Thesis Chapters by N. Gamze Orhon Kılıç
Elliptic curves were being used only for mathematical studies until Miller and Koblitz introduced... more Elliptic curves were being used only for mathematical studies until Miller and Koblitz introduced elliptic curves to
crypto-community in 1985 with independent works. Since then, elliptic curves became one of the most
significant tools in cryptography. Elliptic curve cryptography (ECC) started to be used for commercial purposes
after 1990's.
It provides a better level of security with the same key size than the widely used public key crypto-systems such as
RSA. Nevertheless, time complexity is not at the desired stage. Hence, there have been several studies so far that aims
to increase the time efficiency.
The curve forms that are being used for speed oriented operations came a long way in terms of gathering lower degree
formulas for scalar multiplication which is the core operation of ECC. However, one of the curve forms which is
called Huff curve could not get competitive with the other forms such as Twisted Edwards, Jacobi Quartic,
despite the studies have been made so far. This thesis focuses on increasing the efficiency of Huff form of elliptic
curve by making use of mathematical and computational primitives.
Inversion-free point addition and doubling formulas which are being used in scalar multiplication algorithms, are
proposed for the Huff curve which is defined as
y (1+a x^2) = c x (1+d y^2).
First idea is rather to embed the curve into a different projective space than the preferred for Huff curve previously.
Thus, $\mathbb{P}^1 \times \mathbb{P}^1$ embedding is used instead of $\mathbb{P}^2$ embedding. The second idea is to
make the use of isogenies in order to obtain an alternative doubling formula. Thanks to these two ideas, an improvement
is achieved.
The best algorithm for point doubling on Huff curve was computed with 6M + 5S. The proposed doubling formula in this thesis can be computed with 8M.
Also, operation count of mixed addition is decreased from 10M to 8M. Both sets of formulas are leading to an
effective cost of 2M. Furthermore, they are shown to be 4-way parallel.
Elliptic Curve Cryptography (ECC) has a big role in Information Security. Pollard’s Rho
Attack is... more Elliptic Curve Cryptography (ECC) has a big role in Information Security. Pollard’s Rho
Attack is the only real life threat against elliptic curve based cryptosystems.
Pollard’s
Rho attack solves the so-called Elliptic Curve Discrete Logarithm Problem upon which the
conjectured security is claimed for ECC.
In the context of this thesis, the algorithm realizing Pollard’s Rho attack is initially coded in
Magma programming language as a high level prototype, and then it’s coded in C programming
language for full performance and parallelism. The final code can now run on a TCP/IP
network using n processors(n = 16 in experiments) at their peak to speed-up the attack by a sqrt(n).
Papers by N. Gamze Orhon Kılıç
INTERNATIONAL JOURNAL OF INFORMATION SECURITY SCIENCE, 2023
The increasing demand for secure and anonymous transactions raises the popularity of ring signatu... more The increasing demand for secure and anonymous transactions raises the popularity of ring signatures, which is a digital signature scheme that allows identifying a group of possible signers without revealing the identity of the actual signer. This paper presents efficient supersingular isogeny-based ring signature and linkable ring signature schemes that will find potential
applications in post-quantum technologies. We develop the ring signature scheme by applying the Fiat-Shamir transform on the sigma protocol for a ring which we obtain from the supersingular isogeny-based interactive zero-knowledge identification scheme by adapting the scheme for a ring. We also extend our ring signature protocol with an additional parameter, i.e., a tag that provides to detect if a signer issues two signatures concerning the same ring by preserving anonymity and linkable anonymity. The signature
size of our ring signature protocols increases logarithmically in the size of the ring thanks to the Merkle trees. We show the security proofs and efficiency analyses of the protocols offered. Moreover, we provide the implementation results of the supersingular isogeny-based ring signature, which offers small signature sizes for NIST post-quantum security levels.
International Journal of Information Security Science, 2023
The increasing demand for secure and anonymous transactions raises the popularity of ring signatu... more The increasing demand for secure and anonymous transactions raises the popularity of ring signatures, which is a digital signature scheme that allows identifying a group of possible signers without revealing the identity of the actual signer. This paper presents efficient supersingular isogeny-based ring signature and linkable ring signature schemes that will find potential applications in post-quantum technologies. We develop the ring signature scheme by applying the Fiat-Shamir transform on the sigma protocol for a ring which we obtain from the supersingular isogeny-based interactive zero-knowledge identification scheme by adopting the scheme for a ring. We also extend our ring signature protocol with an additional parameter, i.e., a tag that provides to detect if a signer issues two signatures concerning the same ring by preserving anonymity and linkable anonymity. The signature size of our ring signature protocols increases logarithmically in the size of the ring thanks to the Merkle trees. We show the security proofs and efficiency analyses of the protocols offered. Moreover, we provide the implementation results of the supersingular isogeny-based ring signature, which offers small signature sizes for NIST post-quantum security levels.
linkable ring signature, post-quantum cryptography, ring signature, supersingular isogeny
This paper presents faster inversion-free point addition formulas for the curve y(1 + ax 2) = cx(... more This paper presents faster inversion-free point addition formulas for the curve y(1 + ax 2) = cx(1 + dy 2). The proposed formulas improve the point doubling operation count record 1 from 6M + 5S to 8M and mixed addition operation count record from 10M to 8M. Both sets of formulas are shown to be 4-way parallel, leading to an effective cost of 2M per either of the group operations.
Lecture Notes in Computer Science, 2014
Conference Presentations by N. Gamze Orhon Kılıç
Uluslararası IX. Mantık Çalıştayı, 2019
Pozitif bilimler ve felsefenin birbirine ters düştüğü noktalardan biri zaman kavramıdır. Felsefe ... more Pozitif bilimler ve felsefenin birbirine ters düştüğü noktalardan biri zaman kavramıdır. Felsefe evrensel ve zamansız olarak doğrunun peşindedir. Düşünceler kimi zaman herhangi bir topluma ya da zamana göre değerlendirilse ya da sınıflandırılsa da düşünceler arası karşılaştırılmada zaman bir parametre olarak ele alınmaz . Fakat bu durum pozitif bilim için her koşulda geçerli değildir.
Teoride bir sorunun çözümünü formüle edebiliyor olması, gerçek hayatta bu sorunun kesinlikle çözülebileceği anlamına gelmez. Sözgelimi son yüz elli yılda bilim ve teknolojide gözlemlenen büyük ivme sonucunda ispatlanmış teoremlerin veya benimsenen kuramların uygulamada sorunlarla karşılaştığı görüldü. Dahası matematiksel olarak çözümü olmayan hiçbir kriptografik algoritma mevcut olmasa da pratikte kriptografik algoritmalar vazgeçilmez bir konuma erişti..
Felsefede, özel olarak mantıkta, argümanlar ve önermeler söz konusu olduğunda sağlamlık, geçerlilik, türetilebilirlik gibi ölçütler yeterli görünürken, bilgisayar bilimlerinde ve matematikte zaman parametresinin ele alınması mantıksal ilişkilerin çözümlenmesinde belirleyici bir önem kazanmaktadır. Alternatif teoriler, hesaplamalar ve kuramlar karşılaştırıldığında zaman bakımından avantaj sağlanması pratik bir gerekliliktir. Bu pratik gerekliliğin bir diğer sonucu da bilimsel araştırmanın yönünün belirlenmesidir. Zaman bakımından verimli yolların aranması, mantıksal ilişkilerin incelenmesinde daha verimli yolların bulunmasını kolaylaştırabilir.
Bilgisayar bilimlerindeki bu pratik gereklilik, felsefe bağlamındaki mantık araştırmalarına da ışık tutabilir.
Bu çalışmada, sözü edilen pratik gerekliliğin teorik arkaplanı ve felsefe bağlamındaki mantık araştırmalarında oynayabileceği roller açığa çıkarılmaya çalışılacaktır.
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Thesis Chapters by N. Gamze Orhon Kılıç
crypto-community in 1985 with independent works. Since then, elliptic curves became one of the most
significant tools in cryptography. Elliptic curve cryptography (ECC) started to be used for commercial purposes
after 1990's.
It provides a better level of security with the same key size than the widely used public key crypto-systems such as
RSA. Nevertheless, time complexity is not at the desired stage. Hence, there have been several studies so far that aims
to increase the time efficiency.
The curve forms that are being used for speed oriented operations came a long way in terms of gathering lower degree
formulas for scalar multiplication which is the core operation of ECC. However, one of the curve forms which is
called Huff curve could not get competitive with the other forms such as Twisted Edwards, Jacobi Quartic,
despite the studies have been made so far. This thesis focuses on increasing the efficiency of Huff form of elliptic
curve by making use of mathematical and computational primitives.
Inversion-free point addition and doubling formulas which are being used in scalar multiplication algorithms, are
proposed for the Huff curve which is defined as
y (1+a x^2) = c x (1+d y^2).
First idea is rather to embed the curve into a different projective space than the preferred for Huff curve previously.
Thus, $\mathbb{P}^1 \times \mathbb{P}^1$ embedding is used instead of $\mathbb{P}^2$ embedding. The second idea is to
make the use of isogenies in order to obtain an alternative doubling formula. Thanks to these two ideas, an improvement
is achieved.
The best algorithm for point doubling on Huff curve was computed with 6M + 5S. The proposed doubling formula in this thesis can be computed with 8M.
Also, operation count of mixed addition is decreased from 10M to 8M. Both sets of formulas are leading to an
effective cost of 2M. Furthermore, they are shown to be 4-way parallel.
Attack is the only real life threat against elliptic curve based cryptosystems.
Pollard’s
Rho attack solves the so-called Elliptic Curve Discrete Logarithm Problem upon which the
conjectured security is claimed for ECC.
In the context of this thesis, the algorithm realizing Pollard’s Rho attack is initially coded in
Magma programming language as a high level prototype, and then it’s coded in C programming
language for full performance and parallelism. The final code can now run on a TCP/IP
network using n processors(n = 16 in experiments) at their peak to speed-up the attack by a sqrt(n).
Papers by N. Gamze Orhon Kılıç
applications in post-quantum technologies. We develop the ring signature scheme by applying the Fiat-Shamir transform on the sigma protocol for a ring which we obtain from the supersingular isogeny-based interactive zero-knowledge identification scheme by adapting the scheme for a ring. We also extend our ring signature protocol with an additional parameter, i.e., a tag that provides to detect if a signer issues two signatures concerning the same ring by preserving anonymity and linkable anonymity. The signature
size of our ring signature protocols increases logarithmically in the size of the ring thanks to the Merkle trees. We show the security proofs and efficiency analyses of the protocols offered. Moreover, we provide the implementation results of the supersingular isogeny-based ring signature, which offers small signature sizes for NIST post-quantum security levels.
linkable ring signature, post-quantum cryptography, ring signature, supersingular isogeny
Conference Presentations by N. Gamze Orhon Kılıç
Teoride bir sorunun çözümünü formüle edebiliyor olması, gerçek hayatta bu sorunun kesinlikle çözülebileceği anlamına gelmez. Sözgelimi son yüz elli yılda bilim ve teknolojide gözlemlenen büyük ivme sonucunda ispatlanmış teoremlerin veya benimsenen kuramların uygulamada sorunlarla karşılaştığı görüldü. Dahası matematiksel olarak çözümü olmayan hiçbir kriptografik algoritma mevcut olmasa da pratikte kriptografik algoritmalar vazgeçilmez bir konuma erişti..
Felsefede, özel olarak mantıkta, argümanlar ve önermeler söz konusu olduğunda sağlamlık, geçerlilik, türetilebilirlik gibi ölçütler yeterli görünürken, bilgisayar bilimlerinde ve matematikte zaman parametresinin ele alınması mantıksal ilişkilerin çözümlenmesinde belirleyici bir önem kazanmaktadır. Alternatif teoriler, hesaplamalar ve kuramlar karşılaştırıldığında zaman bakımından avantaj sağlanması pratik bir gerekliliktir. Bu pratik gerekliliğin bir diğer sonucu da bilimsel araştırmanın yönünün belirlenmesidir. Zaman bakımından verimli yolların aranması, mantıksal ilişkilerin incelenmesinde daha verimli yolların bulunmasını kolaylaştırabilir.
Bilgisayar bilimlerindeki bu pratik gereklilik, felsefe bağlamındaki mantık araştırmalarına da ışık tutabilir.
Bu çalışmada, sözü edilen pratik gerekliliğin teorik arkaplanı ve felsefe bağlamındaki mantık araştırmalarında oynayabileceği roller açığa çıkarılmaya çalışılacaktır.
crypto-community in 1985 with independent works. Since then, elliptic curves became one of the most
significant tools in cryptography. Elliptic curve cryptography (ECC) started to be used for commercial purposes
after 1990's.
It provides a better level of security with the same key size than the widely used public key crypto-systems such as
RSA. Nevertheless, time complexity is not at the desired stage. Hence, there have been several studies so far that aims
to increase the time efficiency.
The curve forms that are being used for speed oriented operations came a long way in terms of gathering lower degree
formulas for scalar multiplication which is the core operation of ECC. However, one of the curve forms which is
called Huff curve could not get competitive with the other forms such as Twisted Edwards, Jacobi Quartic,
despite the studies have been made so far. This thesis focuses on increasing the efficiency of Huff form of elliptic
curve by making use of mathematical and computational primitives.
Inversion-free point addition and doubling formulas which are being used in scalar multiplication algorithms, are
proposed for the Huff curve which is defined as
y (1+a x^2) = c x (1+d y^2).
First idea is rather to embed the curve into a different projective space than the preferred for Huff curve previously.
Thus, $\mathbb{P}^1 \times \mathbb{P}^1$ embedding is used instead of $\mathbb{P}^2$ embedding. The second idea is to
make the use of isogenies in order to obtain an alternative doubling formula. Thanks to these two ideas, an improvement
is achieved.
The best algorithm for point doubling on Huff curve was computed with 6M + 5S. The proposed doubling formula in this thesis can be computed with 8M.
Also, operation count of mixed addition is decreased from 10M to 8M. Both sets of formulas are leading to an
effective cost of 2M. Furthermore, they are shown to be 4-way parallel.
Attack is the only real life threat against elliptic curve based cryptosystems.
Pollard’s
Rho attack solves the so-called Elliptic Curve Discrete Logarithm Problem upon which the
conjectured security is claimed for ECC.
In the context of this thesis, the algorithm realizing Pollard’s Rho attack is initially coded in
Magma programming language as a high level prototype, and then it’s coded in C programming
language for full performance and parallelism. The final code can now run on a TCP/IP
network using n processors(n = 16 in experiments) at their peak to speed-up the attack by a sqrt(n).
applications in post-quantum technologies. We develop the ring signature scheme by applying the Fiat-Shamir transform on the sigma protocol for a ring which we obtain from the supersingular isogeny-based interactive zero-knowledge identification scheme by adapting the scheme for a ring. We also extend our ring signature protocol with an additional parameter, i.e., a tag that provides to detect if a signer issues two signatures concerning the same ring by preserving anonymity and linkable anonymity. The signature
size of our ring signature protocols increases logarithmically in the size of the ring thanks to the Merkle trees. We show the security proofs and efficiency analyses of the protocols offered. Moreover, we provide the implementation results of the supersingular isogeny-based ring signature, which offers small signature sizes for NIST post-quantum security levels.
linkable ring signature, post-quantum cryptography, ring signature, supersingular isogeny
Teoride bir sorunun çözümünü formüle edebiliyor olması, gerçek hayatta bu sorunun kesinlikle çözülebileceği anlamına gelmez. Sözgelimi son yüz elli yılda bilim ve teknolojide gözlemlenen büyük ivme sonucunda ispatlanmış teoremlerin veya benimsenen kuramların uygulamada sorunlarla karşılaştığı görüldü. Dahası matematiksel olarak çözümü olmayan hiçbir kriptografik algoritma mevcut olmasa da pratikte kriptografik algoritmalar vazgeçilmez bir konuma erişti..
Felsefede, özel olarak mantıkta, argümanlar ve önermeler söz konusu olduğunda sağlamlık, geçerlilik, türetilebilirlik gibi ölçütler yeterli görünürken, bilgisayar bilimlerinde ve matematikte zaman parametresinin ele alınması mantıksal ilişkilerin çözümlenmesinde belirleyici bir önem kazanmaktadır. Alternatif teoriler, hesaplamalar ve kuramlar karşılaştırıldığında zaman bakımından avantaj sağlanması pratik bir gerekliliktir. Bu pratik gerekliliğin bir diğer sonucu da bilimsel araştırmanın yönünün belirlenmesidir. Zaman bakımından verimli yolların aranması, mantıksal ilişkilerin incelenmesinde daha verimli yolların bulunmasını kolaylaştırabilir.
Bilgisayar bilimlerindeki bu pratik gereklilik, felsefe bağlamındaki mantık araştırmalarına da ışık tutabilir.
Bu çalışmada, sözü edilen pratik gerekliliğin teorik arkaplanı ve felsefe bağlamındaki mantık araştırmalarında oynayabileceği roller açığa çıkarılmaya çalışılacaktır.