Design of lattice-based ElGamal encryption and signature schemes using SIS problem

For the construction of post-quantum digital signature schemes that satisfy the strengthened criterion of resistance to quantum attacks, an algebraic carrier is proposed that allows one to define a hidden commutative group with two-dimensional cyclicity. Formulas are obtained that describe the set of elements that are permutable with a given fixed element. A post-quantum signature scheme based on the considered finite non-commutative associative algebra is described.

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Equidistribution Among Cosets of Elliptic Curve Points in Intervals

Journal of Mathematical Cryptology ◽

10.1515/jmc-2019-0020 ◽

2020 ◽

Vol 14 (1) ◽

pp. 339-345

Author(s):

Taechan Kim ◽

Mehdi Tibouchi

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Character Sums ◽

Fault Analysis ◽

Signature Schemes ◽

Large Subgroup

AbstractIn a recent paper devoted to fault analysis of elliptic curve-based signature schemes, Takahashi et al. (TCHES 2018) described several attacks, one of which assumed an equidistribution property that can be informally stated as follows: given an elliptic curve E over 𝔽q in Weierstrass form and a large subgroup H ⊂ E(𝔽q) generated by G(xG, yG), the points in E(𝔽q) whose x-coordinates are obtained from xG by randomly flipping a fixed, sufficiently long substring of bits (and rejecting cases when the resulting value does not correspond to a point in E(𝔽q)) are close to uniformly distributed among the cosets modulo H. The goal of this note is to formally state, prove and quantify (a variant of) that property, and in particular establish sufficient bounds on the size of the subgroup and on the length of the substring of bits for it to hold. The proof relies on bounds for character sums on elliptic curves established by Kohel and Shparlinski (ANTS–IV).

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An Application of p-Fibonacci Error-Correcting Codes to Cryptography

Mathematics ◽

10.3390/math9070789 ◽

2021 ◽

Vol 9 (7) ◽

pp. 789

Author(s):

Emanuele Bellini ◽

Chiara Marcolla ◽

Nadir Murru

Keyword(s):

Error Correcting Code ◽

Error Correcting Codes ◽

Multiparty Computation ◽

Zero Knowledge ◽

Signature Schemes ◽

Standardization Process ◽

Technology Standardization

In addition to their usefulness in proving one’s identity electronically, identification protocols based on zero-knowledge proofs allow designing secure cryptographic signature schemes by means of the Fiat–Shamir transform or other similar constructs. This approach has been followed by many cryptographers during the NIST (National Institute of Standards and Technology) standardization process for quantum-resistant signature schemes. NIST candidates include solutions in different settings, such as lattices and multivariate and multiparty computation. While error-correcting codes may also be used, they do not provide very practical parameters, with a few exceptions. In this manuscript, we explored the possibility of using the error-correcting codes proposed by Stakhov in 2006 to design an identification protocol based on zero-knowledge proofs. We showed that this type of code offers a valid alternative in the error-correcting code setting to build such protocols and, consequently, quantum-resistant signature schemes.

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