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### Discrete Logarithms in $GF ( P )$ Using the Number Field Sieve

Daniel M. Gordon
1993 SIAM Journal on Discrete Mathematics
GF (p) with heuristic expected running time Lp[1/3; 3 2/3 ].  ...  Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time In this paper we present an algorithm to solve the discrete logarithm problem for  ...  Thanks also to Andrew Odlyzko for several email discussions about discrete logarithms, and Hendrik Lenstra for helpful comments.  ...

### Computing Individual Discrete Logarithms Faster in $${{\mathrm{GF}}}(p^n)$$ with the NFS-DL Algorithm [chapter]

Aurore Guillevic
2015 Lecture Notes in Computer Science
The Number Field Sieve (NFS) algorithm is the best known method to compute discrete logarithms (DL) in finite fields F_p^n, with p medium to large and n ≥ 1 small.  ...  If one can write the target preimage as a product of elements of known (virtual) logarithm, then one can deduce the discrete logarithm of the target.  ...  Let n > 1 and s ∈ F * p n a random element (not in a proper subfield of F p n ). We want to compute its discrete logarithm modulo ℓ, where ℓ | Φ n (p), with Φ n the n-th cyclotomic polynomial.  ...

### Discrete Logarithm in GF(2809) with FFS [chapter]

Razvan Barbulescu, Cyril Bouvier, Jérémie Detrey, Pierrick Gaudry, Hamza Jeljeli, Emmanuel Thomé, Marion Videau, Paul Zimmermann
2014 Lecture Notes in Computer Science
The year 2013 has seen several major complexity advances for the discrete logarithm problem in multiplicative groups of smallcharacteristic finite fields.  ...  This article presents the state of the art with regard to the FFS algorithm, and reports data from a record-sized discrete logarithm computation in a prime-degree extension field.  ...  It brings important data, however, towards the assessment of the feasibility limit of discrete logarithms in GF(2 p ) for prime extension degrees p.  ...

### Solving a 676-Bit Discrete Logarithm Problem in GF(36n)

Takuya HAYASHI, Naoyuki SHINOHARA, Lihua WANG, Shin'ichiro MATSUO, Masaaki SHIRASE, Tsuyoshi TAKAGI
2012 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences
Taking into account the Menezes-Okamoto-Vanstone (MOV) attack, the discrete logarithm problem (DLP) in GF(3 6n ) becomes a concern for the security of cryptosystems using ηT pairings in this case.  ...  Therefore, we first fulfill such an implementation and we successfully set a new record for solving the DLP in GF(3 6n ), the DLP in GF(3 6·71 ) of 676bit size.  ...  discrete logarithm. 1.  ...

### Solving a 676-Bit Discrete Logarithm Problem in GF(36n ) [chapter]

Takuya Hayashi, Naoyuki Shinohara, Lihua Wang, Shin'ichiro Matsuo, Masaaki Shirase, Tsuyoshi Takagi
2010 Lecture Notes in Computer Science
Taking into account the Menezes-Okamoto-Vanstone (MOV) attack, the discrete logarithm problem (DLP) in GF(3 6n ) becomes a concern for the security of cryptosystems using ηT pairings in this case.  ...  Therefore, we first fulfill such an implementation and we successfully set a new record for solving the DLP in GF(3 6n ), the DLP in GF(3 6·71 ) of 676bit size.  ...  discrete logarithm. 1.  ...

### Processing, Lecture notes in Computer Science

1993 АНО НПО «Профессионал», 2005.-480 с. 2. Gordon D. Discrete Logarithms in GF(p) using the Number Field Sieve //SIAM Journal on Discrete Mathematics   unpublished
α : n=38 416 для δ=0,005, p α =0,95.  ...  В табл. 1 содержатся полученные оценки p(A 1 ,c) и p(A 2 ,c) для кода C 1 с указанием приведенных в теореме 1 границ областей компрометации.  ...

### Collision Search for Elliptic Curve Discrete Logarithm over GF(2 m ) with FPGA [chapter]

Guerric Meurice de Dormale, Philippe Bulens, Jean-Jacques Quisquater
Lecture Notes in Computer Science
Indeed, no sub-exponential algorithms are known to solve the underlying hard problem: the Elliptic Curve Discrete Logarithm.  ...  In this last decade, Elliptic Curve Cryptography (ECC) has gained increasing acceptance in the industry and the academic community and has been the subject of several standards.  ...  The underlying hard problem of ECC is the intractability of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Let E (F) be an elliptic curve over a finite field F and let P be a point of E (F).  ...

### An experiment of number field sieve for discrete logarithm problem over $\text{GF}(p^n)$

Kenichiro Hayasaka, Kazumaro Aoki, Tetsutaro Kobayashi, Tsuyoshi Takagi
2014 JSIAM Letters
Using efficient parameters, we have solved the DLP over GF(p 12 ) of 203 bits in about 43 hours using a PC of 16 CPU cores.  ...  The security of the optimal Ate pairing using the BN curves is based on the hardness of the DLP over GF(p 12 ).  ...  Finally, we present an example of the discrete logarithm. Let γ = x 2 + x − 7 be a generator of GF(p 12 ) * = (GF(p)[X]/f 1 (X)) * .  ...

### EdDSA Over Galois Field GF(p^m) for Multimedia Data

Y. N. Shivani, A. Srinivas, B. K. Thanmayi, V. Vignesh, B. V. Srividya
2019 Journal of Engineering Research and Reports
The operations like addition and multiplication in Galois field are different compared to normal addition and multiplication.  ...  The finite field GF (pm) is an indispensable mathematical tool for some research fields such as information coding, cryptology, theory and application of network coding.  ...  Creation of signature is deterministic in EdDSA and it has higher security due to intractability of some discrete logarithm problems.  ...

### An FPGA implementation of a GF(p) ALU for encryption processors

A DALY
2004 Microprocessors and microsystems
Elliptic Curve Cryptosystems over GF(p) have received very little attention to date due to the seemingly more attractive finite field GF(2 m ).  ...  However, we present a GF(p) Arithmetic Logic Unit which can perform 160-bit arithmetic at clock speeds of up to 50MHz.  ...  Unlike the ordinary discrete logarithm problem, no sub-exponential algorithm is known to date to solve the discrete logarithm problem on a suitably chosen elliptic curve.  ...

### An FPGA implementation of a GF(p) ALU for encryption processors

Alan Daly, William Marnane, Tim Kerins, Emanuel Popovici
2004 Microprocessors and microsystems
Elliptic Curve Cryptosystems over GF(p) have received very little attention to date due to the seemingly more attractive finite field GF(2 m ).  ...  However, we present a GF(p) Arithmetic Logic Unit which can perform 160-bit arithmetic at clock speeds of up to 50MHz.  ...  Unlike the ordinary discrete logarithm problem, no sub-exponential algorithm is known to date to solve the discrete logarithm problem on a suitably chosen elliptic curve.  ...

### On Calculating Square Roots in GF(p) [article]

David S. Knight
2016 arXiv   pre-print
This article presents a new method for calculating square roots in GF(p) by exponentiating in GF(p^3) or equivalently modulo irreducible cubic polynomials.  ...  This algorithm is in some ways similar to the Cipolla-Lehmer algorithm which is based on exponentiating in GF(p^2).  ...  But more importantly it has potential applications concerning the integer factorization problem and the discrete logarithm problem in GF (p).  ...

### XTR Extended to GF(p 6m) [chapter]

Seongan Lim, Seungjoo Kim, Ikkwon Yie, Jaemoon Kim, Hongsub Lee
2001 Lecture Notes in Computer Science
Verheul in  proposed a very efficient way called XTR in which certain subgroup of the Galois field GF(p 6 ) can be represented by elements in GF(p 2 ).  ...  At the end of their paper , they briefly mentioned on a method of generalizing their idea to the field GF(p 6m ).  ...  Parameter Selection for Security Consideration Various XTR-based public key systems or key exchange protocols rely their security on the Discrete Logarithm Problem(DLP) in the base g ∈ GF(p 6m ), where  ...

### Sparse Hard Sets for P: Resolution of a Conjecture of Hartmanis

Jin-Yi Cai, D. Sivakumar
1999 Journal of computer and system sciences (Print)
We further prove that if P has a sparse hard set under many-one reductions computable in NC 1 , then P collapses to NC 1 .  ...  Building on a recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non-) existence of sparse sets complete for P under logspace many-one reductions.  ...  Acknowledgments We thank Mitsu Ogihara for showing us his work in a Rochester-Bu alo joint complexity  ...

### Using P systems to Solve the Discrete Logarithm Problem used in Diffie-Hellman Key Exchange Protocol

Xiaojing Ma, Zhitang Li, Hao Tu
2009 International Journal of Computer Network and Information Security
This paper presents a P system with active membranes and strong priority to solve the discrete logarithm problem used in Diffie-Hellman key exchange protocol.  ...  To the best of our knowledge, it's the first time to solve the problem using P systems. Index Terms-P systems, Discrete Logarithm Problem, Diffie-Hellman key exchange protocol Manuscript  ...  In this paper, we describe a P system with active membranes and strong priority to solve the discrete logarithm problem (DLP).  ...
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