# A Kilobit Hidden SNFS Discrete Logarithm Computation

@article{Fried2017AKH, title={A Kilobit Hidden SNFS Discrete Logarithm Computation}, author={Joshua Fried and P. Gaudry and N. Heninger and Emmanuel Thom{\'e}}, journal={ArXiv}, year={2017}, volume={abs/1610.02874} }

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software.

#### 41 Citations

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- 2021

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- 2020

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- Mathematics, Computer Science
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- 2020

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Faster individual discrete logarithms in finite fields of composite extension degree

- Computer Science
- Math. Comput.
- 2019

This work improves the initial splitting phase and applies to any nonprime finite field, and is very efficient when the extension degree is composite. Expand

Threshold Kleptographic Attacks on Discrete Logarithm Based Signatures

- Mathematics, Computer Science
- IACR Cryptol. ePrint Arch.
- 2017

This work combines the notions of threshold scheme and kleptographic attack to construct the first \(\ell \) out of n threshold klePTographic attack on discrete logarithm based digital signatures and prove its security in the standard and random oracle models. Expand

Group-Based Secure Computation: Optimizing Rounds, Communication, and Computation

- Computer Science
- EUROCRYPT
- 2017

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Updating key size estimations for pairings Razvan Barbulescu

- 2019

Recent progress on NFS imposed a new estimation of the security of pairings. In this work we study the best attacks against some of the most popular pairings and propose new key sizes using an… Expand

Sieve algorithms for the discrete logarithm in medium characteristic finite fields. (Algorithmes de crible pour le logarithme discret dans les corps finis de moyenne caractéristique)

- Mathematics, Computer Science
- 2017

This thesis proposes and study two new sieve algorithms allowing us to treat any dimensions, with an emphasis on the three-dimensional case, and provides a complete implementation of the relation collection for some variants of the NFS in three dimensions. Expand

#### References

SHOWING 1-10 OF 65 REFERENCES

A Kilobit Special Number Field Sieve Factorization

- Mathematics, Computer Science
- ASIACRYPT
- 2007

We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 21039 - 1.… Expand

Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the gaussian integer method

- Mathematics, Computer Science
- Math. Comput.
- 2003

It is shown that the number field sieve outperforms the gaussian integer method in the hundred digit range by successfully computing discrete logarithms with GNFS in a large prime field. Expand

On asymptotic complexity of computing discrete logarithms over GF(p)

- Mathematics
- 2003

We analyse the modification of an algorithm for finding discrete logarithms over the field GF(p) (p is a prime number) which has been described by the author previously. It is shown that this… Expand

Designing and Detecting Trapdoors for Discrete Log Cryptosystems

- Mathematics, Computer Science
- CRYPTO
- 1992

Using a number field sieve, discrete logarithms modulo primes of special forms can be found faster than standard primes. This has raised concerns about trapdoors in discrete log cryptosystems, such… Expand

Generating Eecient Primes for Discrete Log Cryptosystems

- 2007

This paper presents a method for generating prime moduli with a special form which can simplify the modular reduction process and reduce the storage requirement. Such moduli will be particularly… Expand

Factorization of a 768-Bit RSA Modulus

- Mathematics, Computer Science
- CRYPTO
- 2010

This paper reports on the factorization of the 768-bit number RSA-768 by the number field sieve factoring method and discusses some implications for RSA.

Modifications to the Number Field Sieve

- Mathematics, Computer Science
- Journal of Cryptology
- 2004

The fact that certain smoothness computations can be reused, and thereby reduce the asymptotic running time of the Number Field Sieve, is used to give a way to precompute tables which will be useful for factoring any integers in a large range. Expand

Polynomial Selection for the Number Field Sieve Integer Factorisation Algorithm

- Mathematics
- 1999

In this thesis we outline new research in integer factorisation with applications to public-key cryptography. In particular, we consider the number field sieve, the newest and fastest knownmethod for… Expand

Discrete logarithms and local units

- Mathematics
- Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences
- 1993

Let K be a number field and (9K its ring of integers. Let l be a prime number and e a positive integer. We give a method to construct leth powers in (9K using smooth algebraic integers. This method… Expand

An L(1/3) Discrete Logarithm Algorithm for Low Degree Curves

- Computer Science, Mathematics
- Journal of Cryptology
- 2010

An algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in X and Y are low with respect to their genera using heuristics similar to the ones used in the number field sieves or the function field sieve is presented. Expand