Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Quantum computers will break today's most popular public-key cryptographic systems, including RSA, DSA, and ECDSA. This book introduces the reader to the next generation of cryptographic algorithms, the systems that resist quantum-computer attacks: in particular, post-quantum public-key encryption systems and post-quantum public-key signature systems. Leading experts have joined forces for the first time to explain the state of the art in quantum computing, hash-based cryptography, code-based cryptography, lattice-based cryptography, and multivariate cryptography. Mathematical foundations and implementation issues are included. This book is an essential resource for students and researchers who want to contribute to the field of post-quantum cryptography.

Post-quantum cryptography

We propose an efficient commutative group action suitable for non-interactive key exchange in a post-quantum setting. Our construction follows the layout of the Couveignes–Rostovtsev–Stolbunov cryptosystem, but we apply it to supersingular elliptic curves defined over a large prime field \(\mathbb F_p\), rather than to ordinary elliptic curves. The Diffie–Hellman scheme resulting from the group action allows for public-key validation at very little cost, runs reasonably fast in practice, and has public keys of only 64 bytes at a conjectured AES-128 security level, matching NIST’s post-quantum security category I.

/pdf/csidh-an-efficient-post-quantum-commutative-group-action-4ximtcsc7w.pdf

CSIDH: an efficient Post-Quantum Commutative Group Action

Thank you for downloading using algebraic geometry. As you may know, people have search numerous times for their chosen readings like this using algebraic geometry, but end up in malicious downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they juggled with some infectious virus inside their computer. using algebraic geometry is available in our book collection an online access to it is set as public so you can download it instantly. Our digital library hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the using algebraic geometry is universally compatible with any devices to read.

Using Algebraic Geometry

Abstract We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the parties to construct a shared commutative square despite the non-commutativity of the endomorphism ring. We give a precise formulation of the necessary computational assumptions along with a discussion of their validity, and prove the security of our protocols under these assumptions. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves. This paper is an extended version of [Lecture Notes in Comput. Sci. 7071, Springer (2011), 19–34]. We add a new zero-knowledge identification scheme and detailed security proofs for the protocols. We also present a new, asymptotically faster, algorithm for key generation, a thorough study of its optimization, and new experimental data.

/pdf/towards-quantum-resistant-cryptosystems-from-supersingular-37t03wpqh1.pdf

Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies

We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the two parties to arrive at a common shared key despite the noncommutativity of the endomorphism ring. Our work is motivated by the recent development of a subexponential-time quantum algorithm for constructing isogenies between ordinary elliptic curves. In the supersingular case, by contrast, the fastest known quantum attack remains exponential, since the noncommutativity of the endomorphism ring means that the approach used in the ordinary case does not apply. We give a precise formulation of the necessary computational assumption along with a discussion of its validity. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves.

/pdf/towards-quantum-resistant-cryptosystems-from-supersingular-3i2d4br24h.pdf

We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the parties to construct a shared commutative square despite the noncommutativity of the endomorphism ring. Our work is motivated by the recent development of a subexponential-time quantum algorithm for constructing isogenies between ordinary elliptic curves. In the supersingular case, by contrast, the fastest known quantum attack remains exponential, since the noncommutativity of the endomorphism ring means that the approach used in the ordinary case does not apply. We give a precise formulation of the necessary computational assumptions along with a discussion of their validity, and prove the security of our protocols under these assumptions. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves. This paper is an extended version of [19]. We add a new zero-knowledge identification scheme, and detailed security proofs for the protocols. We also present a new, asymptotically faster, algorithm for key generation, a thorough study of its optimization, and new experimental data.

/pdf/towards-quantum-resistant-cryptosystems-from-supersingular-2k0o9u35cc.pdf

Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies.

We introduce a new signature scheme, SQISign, (for Short Quaternion and Isogeny Signature) from isogeny graphs of supersingular elliptic curves. The signature scheme is derived from a new one-round, high soundness, interactive identification protocol. Targeting the post-quantum NIST-1 level of security, our implementation results in signatures of 204 bytes, secret keys of 16 bytes and public keys of 64 bytes. In particular, the signature and public key sizes combined are an order of magnitude smaller than all other post-quantum signature schemes. On a modern workstation, our implementation in C takes 0.6s for key generation, 2.5s for signing, and 50ms for verification.While the soundness of the identification protocol follows from classical assumptions, the zero-knowledge property relies on the second main contribution of this paper. We introduce a new algorithm to find an isogeny path connecting two given supersingular elliptic curves of known endomorphism rings. A previous algorithm to solve this problem, due to Kohel, Lauter, Petit and Tignol, systematically reveals paths from the input curves to a `special' curve. This leakage would break the zero-knowledge property of the protocol. Our algorithm does not directly reveal such a path, and subject to a new computational assumption, we prove that the resulting identification protocol is zero-knowledge.

/pdf/sqisign-compact-post-quantum-signatures-from-quaternions-and-464kpzrpqf.pdf

SQISign: Compact Post-quantum Signatures from Quaternions and Isogenies

We present two new Verifiable Delay Functions (VDF) based on assumptions from elliptic curve cryptography. We discuss both the advantages and drawbacks of our constructions, we study their security and we demonstrate their practicality with a proof-of-concept implementation.

/pdf/verifiable-delay-functions-from-supersingular-isogenies-and-tj6avh5b0t.pdf

Luca De Feo

Papers

Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies

Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies

Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies.

SQISign: Compact Post-quantum Signatures from Quaternions and Isogenies

Verifiable Delay Functions from Supersingular Isogenies and Pairings