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Discrete logarithms in quasi-polynomial time in finite fields of fixed characteristic
discrete logarithm problem finite field
2019/6/26
We prove that the discrete logarithm problem can be solved in quasi-polynomial expected time in the multiplicative group of finite fields of fixed characteristic. More generally, we prove that it can ...
Efficient Private Comparison Queries over Encrypted Databases using Fully Homomorphic Encryption with Finite Fields
private queries encrypted database homomorphic encryption
2019/4/3
To achieve security and privacy for data stored on the cloud, we need the ability to secure data in compute. Equality comparisons, ``x=y,x≠yx=y,x≠y'', have been widely studied with many proposals but ...
A new perspective on the powers of two descent for discrete logarithms in finite fields
foundations discrete logarithm problem
2018/7/9
A new proof is given for the correctness of the powers of two descent method for computing discrete logarithms. The result is slightly stronger than the original work, but more importantly we provide ...
ON ISOGENY GRAPHS OF SUPERSINGULAR ELLIPTIC CURVES OVER FINITE FIELDS
foundations Supersingular isogeny graphs
2018/2/8
We study the isogeny graphs of supersingular elliptic curves over finite fields, with an emphasis on the vertices corresponding to elliptic curves of jj-invariant 0 and 1728.
Polynomial multiplication over binary finite fields: new upper bounds
Polynomial multiplication Karatsuba Two-level Seven-way Recursion algorithm
2018/1/29
When implementing a cryptographic algorithm, efficient operations have high relevance both in hardware and software. Since a number of operations can be performed via polynomial multiplication, the ar...
On Fast Multiplication in Binary Finite Fields and Optimal Primitive Polynomials over GF(2)
implementation multiplication modular reduction
2017/9/18
In this paper we present a number of algorithms and optimization techniques to speedup computations in binary extension fields over GF(2). Particularly, we consider multiplication and modular reductio...
On new multivariate cryptosystems based on hidden Eulerian equations over finite fields
Multivariate Cryptography Finite Filds Eulerian equations
2017/2/20
We propose new multivariate cryptosystems over nn-dimensional vector space over a finite field FqFq based on idea of hidden discrete logarithm problem for F∗qF∗q. These cryptosystems are b...
Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields
discrete logarithm problem bilinear pairings cryptanalysis, implementation
2016/12/9
Since 2013 there have been several developments in algorithms for computing discrete logarithms in small-characteristic finite fields, culminating in a quasi-polynomial algorithm. In this paper, we re...
Faster individual discrete logarithms in non-prime finite fields with the NFS and FFS algorithms
finite field discrete logarithm number field sieve
2016/7/13
Computing discrete logarithms in finite fields is a main concern in cryptography. The best algorithms known are the Number Field Sieve and its variants in large and medium characteristic fields (e.g. ...
Improving NFS for the discrete logarithm problem in non-prime finite fields
Discrete logarithm computation NFS
2016/6/12
The aim of this work is to investigate the hardness of the discrete logarithm problem in fields GF(pn) where n is a small integer greater than 1. Though less studied than the small characteristic case...
Extended Tower Number Field Sieve with Application to Finite Fields of Arbitrary Composite Extension Degree
Discrete Logarithm Problem Number Field Sieve Finite Fields
2016/5/30
In a recent work, Kim and Barbulescu~(CRYPTO~2016) proposed an algorithm, called exTNFS, that improves asymptotic complexity for the discrete logarithm problems over Fpn in medium prime case, when the...
Partially homomorphic encryption schemes over finite fields
Homomorphic encryption q-ary functions perfect secrecy
2016/5/3
Homomorphic encryption scheme enables computation in the encrypted domain,
which is of great importance because of its wide and growing range of applications.
The main issue with the known fully (or...
On derivatives of polynomials over finite fields through integration
Finite fields Boolean functions, Derivatives
2016/1/27
In this note, using rather elementary technique and the derived formula that relates
the coefficients of a polynomial over a finite field and its derivative, we deduce many
interesting results relat...
MaxMinMax problem and sparse equations over finite fields
complexity theory equation solving cryptanalysis
2016/1/26
Asymptotical complexity of sparse equation systems over finite field Fq is studied. Let the variable sets belong to a fixed family X={X1,…,Xm} while the polynomials fi(Xi) are taken independently and ...
The Multiple Number Field Sieve for Medium and High Characteristic > Finite Fields
public-key cryptography discrete logarithm problem
2016/1/25
In this paper, we study the discrete logarithm problem in medium and high characteristic finite fields. We propose a variant of the Number Field Sieve (NFS) based on numerous number fields. Our improv...