Discrete Logarithms In Finite Fields Research Articles

A Comparative Study of RSA and ELGAMAL Cryptosystems

Purpose: In this research, a comparative analysis of the RSA and ELGAMAL algorithms is presented using their encryption and decryption speed with different key sizes. ElGamal and RSA are two popular public-key cryptographic algorithms that depend on the mathematical difficulty of computing discrete logarithms in finite fields and factoring large integers, respectively.
 Methodology: In this study, the speed at which messages of various sizes can be encrypted and decrypted using various key sizes is used to gauge how well these algorithms work.
 Findings: RSA usually outperforms ElGamal in terms of encryption and decryption speed, according to the results. However, as the key sizes grow, the performance gap between the two algorithms narrows.
 Unique Contribution to Theory, Policy and Practice: These findings can aid in selecting the appropriate algorithm and key size for different applications that require secure communication and data encryption.

How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits

We significantly reduce the cost of factoring integers and computing discrete logarithms in finite fields on a quantum computer by combining techniques from Shor 1994, Griffiths-Niu 1996, Zalka 2006, Fowler 2012, Ekerå-Håstad 2017, Ekerå 2017, Ekerå 2018, Gidney-Fowler 2019, Gidney 2019. We estimate the approximate cost of our construction using plausible physical assumptions for large-scale superconducting qubit platforms: a planar grid of qubits with nearest-neighbor connectivity, a characteristic physical gate error rate of 10−3, a surface code cycle time of 1 microsecond, and a reaction time of 10 microseconds. We account for factors that are normally ignored such as noise, the need to make repeated attempts, and the spacetime layout of the computation. When factoring 2048 bit RSA integers, our construction's spacetime volume is a hundredfold less than comparable estimates from earlier works (Van Meter et al. 2009, Jones et al. 2010, Fowler et al. 2012, Gheorghiu et al. 2019). In the abstract circuit model (which ignores overheads from distillation, routing, and error correction) our construction uses 3n+0.002nlg⁡n logical qubits, 0.3n3+0.0005n3lg⁡n Toffolis, and 500n2+n2lg⁡n measurement depth to factor n-bit RSA integers. We quantify the cryptographic implications of our work, both for RSA and for schemes based on the DLP in finite fields.

A new perspective on the powers of two descent for discrete logarithms in finite fields

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 a unified geometric argument, eliminating the need to analyse all possible subgroups of $\mathrm{PGL}_2(\mathbb F_q)$. Our approach sheds new light on the role of $\mathrm{PGL}_2$, in the hope to eventually lead to a complete proof that discrete logarithms can be computed in quasi-polynomial time in finite fields of fixed characteristic.

Faster individual discrete logarithms in finite fields of composite extension degree

Computing discrete logarithms in finite fields is a main concern in cryptography. The best algorithms in large and medium characteristic fields (e.g., $\rm {GF}(p^2)$, $\rm {GF}(p^{12})$) are the Number Field Sieve and its variants (special, high-degree, tower). The best algorithms in small characteristic finite fields (e.g., $\rm {GF}(3^{6 \cdot 509})$) are the Function Field Sieve, Joux’s algorithm, and the quasipolynomial-time algorithm. The last step of this family of algorithms is the individual logarithm computation. It computes a smooth decomposition of a given target in two phases: an initial splitting, then a descent tree. While new improvements have been made to reduce the complexity of the dominating relation collection and linear algebra steps, resulting in a smaller factor basis (database of known logarithms of small elements), the last step remains at the same level of difficulty. Indeed, we have to find a smooth decomposition of a typically large element in the finite field. This work improves the initial splitting phase and applies to any nonprime finite field. It is very efficient when the extension degree is composite. It exploits the proper subfields, resulting in a much more smooth decomposition of the target. This leads to a new trade-off between the initial splitting step and the descent step in small characteristic. Moreover it reduces the width and the height of the subsequent descent tree.

Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields

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 report on our successful computation of discrete logarithms in the cryptographically-interesting characteristic-three finite field $\mathbb{F}_{3^{6.509}}$ using these new algorithms; prior to 2013, it was believed that this field enjoyed a security level of 128 bits. We also show that a recent idea of Guillevic can be used to compute discrete logarithms in the cryptographically-interesting finite field $\mathbb{F}_{3^{6.709}}$ using essentially the same resources as we expended on the $\mathbb{F}_{3^{6.509}}$ computation. Finally, we argue that discrete logarithms in the finite field $\mathbb{F}_{3^{6.1429}}$ can feasibly be computed today; this is significant because this cryptographically-interesting field was previously believed to enjoy a security level of 192 bits.

Classifying and generating exact coset representatives of [formula omitted] in [formula omitted

Generating coset representatives of PGL2(Fq) in PGL2(Fq2) is a key ingredient in certain algorithms of determining primitive elements and computing discrete logarithms in finite fields of small characteristic. In this paper, we describe a simple classification of the right cosets of PGL2(Fq) in PGL2(Fq2). Based on the classification, we design a deterministic algorithm that generates the exact coset representatives of PGL2(Fq) in PGL2(Fq2) with O(q3) field operations.

The discrete logarithm problem from a local duality perspective

The discrete logarithm problem is analyzed from the perspective of Tate local duality. Local duality in the multiplicative case and the case of Jacobians of curves over p-adic local fields are considered. When the local field contains the necessary roots of unity, the case of curves over local fields is polynomial time reducible to the multiplicative case, and the multiplicative case is polynomial time equivalent to computing discrete logarithm in finite fields. When the local field does not contains the necessary roots of unity, similar results can be obtained at the cost of going to an extension that contains these roots of unity. There was evidence in the analysis that suggests that the minimal extension where the local duality can be rationally and algorithmically defined must contain the roots of unity. Therefore, the discrete logarithm problem appears to be well protected against an attack using local duality. These results are also of independent interest for algorithmic study of arithmetic duality as they explicitly relate local duality in the case of curves over local fields to the multiplicative case and Tate-Lichtenbaum pairing (over finite fields).

A shorter proof for an explicit formula for discrete logarithms in finite fields

A shorter proof for an explicit formula for discrete logarithms in finite fields is given.

Interpolation of the discrete logarithm in [formula omitted] by Boolean functions and by polynomials in several variables modulo a divisor of q−1

Recently, Shparlinski proved several results on the interpolation of the discrete logarithm in finite prime fields by Boolean functions. In the first part of the paper, these results are extended to arbitrary finite fields of odd characteristic. More precisely, we prove some complexity lower bounds for Boolean functions representing the least significant bit of the discrete logarithm in a finite field. In the second part of the paper we obtain lower bounds on the sparsity and the degree of polynomials over F q in several variables computing the discrete logarithm modulo a prime divisor of q−1. These results are valid for even characteristic, as well.

Interpolation of the Discrete Logarithm in Finite Fields by Boolean Functions

Abstract Recently, Shparlinski proved several results on the interpolation of the discrete logarithm in finite prime fields by Boolean functions. In this extended abstract we present extensions to arbitrary finite fields of odd characteristic. More precisely, we sketch the proofs of some complexity lower bounds for Boolean functions representing the least significant bit of the discrete logarithm in a finite field.

Lower bounds on the linear complexity of the discrete logarithm in finite fields

Let p be a prime, r a positive integer, q=p/sup r/, and d a divisor of p(q-1). We derive lower bounds on the linear complexity over the residue class ring Z/sub d/ of a (q-periodic) sequence representing the residues modulo d of the discrete logarithm in F/sub q/. Moreover, we investigate a sequence over F/sub q/ representing the values of a certain polynomial over F/sub q/ introduced by Mullen and White (1986) which can be identified with the discrete logarithm in F/sub q/ via p-adic expansions and representations of the elements of F/sub q/ with respect to some fixed basis.

Immunizing public key cryptosystems against chosen ciphertext attacks

Three methods for strengthening public key cryptosystems in such a way that they become secure against adaptively chosen ciphertext attacks are presented. In an adaptively chosen ciphertext attack, an attacker can query the deciphering algorithm with any ciphertext except for the exact object ciphertext to be cryptanalyzed. The first strengthening method is based on the use of one-way hash functions, the second on the use of universal hash functions, and the third on the use of digital signature schemes. Each method is illustrated by an example of a public key cryptosystem based on the intractability of computing discrete logarithms in finite fields. Security of the three example cryptosystems is formally proved. Two other issues, namely, applications of the methods to public key cryptosystems based on other intractable problems and enhancement of information authentication capability to the cryptosystems, are also discussed. >

A note on discrete logarithms in finite fields

In this note we provide a number of polynomial formulas for the discrete logarithm problem in an arbitrary multiplicative subgroup of a finite field. These formulas generalize previous formulas for computing logarithms in the multiplicative group of all nonzero elements.

A short proof for explicit formulas for discrete logarithms in finite fields

A short proof for explicit formulas for discrete logarithms in finite fields