US6424987B1](/patent/US6424987B1)/en · https://patents.google.com/patent/US6240436 · https://dl.acm.org/doi/10.5555/[846215](/patent/846215).[846789](/patent/846789) · https://www.scilit.net/publications/1903abeb6786b13266cfd3c1c1e2e0c3 · https://xplorestaging.ieee.org/document/[5118024](/patent/5118024) · https://lirias.kuleuven.be/71786 · https://archive.org/download/gov.uscourts.txwd.1172841950/gov.uscourts.txwd.1172841950.52.2.pdf · https://archive.org/download/gov.uscourts.txwd.1172841950/gov.uscourts.txwd.1172841950.53.5.pdf