Abstract: Disclosed are methods and systems to encrypt data with SomeWhat Homomorphic Encryption (SWHE) properties for submission to a distributed ledger/blockchain that allows further open operations retained in the distributed ledger/blockchain on the encrypted data that will be properly reflected when the encrypted result is decrypted by the data owner. The somewhat homomorphic properties include addition and scalar division. Also disclosed is an ability to update a secret key applied for a ciphertext such that a single piece of data may be provided on the distributed ledger/blockchain by a data owner to a new data owner without also exposing other data encrypted with the original secret key of the original data owner.

Abstract: Disclosed are methods and systems to provide distributed computation within a Fully Homomorphic Encryption (FHE) system by using g-adic properties to separate a ciphertext into multiple ciphertexts for each Hensel digit level. A number t of computation units may individually perform addition and/or multiplication of each Hensel digit level on each of the computation units and then reconstruct the resulting value from the result ciphertext of each computation unit using p-adic and g-adic operations. Accordingly, computation burdens may be distributed to several computation units.

Abstract: Disclosed are methods and systems to use p-adic numbers to permit a RSA cryptosystem to send rational numbers or to add randomness to the RSA cryptosystem. An embodiment may convert at the source device a rational number to an integer as p-adic based Hensel code representation of the rational number at the source device and then recover the rational number at the destination device by reversing the Hensel code back to the original rational number. Another embodiment may use a g-adic inverse of a message value together with a random number to obtain a different rational number to encrypt for each different random number resulting in different ciphertexts representing the same message value while still recovering the original message value despite having a different ciphertexts for the same message value. The various embodiments further retain the multiplicative homomorphism of the RSA cryptosystem since the p-adic Hensel codes are also multiplicative homomorphic.

Abstract: Disclosed are methods and systems to provide public and private-key leveled Fully Homomorphic Encryption (FHE) systems using Hensel Codes and p-adic and g-adic properties for encryption and decryption that also provide for homomorphic arithmetic operations on encrypted ciphertexts. A source device may encrypt the ciphertext of a message using Hensel Codes, then deliver the ciphertext to either a destination device or an intermediary device. When the intermediary device receives the ciphertext from the source device, the intermediary device may homomorphically perform Hensel Code arithmetic computations with the ciphertext and at least one additional ciphertext and send the result ciphertext to the destination device. The destination device decrypts the ciphertext, giving the original message when no computations have been performed by the intermediary device, or the unencrypted result equivalent to the unencrypted computations performed on the ciphertexts by the intermediary device.

Abstract: A device stack for an electronic memory or other device includes a substrate and first and second layers of insulating material. The first layer of insulating material is supported by the substrate. A semiconducting ferroelectric layer is positioned and electrically isolated between the first and second layers of insulating material. A stress layer capable of converting a ferroelectric or semiconductor material into a semiconducting ferroelectric material can be positioned in contact with the semiconducting ferroelectric layer. In some embodiments, the device is a Metal-Insulator-FeS-Insulator-Semiconductor (MIFIS) device that allows for controlled switching of the semiconducting ferroelectric (FeS) layer between various polarization states. Switching polarization states is enabled by application of an electric field by a semiconducting electrode.

Abstract: A device stack for an electronic memory or other device includes a substrate and first and second layers of insulating material. The first layer of insulating material is supported by the substrate. A semiconducting ferroelectric layer is positioned and electrically isolated between the first and second layers of insulating material. An electrode is positioned onto or above the second layer of insulating material. In some embodiments, the device is a Metal-Insulator-FeS-Insulator-Semiconductor (MIFIS) device that allows for controlled switching of the semiconducting ferroelectric (FeS) layer between various polarization states. Switching polarization states is enabled by application of an electric field by the electrode.

Abstract: Non-volatile memory devices utilizing polarizable ferroelectric-semiconductor materials as the floating gate in floating-gate field-effect metal oxide transistors are described. Such materials can be annealed at temperatures less than 450° C., and fields below about 250 kV/cm can be used for changing polarization of the ferroelectric semiconductor materials, leading to devices capable of high endurance (>1010 cycles).

Abstract: Disclosed are methods and systems to encrypt/decrypt a data message using Geometric Algebra and Hensel encoding (i.e., finite p-adic arithmetic). The security key(s), message data, and ciphertext are all represented as Geometric Algebra multivectors where a sum of the coefficients of an individual multivector is equal to the numeric value of the corresponding message or security key. Various Geometric Algebra operations with the message and security key multivectors act to encrypt/decrypt the message data. Each coefficient of the security key and message multivectors is further Hensel encoded to provide additional confusion/diffusion for the encrypted values. The Geometric Algebra operations permit homomorphic operations for adding, subtracting, multiplication and division of ciphertext multivectors such that the resulting ciphertext, when decrypted, is equal to corresponding mathematical operations using the unencrypted values.

Abstract: Disclosed are methods and systems to conceal (encrypt) & recover (decrypt) a data message using Geometric Algebra using Modular Concealment (MC) between a first computing device and a second computing device over a network communication connection. The security key(s), message data, and ciphertext are all represented as Geometric Algebra multivectors. The MC concealment provides for both additive and multiplicative homomorphism. Further data representations are presented for multivector packing schemes including Clifford Eigenvalue Packing (CEP) and Complex Magnitude Squared Packing (CMSP). The CEP and CMSP data representations also provide support for additive and multiplicative homomorphism. To assist in security key exchange, a key exchange protocol is also presented for the creation and transfer of security key multivectors.

Abstract: Disclosed are methods and systems to provide distributed computation within a Fully Homomorphic Encryption (FHE) system by using g-adic properties to separate a ciphertext into multiple ciphertexts for each Hensel digit level. A number t of computation units may individually perform addition and/or multiplication of each Hensel digit level on each of the computation units and then reconstruct the resulting value from the result ciphertext of each computation unit using p-adic and g-adic operations. Accordingly, computation burdens may be distributed to several computation units.

Abstract: Disclosed are methods and systems to encrypt data with SomeWhat Homomorphic Encryption (SWHE) properties for submission to a distributed ledger/blockchain that allows further open operations retained in the distributed ledger/blockchain on the encrypted data that will be properly reflected when the encrypted result is decrypted by the data owner. The somewhat homomorphic properties include addition and scalar division. Also disclosed is an ability to update a secret key applied for a ciphertext such that a single piece of data may be provided on the distributed ledger/blockchain by a data owner to a new data owner without also exposing other data encrypted with the original secret key of the original data owner.

Abstract: Disclosed are methods and systems to encrypt/decrypt a data message using Geometric Algebra and Hensel encoding (i.e., finite p-adic arithmetic). The security key(s), message data, and ciphertext are all represented as Geometric Algebra multivectors where a sum of the coefficients of an individual multivector is equal to the numeric value of the corresponding message or security key. Various Geometric Algebra operations with the message and security key multivectors act to encrypt/decrypt the message data. Each coefficient of the security key and message multivectors is further Hensel encoded to provide additional confusion/diffusion for the encrypted values. The Geometric Algebra operations permit homomorphic operations for adding, subtracting, multiplication and division of ciphertext multivectors such that the resulting ciphertext, when decrypted, is equal to corresponding mathematical operations using the unencrypted values.

Abstract: Disclosed are methods and systems to encrypt and decrypt a data message using Geometric Algebra. The encrypt operation performed on a source computing device uses the geometric product (Clifford Product) of a multivector created from plain text/data of the data message with one or more other multivectors that carry encryption keys, the identity of the source and/or other data-centric information. The source computing device sends the encrypted message to a destination computing device. The decrypt operation performed on the destination computing devices recovers the original message multivector, and ultimately the original data message by employing geometric algebra operations such as multivector inverse, Clifford conjugate and others along with the geometric product.

Abstract: Disclosed are methods and systems for encrypting numeric messages using Geometric Algebra on at least one source device and then storing and sorting the encrypted messages on an intermediary system without decrypting the encrypted numeric messages on the intermediary system and/or on a sort request device requesting the sort. Both the intermediary and sort request devices/systems do not need to have knowledge of the encryption security keys. A sort result (sorted group of cryptotext multivectors) may be sent to a destination device. The sorted encrypted data may be decrypted and kept in sorted order on the destination device. Encrypt operations use the geometric product (Clifford Product) of multivectors created from plain text/data with one or more other multivectors that carry encryption keys. Decrypt operations decrypt encrypted data by employing geometric algebra operations such as multivector inverse, Clifford conjugate and others along with the geometric product.

Abstract: Disclosed are methods and systems for encrypting numeric messages using Geometric Algebra on at least one source device and then storing and searching the encrypted messages on an intermediary system without decrypting the encrypted numeric messages on the intermediary system and/or on a search request device requesting the search. Both the intermediary and search request devices/systems do not need to have knowledge of the encryption security keys. A search result (FOUND/NOT-FOUND) may be sent to a destination device. The FOUND encrypted data, as well as other encrypted data linked to the FOUND encrypted data, may be sent to a destination device and decrypted. Encrypt operations use the geometric product (Clifford Product) of multivectors created from plain text/data with one or more other multivectors that carry encryption keys. Decrypt operations decrypt encrypted data by employing geometric algebra operations such as multivector inverse, Clifford conjugate and others along with the geometric product.

Abstract: Disclosed are methods and systems for encrypting a numeric message using Geometric Algebra on a source computing device, performing scalar-vector multiplication on the encrypted numeric message and an unencrypted scalar data value to get an encrypted scalar multiplicative result without decrypting the encrypted numeric message on an intermediary computing system that does not have knowledge of the encryption security keys, and decrypting using Geometric Algebra the encrypted scalar multiplicative result on a destination computing device such that the decrypted result is equal to multiplication of the unencrypted numeric message and the scalar data value. Encrypt operations use the geometric product (Clifford Product) of multivectors created from plain text/data of the numeric data message with one or more other multivectors that carry encryption keys.

Abstract: Disclosed are methods and systems to encrypt and decrypt a data message using Geometric Algebra. The encrypt operation performed on a source computing device uses the geometric product (Clifford Product) of a multivector created from plain text/data of the data message with one or more other multivectors that carry encryption keys, the identity of the source and/or other data-centric information. The source computing device sends the encrypted message to a destination computing device. The decrypt operation performed on the destination computing devices recovers the original message multivector, and ultimately the original data message by employing geometric algebra operations such as multivector inverse, Clifford conjugate and others along with the geometric product.

Abstract: A resistive switching memory comprising a first electrode and a second electrode; an active resistive switching region between the first electrode and the second electrode, the resistive switching region comprising a transition metal oxide and a dopant comprising a ligand, the dopant having a first concentration; a first buffer region between the first electrode and the resistive switching material, the first buffer region comprising the transition metal oxide and the dopant, wherein the dopant has a second concentration that is greater than the first concentration. In one embodiment, the second concentration is twice the first concentration. In one embodiment, the first buffer region is thicker than the active resistive switching region.

Abstract: A resistive switching memory comprising a first electrode and a second electrode; an active resistive switching region between the first electrode and the second electrode, the resistive switching region comprising a transition metal oxide and a dopant comprising a ligand, the dopant having a first concentration; a first buffer region between the first electrode and the resistive switching material, the first buffer region comprising the transition metal oxide and the dopant, wherein the dopant has a second concentration that is greater than the first concentration. In one embodiment, the second concentration is twice the first concentration. In one embodiment, the first buffer region is thicker than the active resistive switching region.

Abstract: A re-programmable antifuse field programmable gate array (FPGA) integrated circuit, the FPGA comprising: a plurality of CeRAM resistive switching elements forming a connection block, the switching elements capable of being switched from a conductive (ON) state to a non-conductive (OFF) state and back to a conductive (ON) state; a plurality of logic elements forming a logic block; and a programming circuit for turning the CeRAM switching elements OFF and ON to connect the logic elements to form the FPGA.