An Attack Bound for Small Multiplicative Inverse of <i>φ</i>(<i>N</i>) mod <i>e</i> with a Composed Prime Sum <i>p</i> + <i>q</i> Using Sublattice Based Techniques

• Main Authors: Pratha Anuradha Kameswari, Lambadi Jyotsna
• Format: Article
• Language: English
• Published: MDPI AG 2018-11-01
• Series: Cryptography
• Subjects: RSA; Cryptanalysis; lattices; LLL (Lenstra–Lenstra–Lovász) algorithm; Coppersmith’s method
• Online Access: https://www.mdpi.com/2410-387X/2/4/36

In this paper, we gave an attack on RSA (Rivest&#8315;Shamir&#8315;Adleman) Cryptosystem when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#966;</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> has small multiplicative inverse modulo <i>e</i> and the prime sum <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mrow> </semantics> </math> </inline-formula> is of the form <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> </semantics> </math> </inline-formula>, where <i>n</i> is a given positive integer and <inline-formula> <math display="inline"> <semantics> <msub> <mi>k</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>k</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> are two suitably small unknown integers using sublattice reduction techniques and Coppersmith&#8217;s methods for finding small roots of modular polynomial equations. When we compare this method with an approach using lattice based techniques, this procedure slightly improves the bound and reduces the lattice dimension. Employing the previous tools, we provide a new attack bound for the deciphering exponent when the prime sum <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <msub> <mi>k</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> </semantics> </math> </inline-formula> and performed an analysis with Boneh and Durfee&#8217;s deciphering exponent bound for appropriately small <inline-formula> <math display="inline"> <semantics> <msub> <mi>k</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>k</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula>.