No abstract

El Gamal encryption was introduced in 1985 and is still commonly used today. Its hardness is based on a discrete logarithm problem defined over the finite abelian cyclic group group chosen in the original paper was but later it was proven that using the group of Elliptic Curve points could significantly reduce the key size required. The modified El Gamal encryption is dubbed its analog version. This analog encryption bases its hardness on Elliptic Curve Discrete Logarithm Problem (ECDLP). One of the fastest attacks in cracking ECDLP is the Pollard Rho algorithm, with the expected number of iterations where is the number of points in the curve. This paper proposes a modification of the Pollard Rho algorithm using a negation map. The experiment was done in El Gamal analog encryption of elliptic curve defined over the field  with different values of small digit . The modification was expected to speed up the algorithm by  times. The average of speed up in the experiment was 1.9 times.

Serangan aljabar dapat dilakukan dalam dua tahapan yaitu mendapatkan sistem persamaan polinomial dan menentukan solusi dari sistem persamaan polinomial tersebut. Pada penelitian ini dilakukan serangan aljabar pada S-IDEA. Proses enkripsi satu round S-IDEA terdiri dari 14 langkah sedangkan sampai dengan Langkah ke-7 sudah diperoleh persamaan polinomial yang besar yaitu terdiri dari 4.721 monomial. Oleh karena keterbatasan sumber daya, dibuat miniatur S-IDEA agar serangan aljabar dapat dilakukan pada setiap langkah secara utuh. Algoritme miniatur S-IDEA terdiri dari 2,5 round yang setiap round-nya terdiri dari 14 langkah seperti halnya S-IDEA. Proses serangan aljabar pada miniatur S-IDEA menghasilkan 8 persamaan polinomial dengan monomial maksimal yang diperoleh yaitu sebanyak 1.109 monomial. Penentuan solusi dari persamaan polinomial yang diperoleh dilakukan dengan metode XL algorithm dan basis Gröbner. Metode XL algorithm dilakukan sampai tahap 4 dari 5 tahap, yaitu tahap linierisasi. Tahap linierisasi menghasilkan 136 persamaan yang didalamnya terdapat 1512 monomial. Konstanta dari persamaan linier tersebut dapat direpresentasikan ke dalam bentuk matriks berukuran 1512×136. Besarnya sistem persamaan hasil linierisasi yang diperoleh menyebabkan nilai kunci belum bisa didapatkan secara langsung melainkan harus dilakukan analisis lebih lanjut mengenai persamaan mana saja yang perlu digunakan untuk tahap selanjutnya pada XL algorithm. Sementara itu penentuan solusi dengan basis Gröbner menghasilkan 34 persamaan baru yang cukup panjang, sehingga nilai kunci belum dapat diperoleh secara langsung.

Digital signature has an important role in the digital era, where more and more people are joining the paperless life. Many cryptographic researchers support digital development by creating cryptographic schemes that are safe to use, and one of them is digital signature. This paper proposes a digital signature scheme based on an elliptic curve defined over with , where  and   are private keys of prime number elements. This scheme utilizes the advantages of elliptic curve cryptography in terms of security by using points that satisfy the elliptic curve equation. Additionally, the shorter key size increases the speed, making this scheme faster in signature values generation and verification process.This research was conducted to determine the differences between the modified ESIGN scheme based on elliptic curve and the original ESIGN scheme. The process of finding the point on the ring , with a large , resulted in a more complex key generation algorithm. However, the selection of two points in this key generation is precomputed. This means the actual signature value generation algorithm took significantly less time than the original. This is one of the advantages of the proposed scheme.