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Quantum Cryptoanalysis of Elliptic Curve Systems

Juan Manuel Garcia Garcia, Rolando. Menchaca Garcia

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Abstract

THE SECURITY OF ELLIPTIC CURVE CRYPTOSYSTEMS IS BASED IN THE INTRACTABILITY OF THE ELLIPTIC CURVE DISCRETE LOGARITHM PROBLEM. THE BEST CLASSICAL ALGORITHM KNOWN UNTIL DQTE TO SOLVE THIS PROBLEM IS FULLY EXPONENTIAL IN TIME. THIS IS THE REASON WHY THE ELLIPTIC CURVE PUBLIC KEY CRYPTOGRAPHY IS CONSIDERED THE MOST SECURE KNOWN UNTIL DATE. WE PRESENT IN THIS PAPER AN ALGORITHM THAT RUNNING ON A QUANTUM COMPUTER, CAN SOLVE IN POLYNOMIAL TIME THE ELLIPTIC CURVE DISCRETE LOGARITHM PROBLEM. THEN, IF A FUNCTIONAL QUANTUM COMPUTER ISEVER BUILD, ALL THE ELLIPTIC CURVE CRYPTOSYSTEMS WOULD BECOME INSECURE.

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What this paper is about

THE SECURITY OF ELLIPTIC CURVE CRYPTOSYSTEMS IS BASED IN THE INTRACTABILITY OF THE ELLIPTIC CURVE DISCRETE LOGARITHM PROBLEM. THE BEST CLASSICAL ALGORITHM KNOWN UNTIL DQTE TO SOLVE THIS PROBLEM IS FULLY EXPONENTIAL IN TIME. THIS IS THE REASON WHY THE ELLIPTIC CURVE PUBLIC KEY CRYPTOGRAPHY IS CONSIDERED THE MOST SECURE KNOWN UNTIL DATE. WE PRESENT IN THIS PAPER AN ALGORITHM THAT RUNNING ON A QUANTUM COMPUTER, CAN SOLVE IN POLYNOMIAL TIME THE ELLIPTIC CURVE DISCRETE LOGARITHM PROBLEM. THEN, IF A FUNCTIONAL QUANTUM COMPUTER ISEVER BUILD, ALL THE ELLIPTIC CURVE CRYPTOSYSTEMS WOULD BECOME INSECURE.

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Available abstract

THE SECURITY OF ELLIPTIC CURVE CRYPTOSYSTEMS IS BASED IN THE INTRACTABILITY OF THE ELLIPTIC CURVE DISCRETE LOGARITHM PROBLEM. THE BEST CLASSICAL ALGORITHM KNOWN UNTIL DQTE TO SOLVE THIS PROBLEM IS FULLY EXPONENTIAL IN TIME. THIS IS THE REASON WHY THE ELLIPTIC CURVE PUBLIC KEY CRYPTOGRAPHY IS CONSIDERED THE MOST SECURE KNOWN UNTIL DATE. WE PRESENT IN THIS PAPER AN ALGORITHM THAT RUNNING ON A QUANTUM COMPUTER, CAN SOLVE IN POLYNOMIAL TIME THE ELLIPTIC CURVE DISCRETE LOGARITHM PROBLEM. THEN, IF A FUNCTIONAL QUANTUM COMPUTER ISEVER BUILD, ALL THE ELLIPTIC CURVE CRYPTOSYSTEMS WOULD BECOME INSECURE.

Key concepts: Elliptic curve point multiplication, Elliptic curve cryptography, Hessian form of an elliptic curve, Schoof's algorithm, Post-quantum cryptography, Tripling-oriented Doche–Icart–Kohel curve, Counting points on elliptic curves, Discrete logarithm

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