Encryption scheme based on expanded Reed-Solomon codes

Karan Khathuria; Joachim Rosenthal; Violetta Weger

doi:10.3934/amc.2020053

Article Contents

2021, Volume 15, Issue 2: 207-218. Doi: 10.3934/amc.2020053 open access

This issue Previous Article Next Article On the dimension of the subfield subcodes of 1-point Hermitian codes

Encryption scheme based on expanded Reed-Solomon codes

Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

Received: June 2019

Revised: October 2019

Early access: January 2020

Published: May 2021

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Abstract

We present a code-based public-key cryptosystem, in which we use Reed-Solomon codes over an extension field as secret codes and disguise it by considering its shortened expanded code over the base field. Considering shortened expanded codes provides a safeguard against distinguisher attacks based on the Schur product. Moreover, without using a cyclic or a quasi-cyclic structure we obtain a key size reduction of nearly $ 45 \% $ compared to the classic {McE}liece cryptosystem proposed by Bernstein et al.

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Mathematics Subject Classification: 14G50, 94A60, 11T71.

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Table 1. Comparing key sizes of the proposed cryptosystem with $ m = 3 $ and $ \lambda = 2 $ reaching a $ 256 $-bit security level against the modified ISD algorithm

Rate	$ q $	$ n $	$ k $	$ t $	Key Size (bits)
0.60	13	1382	829	277	6783627
0.65	13	1270	825	223	5952804
0.70	13	1207	844	182	5339456
0.75	13	1192	894	149	4929077
0.80	13	1230	984	123	4702652
0.82	13	1258	1031	114	4624198
0.85	13	1340	1139	101	4634545
0.87	13	1420	1235	93	4692805
0.90	13	1602	1441	81	4863276

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Table 2. Comparing key sizes of the proposed cryptosystem with $ m = 4 $ and $ \lambda = 2 $ reaching a $ 256 $-bit security level against the modified ISD algorithm

Rate	$ q $	$ n $	$ k $	$ t $	Key Size (bits)
0.65	7	2360	1534	413	13134108
0.70	7	1945	1361	292	10191102
0.75	7	1738	1303	218	8480009
0.80	7	1662	1329	167	7448878
0.85	7	1700	1445	128	6815134
0.87	7	1770	1539	116	6785893
0.89	7	1872	1666	103	6754721
0.91	7	2024	1841	92	6814326

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Table 3. Comparing the key sizes of the proposed parameters against different cryptosystems

		$ q $	$ m $	$ n $	$ k $	Key Size (in bits)
Proposed system	Type Ⅰs	13	3	1258	1031	4624198
Proposed system	Type Ⅱ	7	4	1872	1666	6754721
classical McEliece		2	13	6960	5413	8373911
BBCRS based schemes	$ w=1.708 $ and $ z=1 $	1423	1	1422	786	5113520
	$ w=1.2 $ and $ z=10 $	1163	1	1162	928	2274160
	$ w=2 $ and $ z=0 $	1993	1	1992	1593	6966714

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