Abstract: Systems and methods for digitally certified stationery are described. In one aspect, a stationery granting authority (SGA) receives a request from a user to generate a document. If the user is authorized for the requested document, the SGA generates a certificate with credentialing information from data in the request. The SGA generates a first digital signature from some of the credentialing information. The SGA communicates the certificate to the user for editing and distribution as the document. A recipient of the document determines whether the document is “official” by contacting a specified service to provide certain information from the document. The verification service computes a second digital signature from the provided information for comparison to the first digital signature. If there is a match, the service notifies the recipient that the document is valid/official. Otherwise, the recipient is notified that the document is not valid.

Abstract: Pseudorandom numbers may be generated from input seeds using expander graphs. Expander graphs are a collection of vertices that are interconnected via edges. Generally, a walk around an expander graph is determined responsive to an input seed, and a pseudorandom number is produced based on vertex names. Specifically, a next edge, which is one of multiple edges emanating from a current vertex, is selected responsive to an extracted seed chunk. The next edge is traversed to reach a next vertex. The name of the next vertex is ascertained and used as a portion of the pseudorandom number being produced by the walk around the expander graph.

Abstract: Arrangements are provided for efficient erasure coding of files to be distributed and later retrieved from a peer-to-peer network, where such files are broken up into many fragments and stored at peer systems. The arrangements further provide a routine to determine the probability that the file can be reconstructed. The arrangements further provide a method of performing the erasure coding in an optimized fashion, allowing fewer occurrences of disk seeks.

Abstract: Digital signatures for network coding are described. In one aspect, digital signatures for network coding are described. In one aspect, segmented blocks of content for distribution are digitally signed using homomorphic digital signatures generated from an elliptic curve. A linear combination of packets comprising the digitally signed content is distributed to a destination device according to an implemented distribution scheme. The linear combination of packets includes public information when digitally signing the segmented blocks. The homomorphic digital signatures and the public information allow a device receiving one or more packets of the linear combination of packets to verify and authenticate content associated with the one of our packets independent of secure transmission of secret keys and hash digests used to digitally sign the one or more packets.

Abstract: Systems and methods for elliptic curve octupling for weighted projective coordinates are described. In one aspect, a weighted projective point P on an elliptic curve is identified. 8P is computed from P independent of repeated doubling operations using fewer field multiplications.

Abstract: A method and system is provided for monitoring the availability of a peer in a P2P system that is used to provide remote storage or remote processing power. In one illustrative example, a recipient peer requests access to a service provisioned by another peer in a peer-to-peer network. The request may be a request to access a file or a file fragment that is being stored on the other peer. In order to make use of the accessed service, after receiving access to the service provisioned by the peer, the recipient peer needs to report to a central server that the service has been rendered. For instance, in some cases the file fragment accessed by the recipient peer may be encrypted, in which case the central server will send the recipient peer a decryption key after receiving the report that the service has been rendered.

Abstract: Techniques are disclosed for representing and evaluating large prime degree isogenies for use in cryptographic signature and encryption schemes. An isogeny of prime degree 1 may be represented as an ideal in the form (1, A*alpha+B), where 1 comprises the degree of a prime number, the prime number is split into integers a and b, and alpha is a known endomorphism. For a given degree 1, integers a and b define a unique isogeny, allowing the isogeny to be stored with 3 log(1) bits of information. Techniques are also disclosed to evaluate the isogeny at a given point by decomposing the isogeny into an integer and a plurality of smaller degree isogenies, evaluating the smaller degree isogenies at the point with traditional means, and multiplying the results of the evaluations together and with the integer.

Abstract: Function properties may be approximated using an expander graph. For example, an approximate average of a function may be determined by randomly exploring an expander graph. Values of the function are associated with vertices of the expander graph. The expander graph is randomly explored by traversing edges and encountering vertices. The exploration may comprise a crawl, a walk, and so forth. An approximate average of the function is determined based on the function values that are associated with encountered vertices.

Abstract: Systems and methods for elliptic curve octupling using Single Instruction Multiple Data (SIMD) processing are described. In one aspect, a weighted projective point P on an elliptic curve, P having coordinates (x, y, z) is identified. Value 8P is computed from P with 12 sets of field multiplications using SIMD processing. Each set of field multiplications includes one to four respective field multiplications. Each set of field multiplications is performed in parallel according to an assigned time-step.

Abstract: An exemplary method includes receiving a request to register a peer in a peer-to-peer system; generating or selecting a transaction key for the peer; storing the transaction key in association with registration information for the peer; transmitting the transaction key to the peer and, in response to a request to perform a desired peer-to-peer transaction by another peer, generating a token, based at least in part on the transaction key. Such a token allows for secure transactions in a peer-to-peer system including remote storage of data and retrieval of remotely stored data. Other exemplary techniques are also disclosed including exemplary modules for a peer-to-peer server and peers in a peer-to-peer system.

Abstract: An exemplary method for verifying the integrity of remotely stored data includes providing a key; providing a fingerprint, the fingerprint generated using the key in a keyed cryptographic hash function as applied to data of known integrity; sending the key to a remote storage location that stores a copy of the data of known integrity; receiving a fingerprint from the remote storage location, the fingerprint generated using the key in a keyed cryptographic hash function as applied to the remotely stored copy of the data; and verifying the integrity of the remotely stored copy of the data based at least in part on comparing the provided fingerprint to the received fingerprint. Other exemplary methods, systems, etc., are also disclosed.

Abstract: Systems and methods for computing modular polynomials modulo large primes are described. In one aspect, the systems and methods generate 1-isogenous elliptic curves. A modular polynomial modulo a large prime p is then computed as a function of 1-isogenous elliptic curves modulo p.

Abstract: Systems and methods for computing modular polynomials modulo large primes are described. In one aspect, the systems and methods generate l-isogenous elliptic curves. A modular polynomial modulo a large prime p is then computed as a function of l-isogenous elliptic curves modulo p.

Abstract: Systems and methods for cryptographically processing data as a function of a Cartier pairing are described. In one aspect, a Cartier pairing is generated from two different abelian varieties or abelian varieties and an isogeny between them. Data is cryptographically processed based on the Cartier pairing.

Abstract: Efficient data storage and retrieval (e.g., in terms of time and space requirements) is facilitated by implementing an indexing structure comprising an indexing array. That is, a functional relationship between elements of a source set and elements of a query result set can be stored in the indexing structure. This allows, for example, a query regarding whether an element is a member of a set (e.g., whether a particular website or Uniform Resource Locator (URL)) has been visited before) as well as a relationship between the member set and the query (e.g., the number of hyperlinks in the website the last time it was visited) to be resolved efficiently.

Abstract: Systems and methods to detect unauthorized changes to a printed document are described. In one aspect, a digital signature of original content associated with electronic document is embedded into the original content to create a content signed document. The systems and methods use the embedded digital signature to automatically determine whether text-based content associated with a printout of the content signed document was changed from the original content associated with the electronic document.

Abstract: Systems and methods for security against corruption for networked storage are described. In one aspect, a destination device receives a linear combination of packets. The linear combination of packets represents digitally signed blocks of content and public information used to digitally sign segmented blocks of content. The destination device recombines the linear combination of packets to compute new valid signatures as linear combinations of received homomorphic digital signatures. The new valid signatures are for verifying by a requesting computing device recovering data associated with at least a subset of the linear combination of packets, integrity of the at least a subset. This provides the requesting node with security against corruption of data in networked storage.

Abstract: Systems and methods for secure signatures are described. In one aspect, a secure signature is generated. The secure signature strongly binds an image of an electronic signature (an “electronic signature”) to content in either electronic or printed form. Responsive to receiving a request from a user, the systems and methods determine whether an electronic signature associated with a printed page represents a secure signature. If so, the systems and methods determine and notify the user of whether the secure signature was cryptographically bound by a signer of the electronic signature to the content being signed.

Abstract: Systems and methods for digitally certified stationery are described. In one aspect, a stationery granting authority (SGA) receives a request from a user to generate a document. If the user is authorized for the requested document, the SGA generates a certificate with credentialing information from data in the request. The SGA generates a first digital signature from some of the credentialing information. The SGA communicates the certificate to the user for editing and distribution as the document. A recipient of the document determines whether the document is “official” by contacting a specified service to provide certain information from the document. The verification service computes a second digital signature from the provided information for comparison to the first digital signature. If there is a match, the service notifies the recipient that the document is valid/official. Otherwise, the recipient is notified that the document is not valid.

Abstract: Hash function constructions from expander graphs are described. In one aspect, an expander graph is walked to compute a hash function. The expander graph is walked using respective subsets of an input message. A label of a last vertex walked is an output of the hash function.